Analysis as a Life: Dedicated to Heinrich Begehr on the Occasion of his 80th Birthday [1st ed.] 978-3-030-02649-3, 978-3-030-02650-9

Table of contents :
Front Matter ....Pages i-xxii
Deformation of Complex Structures and Boundary Value Problem with Shift (G. Akhalaia, G. Giorgadze, G. Makatsaria, N. Manjavidze)....Pages 1-17
Dirichlet Problem for Poisson and Bi-Poisson Equations in Clifford Analysis (Ümit Aksoy)....Pages 19-37
Boundary Eigenvalues of Pluriharmonic Functions for the Third Boundary Condition on the Unit Polydiscs (Alip Mohammed)....Pages 39-45
Survey of Some General Properties of Meromorphic Functions in a Given Domain (G. Barsegian)....Pages 47-67
Boundary Value Problems in Polydomains (Ahmet Okay Çelebi)....Pages 69-92
Completeness Theorems on the Boundary in Thermoelasticity (Alberto Cialdea)....Pages 93-115
A Circle Pattern Algorithm via Combinatorial Ricci Flows (Dong-Meng Xi, Shi-Yi Lan, Dao-Qing Dai)....Pages 117-138
Strong Asymptotic Analysis of OLPs on the Unit Circle by Riemann-Hilbert Approach (Yufeng Wang, Yifeng Lu, Jinyuan Du)....Pages 139-169
Time Dependent Solutions for the Biot Equations (Robert P. Gilbert, George C. Hsiao)....Pages 171-191
Schwartz-Type Boundary Value Problems for Monogenic Functions in a Biharmonic Algebra (S. V. Gryshchuk, S. A. Plaksa)....Pages 193-211
The String Equation for Some Rational Functions (Björn Gustafsson)....Pages 213-235
Newtonian and Single Layer Potentials for the Stokes System with L∞ Coefficients and the Exterior Dirichlet Problem (Mirela Kohr, Sergey E. Mikhailov, Wolfgang L. Wendland)....Pages 237-260
Special Functions Method for Fractional Analysis and Fractional Modeling (S. V. Rogosin, M. V. Dubatovskaya)....Pages 261-278
On Elliptic Systems of Two Equations on the Plane (A. P. Soldatov)....Pages 279-301
Real Variable Inverse Laplace Transform (Vu Kim Tuan, A. Boumenir, Dinh Thanh Duc)....Pages 303-318

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Trends in Mathematics

Sergei Rogosin Ahmet Okay C¸ elebi Editors

Analysis as a Life Dedicated to Heinrich Begehr on the Occasion of his 80th Birthday

Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference.

More information about this series at http://www.springer.com/series/4961

Sergei Rogosin • Ahmet Okay Çelebi Editors

Analysis as a Life Dedicated to Heinrich Begehr on the Occasion of his 80th Birthday

Editors Sergei Rogosin Department of Economics Belarusian State University Minsk, Belarus

Ahmet Okay Çelebi Mathematics Department Yeditepe University Ata¸sehir, Istanbul, Turkey

ISSN 2297-0215 ISSN 2297-024X (electronic) Trends in Mathematics ISBN 978-3-030-02649-3 ISBN 978-3-030-02650-9 (eBook) https://doi.org/10.1007/978-3-030-02650-9 Library of Congress Control Number: 2018967001 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The book contains several important research contributions from the different branches of analysis. These papers are combined due to the common idea. The authors express their gratitude to Professor Heinrich Begehr for his help in their research career and dedicate articles to Begehr’s 80th birthday. Scientific achievements by Prof. H. Begehr are well known (see a short biographical paper below as well as the paper Vaitekhovich T.: Heinrich Begehr: Citation for his 70th birthday, Analysis, v. 30 (1), pp. 1–26). More important for analytic society is his numerous collaborations with people all around the world. Many mathematicians from different countries are thankful to him for the role he played in their scientific life. Professor H. Begehr is known also for his initiatives. Among them is his role in the creation and further development of the International Society for Analysis, its Applications and Computation (ISAAC). With this volume the colleagues and friends of Professor Heinrich Begehr would like to thank him and wish to have a long life in analysis. Minsk, Belarus Istanbul, Turkey August 2018

Sergei Rogosin Ahmet Okay Çelebi

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Analysis as a Life: To the 80th Birthday of Professor H. Begehr

Professor Heinrich Gottfried Werner Begehr has celebrated his 80th birthday a few months earlier than this book is appeared. We would like to present here a few biographic data and briefly describe his research career. Also we have given some reminiscence about his life. More information can be found at Prof. H.Begehr’s web-site at FU-Berlin: http://page.mi.fu-berlin.de/begehrh/. He was born in Halle/ Germany on April 17, 1939. He has obtained his M.Sc. degree in Freie Universtität (FU) Berlin in 1966. He has completed his Dr. rer. Nat. degree in 1968 under the supervision of Professor Alexander Dinghas and his habilitation in 1970 (just in 2 years) in FU Berlin. He worked as a Professor in FU from 1970 to 2004, that is, until the date he is retired. In 1972 he has been elected as the Director of I. Math. Inst. FU Berlin. He held this position with some interruptions, for about 18 years. In 1975 Professor R. P. Gilbert visited Hahn-Meitner Institute in Berlin by an Alexander von Humboldt award for a semester. During this period Professor Begehr and Professor Gilbert have interchanged many ideas. This cooperation have ended up with many joint papers, the book project “Transformations, Transmutations, and Kernel Functions” composed of two volumes, together with the first steps towards the foundation of the journal “Complex Analysis, Theory and Applications”. Later the name of this journal has evolved as “Complex Variables and Elliptic Equations”. The next important outcome of this cooperation was founding the “International Society for Analysis, its Applications and Computation (ISAAC)”. This society was founded in 1995. The sequence of International ISAAC Congress’s, which are held in every 2 years, has started in 1997 at the University of Delaware. He was the secretary and treasurer of newly founded ISAAC. In 2001, Professor Begehr was the first elected president of the Society. The other presidents were Man Wah Wong (2005–2009), Michael Ruzhansky (2009–2013), Luigi Rodino (2013–2017) and Michael Reissig (2017–). After 2005 he was secretary and treasurer until the elections in 2017. ISAAC has supported more than 35 conferences all around the world, besides 11 ISAAC Congresses. Begehr’s first book with title “Topics in complex analysis: Four lectures given at the University of Delaware in January and February 1977” has appeared in 1977. vii

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Analysis as a Life: To the 80th Birthday of Professor H. Begehr

Now he has seven published books. He was one of the editors of 10 proceedings of the ISAAC congresses. He has translated or edited 30 books. He is the co-editor of – Berliner Studienreihe zur Mathematik, Heldermann Verlag, since 2004; (together with R. Gorenflo). – Series on Analysis, Applications and Computation, World Scientific, since 2005; (together with R. P. Gilbert, M. W. Wong). – Complex Variables and Elliptic Equations, Taylor and Francis, since 2008; (senior editor together with M. Lanza de Cristoforis and A. Pankov) He is also in the editorial board of the following journals and series of monographs: – Complex Variables and Elliptic Equations, since 1982; Taylor and Francis, – Monograph and Surveys in Pure and Applied Mathematics, since 1997; Chapman and Hall/CRC-Press, – Research Notes in Mathematics, since 1997; Chapman and Hall/CRC-Press, – International Society of Analysis, its Applications and Computation, during 1997–2004; Kluwer Academic publisher, – General Mathematics, since 2001; Lucian Blaga, Univ. of Sibiu, Romania, – Journal of Applied Functional Analysis, since 2004; NOVA Publ., – Journal of Analysis and Applications, since 2005; SAS Intern. Publ., – Advances in Algebra and Analysis, since 2005; Urmi Scientific Vision, – International Journal of Mathematics and Applications (IJMA), since 2005; Global Research Publications Serials, India – International Journal of Mathematics and Mathematical Sciences (IJMMS), since 2006; Hindawi Publ., – Advances in Pure and Applied Mathematics, since 2008; Hedermann Verlag, Lemgo. He has been invited to many foreign universities as “Guest Professor” and “Guest Research Professor”. He is a member of the following academic societies: – – – – – –

Berlin Mathematical Society, since 1984; International Society of Analysis, its Applications and Computation, since 1996; Member of Russian Academy of Natural Sciences, Armenian Branch, since 2002; Member of European Acad. Sci., since 2004; Foreign Member of Russian Academy of Natural Sciences, since 2003; Honorary Member of Europaische Akad. Naturwiss. in Hanover, since 2003.

The number of his research papers are more than 180. He is a very productive Ph.D. adviser. He has advised at least 26 students from nine different countries. He was a devoted teacher and continued to hold the weekly seminars after his retirement, too. Lastly, tracing his research papers and books, anybody can observe that Heinrich Begehr is an excellent co-worker.

Analysis as a Life: To the 80th Birthday of Professor H. Begehr

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We are sure that his contributions to complex partial differential equations and generally to the world of Mathematics will continue through himself and his 26 students. Minsk, Belarus Istanbul, Turkey

Berlin 2001

Ankara in April 2004

Sergei Rogosin Ahmet Okay Çelebi

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Ankara 2007

Analysis as a Life: To the 80th Birthday of Professor H. Begehr

Analysis as a Life: To the 80th Birthday of Professor H. Begehr

Ankara 2007

Moscow 2011

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Analysis as a Life: To the 80th Birthday of Professor H. Begehr

Collage for Heinrich Begehr’s 70th birthday

Analysis as a Life: To the 80th Birthday of Professor H. Begehr

ISAAC 2009 in London

Krakow 2013

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Krakow 2013

Macao 2015

Analysis as a Life: To the 80th Birthday of Professor H. Begehr

Analysis as a Life: To the 80th Birthday of Professor H. Begehr

Heinrich Begehr and Okay Çelebi at the ISSAC 2017 in Växjö, Sweden

Jan Begehr

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Heinrich Begehr

Begehr in Yerevan in 2002

Analysis as a Life: To the 80th Birthday of Professor H. Begehr

Analysis as a Life: To the 80th Birthday of Professor H. Begehr

Catania 2005

Graz 1997

xvii

Few Stories from the Scientific Life of Professor Heinrich Begehr

Heinrich was visiting me at my beach house. The wind was very heavy as was the ocean. We decided to go swimming anyway. The water was too rough for me so I headed back to shore after a while. Heinrich continued to swim on. When I was on land the waves had gotten really rough and I started to worry about Heinrich. I became really worried that he might drown and did not know what to do. I could not see him because of the high waves. Some sailors came to shore and I asked had they seen someone swimming; they said yes he was still swimming out to sea. Eventually, when I had given up hope, Heinrich steps out of the sea. It seems he had wanted to swim to one of the control lights, which is a good swim in calm water but unbelievable in the rough sea we were experiencing now. Newark, DE, USA

Robert Gilbert

I hope all is fine and well in Berlin, or wherever your travels have happened to take you at the moment. It saddens me to realize it must already be 12 years since we met last—on the conference in Shantou, China, where I remember that unhappy baywatch when he realized that he cannot stop you from swimming about 1 km further out than permitted! Prague, Czech Republic

Miroslav Englis

I would like to recall a very important phase in our relationship with Heinrich, which represents him not only as a great scientist, but a high-quality person. In 1991 Heinrich together with the world’s best mathematicians arrived in Tbilisi for the symposium dedicated to the 100th anniversary of Niko Muskhelishvili. This was the time of high fulfillment for Georgia: we declared independence, elected our first president, set on to building the free, democratic society. The guests met with the president and returned home impressed with the encounter. At home they were hosted by Giorgi Manjavidze, I have wonderful memories of these days. . . Sadly, Georgia had to face some hard times soon after that, legitimately appointed president was overthrown by the communist junta, a period of real hardship was xix

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Few Stories from the Scientific Lifeof Professor Heinrich Begehr

underway. This period lasted long, nothing positive could be seen on the horizon. When Heinrich learned that our family was also experiencing difficulties, and shy to offer direct help to my father, he came up with an idea to support us in a very delicate way: an offer to the young people in the Complex Analysis group, to form a scholarship team. This was the first scientific scholarship possibility in Georgia. I will never forget how caring Heinrich was towards our group and especially to me, after my father’s death. Without his corrections and advice, it would have been impossible for us to publish articles or to participate in conferences and congresses. He empowered us to lead my father’s way and the traditions of Complex Analysis that had rich history in Georgia. Tbilisi, Georgia

Nino Manjavidze

This story happened in winter 2001 at the Conference AMADE in Minsk. It is wellknown that Heinrich Begehr prefers the sport style of life, practically does not drink alcohol. At AMADE-2001 we organized an excursion to Khatyn’, the memorial complex dedicated to villages burned by fascists together with people lived in these villages. The complex was built in the following manner: an empty well and a bell are built on the place of each burned house of the Khatyn’ village. Winter 2001 in Minsk was fairly cold and snowy. In these circumstances the bells sounded especially eerie and impressive. When participants of the Conference returned to the bus, there was rather long silence. Then Heinrich said “give me a glass of vodka. . .” For him as a German it was even harder than for all others. Minsk, Belarus

Sergei Rogosin

When I was working on my PHD with Professor Haack at the TU-Berlin, I was interested in getting the book on Potential Theory by Josip Plemelj which was in the library of Professor A. Dinghas at the Free University Berlin. Professor Dinghas was a not simple person and usually did not lend any of his books in particular to some guy from the TU. He was always in the center of a cloud of lady students. However, Heinrich Begehr was also one of the students of Dinghas. So, I asked him for help and indeed Heinrich could get Plemelj’s book and within very few days I copied the whole book which till this day is one of the juwels in my library. The book is wonderful and helped me significantly for my thesis. Till this day I am grateful to Heinrich Begehr. Stuttgart, Germany

Wolfgang Wendland

Contents

Deformation of Complex Structures and Boundary Value Problem with Shift . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . G. Akhalaia, G. Giorgadze, G. Makatsaria, and N. Manjavidze

1

Dirichlet Problem for Poisson and Bi-Poisson Equations in Clifford Analysis . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Ümit Aksoy

19

Boundary Eigenvalues of Pluriharmonic Functions for the Third Boundary Condition on the Unit Polydiscs. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Alip Mohammed

39

Survey of Some General Properties of Meromorphic Functions in a Given Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . G. Barsegian

47

Boundary Value Problems in Polydomains. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Ahmet Okay Çelebi

69

Completeness Theorems on the Boundary in Thermoelasticity . . . . . . . . . . . . Alberto Cialdea

93

A Circle Pattern Algorithm via Combinatorial Ricci Flows.. . . . . . . . . . . . . . . . 117 Dong-Meng Xi, Shi-Yi Lan, and Dao-Qing Dai Strong Asymptotic Analysis of OLPs on the Unit Circle by Riemann-Hilbert Approach .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 139 Yufeng Wang, Yifeng Lu, and Jinyuan Du Time Dependent Solutions for the Biot Equations . . . . . . .. . . . . . . . . . . . . . . . . . . . 171 Robert P. Gilbert and George C. Hsiao Schwartz-Type Boundary Value Problems for Monogenic Functions in a Biharmonic Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 193 S. V. Gryshchuk and S. A. Plaksa xxi

xxii

Contents

The String Equation for Some Rational Functions . . . . . . .. . . . . . . . . . . . . . . . . . . . 213 Björn Gustafsson Newtonian and Single Layer Potentials for the Stokes System with L∞ Coefficients and the Exterior Dirichlet Problem.. . . .. . . . . . . . . . . . . . . . . . . . 237 Mirela Kohr, Sergey E. Mikhailov, and Wolfgang L. Wendland Special Functions Method for Fractional Analysis and Fractional Modeling . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 261 S. V. Rogosin and M. V. Dubatovskaya On Elliptic Systems of Two Equations on the Plane. . . . . .. . . . . . . . . . . . . . . . . . . . 279 A. P. Soldatov Real Variable Inverse Laplace Transform .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 303 Vu Kim Tuan, A. Boumenir, and Dinh Thanh Duc

Deformation of Complex Structures and Boundary Value Problem with Shift G. Akhalaia, G. Giorgadze, G. Makatsaria, and N. Manjavidze

Dedicated to H. Begehr

Abstract In this paper we consider so called Beltrami parametrization of Riemann surfaces and show that the Riemann-Hilbert boundary value problem with shift is equivalent to classical Riemann-Hilbert boundary value problem with respect to the complex structures defined by Beltrami parametrization induced from shift operator. Keywords Beltrami equation · Conformal structure · Shift operator · Holomorphic bundle · Riemann surface Mathematics Subject Classification (2010) Primary 30E25, 30G35; Secondary 31A30

1 Heinrich Begehr and Complex Analysis Group in Georgia Heinrich Begehr is an honorable representative of the German mathematical school, under the auspices of which complex analysis was born. Riemann, Weierstrass, Hilbert problems were the subjects of research of the founders of Georgian mathematical school. Great German mathematician such as F. Hirzebruch and H. Rörhl were well acquainted with and often quoted the representatives of the Georgian mathematicians N.Mushkhelishvili, I.Vekua, B. Khvedelidze, G. Manjavidze, R. Gamkrelidze.

G. Akhalaia · G. Giorgadze · G. Makatsaria I. Vekua Institute of Applied Mathematics, Tbilisi, Georgia e-mail: [email protected]; [email protected] N. Manjavidze () Ilia State University, Tbilisi, Georgia e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_1

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G. Akhalaia et al.

It is a great pleasure for our group of complex analysis to make tribute of this work to Heinrich Begehr on the occasion of his 80th birthday. Many of his works are dedicated to the same problems as he got familiar with the works of Georgian mathematicians Muskhelishvili, Vekua and Bitsadze long time ago. He maintained his scientific ties and friendship with the peers of the next generation, among them with Prof. Obolashvili in particular. Our friendship became even stronger during Heinrich’s visit to Tbilisi along with many other prominent mathematicians at the event of 100th anniversary of Niko Muskhelishvili. This was in 1991, when he was hosted in the family of Prof. Manjavidze. Almost three decades have passed since then and he has made enormous contribution in the scientific advancement of each of us. We are greatly honored to be the permanent members of the ISAAC organization founded by Heinrich and R. Gilbert. In 2007 under the guidance of Heinrich and Prof. Jaiani the ISSAC conference dedicated to the 100th anniversary of Ilia Vekua was held in Tbilisi. We have participated in almost all meetings of the ISAAC congress. It is our utmost joy and pleasure that this collaboration and friendship remains fruitful and warm until today. Heinrich is a special, unique person for everyone, he tries to help others to solve their problems both in mathematics and in life. It is both a great privilege and responsibility to have scientific partnership with Heinrich as we are witnessing how these traditions are being carried on in his humble and collegial attitude towards us. Many thanks and sincere wishes to our charming, talented and highly revered person.

2 Boundary Value Problem with Shift Suppose that X is a compact Riemann surface with a distinguished point x∞ and a given local parameter z−1 around x∞ : thus z is a holomorphic map from a neighbourhood of x∞ to a neighbourhood of ∞ in the Riemann sphere CP 1 . We shall assume that z(x∞ ) = ∞, and that z is an isomorphism between a neighbourhood of x∞ and the region |z| > 1/2 on the Riemann sphere. The standard circle S 1 can then be identified with the circle |z| = 1 around x∞ on X. We shall denote the part of X where |z| > 1 by X∞ , and the complement of the region where |z| ≥ 1 by X0 . Thus X 0 ∩ X ∞ = S 1 , where X 0 , X ∞ are the closure of X0 and X∞ , respectively. Let H (n) = L2 (S 1 , C n ) Hilbert space of square-summable C n -valued functions (n) on circle. Denote by HX closed subspace of H (n) consisting of the boundary values of holomorphic maps X0 → C n . The loop group LGLn (C) of all continuous maps S 1 → GLn (C) acts on the space H (n) . The stabilizer of HXn in LGLn (C) is the group L+ X GLn (C) of loops which are the boundary values of holomorphic maps X0 → GLn (C). The point of quotient space Gr n,X ∼ = LGLn (C)/L+ X GLn (C) can be identified with an isomorphism class of pair (E, α), where E is a holomorphic

Deformation of Complex Structures

3

vector bundle on X and α is trivialization of E|X∞ which extends smoothly to X∞ . Denote by L− X GLn (C) the space of loops which are the boundary values of holomorphic maps X∞ → GLn (C). Then the natural action of L− X GLn (C) on Gr n,X permutes the trivializations α transitively and obtain the following proposition. Proposition 2.1 ([13]) The set of double cosets + L− X GLn (C) \ LGLn (C)/LX GLn (C)

is the set of isomorphism classes of n-dimensional holomorphic vector bundles on X. Let  be a smooth closed positively oriented curve in Riemann surface of genus g = 0, i.e. in CP 1 , which separates CP 1 into two connected domains U+ and U− . Suppose 0 ∈ U+ and ∞ ∈ U− . Let us denote by  the space of all Höldercontinuous matrix functions f :  → GLn (C) with the natural topology. Let + = {f ∈  : f is the boundary value of the matrix function holomorphic in + U }. − = {f ∈  : f is the boundary value of the matrix function holomorphic in − U and is regular at infinity f (∞) = 1}. Above proposition is generalization of the following Birkhoff Factorization theorem. Proposition 2.2 Any matrix function f ∈  can be represented as f (t) = f − (t)dK f + (t),

(2.1)

where f ± ∈ ± and dK is a diagonal matrix dK = diag(t k1 , . . . , t kn ) satisfying the condition k1 ≥ . . . ≥ kn . The diagonal matrix dK will be called the characteristic loop of the corresponding matrix function, K = (k1 , k2 , . . . , kn ) called the characteristic multi-index or partial indices of f . Two matrix functions f, g ∈  will be called equivalent, if f and g have identical characteristic multi-indices. For K = (k1 , k2 , . . . , kn ), denote by K the set of equivalence classes of loops . Consider the holomorphic vector bundle on CP 1 which is obtained by the covering of the Riemann sphere CP 1 by three open sets {U + , U − , U3 = CP 1 \ {0, ∞}}, with transition functions g13 = f + : U + ∩ U3 → GLn (C), g23 = f − dK : U − ∩ U3 → GLn (C). It is denoted by E → CP 1 (see [5]).

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Theorem 2.3 ([13]) Every holomorphic vector bundle splits into direct sum of the line bundles E∼ = E(k1 ) ⊕ . . . ⊕ E(kn ).

(2.2)

The numbers k1 ,. . . ,kn are the Chern numbers of the line bundles E(k1 ),. ..,E(kn ) and satisfy the conditions k1 ≥. . . ≥ kn . The integer-valued vector K = (k1 , . . . , kn ) ∈ Z n is called the splitting type of the holomorphic vector bundle E. It defines uniquely the holomorphic type of the bundle E. Connection between partial indices of the boundary value problem, characteristic multi-index of the matrix-function f ∈  and splitting type of the holomorphic vector bundle E are presented in the following summarizing theorem: Theorem 2.4 There is a one-to-one correspondence between the strata K and holomorphic vector bundles on CP 1 . Consider the following Riemann-Hilbert boundary value problem: RHBVP: Find a piecewise holomorphic vector function (t) in U+ ∪U− , which admits continuous boundary values on  and satisfies on  the boundary condition + (t) = f (t)− (t), t ∈  and has finite order at ∞. Denote by O(E) the sheaf of germs of holomorphic sections of the bundle E, then the solutions of the RHBVP are elements of the zeroth cohomology group H 0 (CP 1 , O(E)), therefore the number l of the linearly independent solutions of problem is dim H 0 (CP 1 , O(E)). Since the Chern number c1 (E) of the bundle E is equal to the index of det f (t), we have obtained the known criterion of solvability of the RHBVP. In particular the following theorem is true: Theorem 2.5 The Riemann-Hilbert boundary problem has solutions if and only if c1 (E) ≥ 0, and the number l of linearly independent solutions is l = dim H 0 (CP 1 , O(E)) =



ki + 1.

ki >0

Consider the system of 2n elliptic partial differential equations presented in complex form ∂− W (z) = A(z)W (z), z

(2.3)

where A(z) is bounded matrix function on a domain U ⊂ C and W (z) = (w1 (z), . . . , wn (z)) is an unknown vector function [14]. Let C(t) be any matrix function on  and C(t) ∈ , which has a holomorphic extension to U + , not necessarily nonsingular everywhere, and let 1 −1 C) = 0, then there exists an extension of f −1 C to U + . Denote 2π  argdet(f

Deformation of Complex Structures

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by P (z) this extension and let (z) be some holomorphic solution of the RHBVP. Consider the substitution w(z) = P (z)(z) on z ∈ U + ; w(z) = (z) on z ∈ U − . Proposition 2.6 The matrix function (z) is holomorphic in U + ∪ U − iff w is a solution of the system ∂z w = Aw,

(2.4)

where A(z) = ∂z P P −1 , for z ∈ U + and A(z) = 0, for z ∈ U − . We investigate the following generalization of Riemann-Hilbert boundary value problem with shift ϕ + [α(t)] = a(t)ϕ − (t)

(2.5)

where α is an orientation preserving diffeomorphism of boundary curve. The study of this problem can be reduced to the study of Riemann-Hilbert boundary value problem without shift for the elliptic system of the form ∂z w − q(z)∂z w = 0.

(2.6)

Last equation is Beltrami equation and defines complex structure on Riemann surface, therefore boundary value problem with shift is usual Riemann-Hilbert boundary value problem respect to complex structure, defined by corresponding Beltrami equation. From this follows, such type boundary value problems (in particular, Hasseman or Carleman boundary value problems) can be consider as Riemann-Hilbert boundary value problem with appropriate complex structure. The complete analysis of the problem (2.5) was done by D. Kveselava. Furthermore this problem and its variations were the object of research of many leading mathematicians (Vekua N., Korzadze R., Litvinchuk G., Orth D., Zverovich E., Chibrikova L., Manjavidze G. and others). For more detailed historical survey of this problem refer to [10]. In our opinion Bojarski was the first who stated that the problem (2.5) is equivalent to the classical Riemann-Hilbert boundary value problem (i.e. when a(t) ≡ t) for the solution of the Beltrami equation, where the Beltrami coefficient is defined up to equivalence by the given homeomorphism. This idea was completely implemented by Manjavidze G. In particular he reduced this problem to the classical Riemann-Hilbert problem for the solution of the Beltrami equation and expressed the Beltrami coefficient by the given homeomorphism. He also investigated the problem (2.5) for the solution of general elliptic systems, which involve the Beltrami and Carleman- Bers-Vekua equations as particular case. He obtained calculation formulas for the canonical matrix of the problem (2.5) and its general solution. He studied the dependence of the partial indices of the problem on the shift operator (see [11, 12]).

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Presently the deformation of conformal structures of the Riemann surfaces and of the complex structures of homomorphic bundles, Teichmüller space and corresponding Beltrami equation are the subjects of interest of theoretical physics due to the diverse applications in the conformal field theory [2], string theory [2] and in exactly solvable models of two dimensional quantum mechanics as well (see [8]). Therefore it is challenging to analyze the connection of the Riemann-Hilbert boundary value problem with abovementioned objects of algebraic topology.

3 Beltrami Parametrization and Boundary Value Problem with Shift The complex structure on X is given by the complex covering (Uα , zα ) of X. We dzα will denote by K the canonical bundle of X defined by the 1-cocycle kαβ = dz β with respect to this fixed local complex coordinates zα , and we will set ∂zα = ∂z∂α and ∂zα = ∂z∂α . We introduce Beltrami differential, denoted by μ, namely a (1, 0)vector valued (0, 1)-form with |μ| < 1. The Beltrami differential μ can be viewed as a smooth section of the bundle K −1 ⊗ K, i.e. −1 μα = kαβ k αβ μβ

in Uα ∩ Uβ such that μα < 1. Let B(Xn ) denote the space of smooth Beltrami differentials on X. By the Beltrami parametrization [9] of complex structures over X we mean to find the holomorphic coordinates {wα } with respect to the complex structure parametrized by μ = μα , |μα | < 1. This comes to solve locally the following Pfaff system: dwα = λα (dzα + μα dzα ),

(3.1)

in Uα , from this we get ∂wα =

∂α − μα ∂ α . λα (1 − μα μα )

To solve Eq. (3.1) is equivalent to solve locally in Uα the Beltrami equation ∂ α wα − μα ∂α wα = 0.

(3.2)

Beltrami equation (3.2) always admits as a solution a quasiconformal mapping with dilatation coefficient μα , |μα | < 1. One thus remarks that wα is a holomorphic functional of μα , which will be denoted for a while by wμα , and will play the role of the complex coordinate. Therefore, the solution of the Beltrami equation

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is a mapping (zα , zα ) → (wα (zα , zα ), w α (zα , zα )) in Uα , which preserves the orientation, and is thus locally invertible, so that wμα defines a new complex coordinate mapping on Uα . Moreover, in the intersection Uα ∩ Uβ , it follows from the patching law of μ, that solving the Beltrami equation in the overlap shows that the transition function wμα (wμβ )−1 is holomorphic and depends holomorphically on μ. Hence, the covering {(Uα , wμα }) defines a new μ complex structure on X, the one given by μ and denoted by Xn . The fibered complex manifold B(X) × X, with local complex coordinates (μ, wμ ), defines a complex analytic family of compact Riemann surfaces. We will say that the Riemann surface Xμ is different from the Riemann surface X, (μ = 0 corresponds to the standard complex structure), with local complex coordinates {(Uα , zα )}. Let ϕ be a smooth diffeomorphism of X homotopic to the identity map. Let μ be a given smooth Beltrami differential on X, represented in a chart, let say (Uα , zα ), by the smooth function μ. One can see that diffeomorphisms act holomorphically on the coordinates (μ, wμ ), and therefore μ and μϕ define equivalent complex structures. Let S 1 be a unit circle dividing the extended complex plane CP 1 = C ∪ ∞ into connected components: U + and U − . We consider the following boundary value problem: Find a piecewise analytic function in CP 1 with jump line S 1 satisfying the boundary conditions ω+ (α(t)) = ω− (t),

t ∈ S1,

ω+ (z) = z + O(z−1 ).

(3.3) (3.4)

Consider the function 

z α(z) = |z|α |z|

 (3.5)

from S 1 to U + . Let  w(z) =

ω(α(z)), z ∈ U + ∪ S 1 , ω(z), z ∈ U −,

then w(z) is continuous function on complex plane and satisfies (2.6) Beltrami equation with coefficient

q(z) = 0, |z| > 1; q(z) = e2iθ

1 − ν (θ ) , z = |z|eiθ ∈ U +

1 + ν (θ )

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and α(t) = α(eiθ ) = eiν(θ) , where ν(θ ) is increasing function on [0, 2π], ν(2π) = ν(0) + 2π. Indeed, 

z zα

  z z

− α˙

  z

z ∂z α ˙ iθ ) eiθ α(eiθ ) − α(e ∂z w = =    = q(z, z) =   = −iθ ∂z w ∂z α e α(eiθ ) + α(e ˙ iθ ) z z z ˙ zα z +α z

= e2iθ

1 − ν (θ ) .

1 + ν (θ )

Denote by w0 main homeomorphism of Beltrami equation and suppose w0 images S 1 on Liapunov curve . Then there exist the solution ω of the problem (3.3), (3.4) and functions ω+ (z) and ω− (z) which are solutions of boundary value problem (3.3), (3.4) transforming one-to-one the domains U + and D − into the domains D + and D − , respectively. The inverse function of ω− reduces the boundary value problem with shift to the Riemann-Hilbert boundary value problem. Similar result is true for matrix boundary value problem and by analogy of one dimensional case the theory of Q-holomorphic vectors is responsible for the complex structure of vector bundles on Riemann surfaces. Therefore the RiemannHilbert boundary value problem for generalized analytic vectors is a general problem in this class of problems.

4 Almost Complex Structure Let X be a two-dimensional connected smooth manifold. By definition two complex atlases U and V are equivalent if their union is a complex atlas. A complex structure on X is an equivalence class of complex atlases. Riemann surface is a connected surface with a complex structure. A differential 1-form on X with respect to a local coordinate z can be represented in the form ω = αdz + βdz. Thus, ω has bidegree (1,1) and is a sum of the forms ω1,0 = αdz and ω0,1 = βdz of bidegree (1,0) and (0,1)respectively. The change of local coordinate z → iz induces on the differential forms the mapping given by ω → i(αdz − βdz) = iω1,0 − iω0,1 . Denote by J the operator defined on 1-forms by the rule J ω = iω1,0 − iω0,1. This operator does not depend on the change of the local coordinate z and J 2 = −1, where 1 denotes the identity operator. Therefore, the splitting 1 = 1,0 + 0,1 is the decomposition of the space of differential 1-forms into eigenspaces of J : T ∗ (X)C → T ∗ (X)C . On the tangent space T X the operator J acts via ω(J v) = (J ω)(v), for every vector ∂ field v ∈ T X. If z = x + iy and taking v = ∂x , one has dz(J v) = idz(

∂ ∂ ∂ ∂ ∂ ∂ ) = i = dz( ) ⇒ J = ,J =− . ∂x ∂y ∂x ∂y ∂y ∂x

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  0 −1 It means that on the basis of T X the operator J is given by . 1 0 Therefore, by the complex structure defined from local coordinates defines the operator J : T ∗ (X)C → T ∗ (X)C , with the property J 2 = −1. This operator is called almost complex structure. Conversely, let X be a smooth surface and let J : Tx (X) → Tx (X), x ∈ X, be such an operator, i. e. J 2 = −1. The pair (X, J ) is called a pseudoanalytic surface. As above, by duality it is possible to define J on 1-forms on X. The space of 1forms 1 decomposes into eigenspaces corresponding to the eigenvalues ±i of J and 1 = J1,0 + J0,1 . In particular, J J1,0 = iJ1,0 and J J0,1 = −iJ0,1. Let f be a smooth function, then df ∈ 1 and decomposes by bidegree as df = ∂J f + ∂ J f, where ∂J f := (df )J1,0 and ∂ J f := (df )J1,0 . By definition f is J -holomorphic if it satisfies the Cauchy-Riemann equation ∂ J f = 0. Let (X, J ) be a pseudoanalytic surface. In the neighborhood of every point x ∈ X it is possible to change the local coordinate in such a way that dz will be of (1, 0)J type. Then the decomposition of dz by bidegree is dz = ω + δ, where ω, δ are forms of bidegree (1, 0)J . Because the fibre of TJ1,0 X is a one-dimensional complex space, we have δ = μω, where μ is some smooth function μ(0) = 0. From this it follows ∂ ∂ ( ∂x , ∂y )

dz = ω + μω and dz = ω + μω.

(4.1)

Therefore for every smooth function f in the neighborhood of x ∈ X we have df = (∂f + μ∂f )ω + (∂f + μ∂f )ω = ∂J f + ∂ J f From this it follows that f is J -holomorphic iff ∂ J f = 0, i.e. ∂f + μ∂f = 0,

(4.2)

or in old notations ∂z f + ν∂z f = 0. Equation (4.2) is the Beltrami equation. Thus a smooth function defined on a pseudo-analytic surface (X, J ) is J -holomorphic iff it satisfies the Beltrami equation. Suppose f is J -holomorphic and let f = ϕ + iψ, where ϕ and ψ are real-valued functions. Consider the complex-valued function w defined by the identity w = ϕF + ψG, where F, G are complex-valued Hölder continuous functions satisfying the condition I m(F G) > 0 [3, 8].

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Proposition 4.1 The function w = ϕF + ψG is (F, G)-pseudo-analytic. Indeed, w = ϕF + ψG = iG−F 2 f + solution of the Beltrami equation

−iG−F 2

f , from which it follows that f is a

(iG − F )∂f − (iG + F )∂f = 0 iff w is a solution of the Carleman-Bers-Vekua equation ∂w +

F ∂G − ∂F G FG − FG

w+

F ∂G − ∂F G FG − FG

w = 0.

In D ⊂ C every metric has the form λ|dz + μdz|, where λ > 0 and the complex function μ satisfies |μ| < 1, from which it follows, that J is defined uniquely by the 1-form ω = dz + μdz on D with properties J ω = iω, J ω = −iω. The forms of this type are forms of bidegree (1,0) with respect to J (the space of such forms has been denoted above by J1,0 ). If δ ∈ J1,0 , then δ = αω + βω and it is proportional to ω. Therefore J is determined uniquely up to a constant multiplier (1, 0)J by the form ω. Functions holomorphic with respect to J have differentials proportional to ω. Indeed, if df + iJ (df ) = 0, then J (df ) = idf and from the representation df = αω + βω we obtain, that βω = 0. Since df = αω + βω, in D ⊂ C the Cauchy-Riemann equation with respect to J with base form ω = dz + μdz can be represented as the Beltrami equation ∂f = μ∂f. This equation has a solution f such that it is a biholomorphic map from (D, J ) to f (D), Jst , where Jst is the standard conformal structure on C. Therefore we have proved the following proposition. Proposition 4.2 (See [1]) On simply connected domains there exists only one complex structure and conformal structures are in one-to-one correspondence with complex functions μ with |μ| < 1. From this proposition and Theorem 4.1 it follows the proposition Proposition 4.3 (See [1]) There exists a one-to-one correspondence between the space of conformal structures and the space of generalized analytic functions on each simply connected open domain of the complex plane.

5 The Holomorphic Discs Equation Let D be the unit disc in the complex plane C with the standard complex structure . Jst and the coordinate function ζ. Jst is uniquely determined by the form dζ ∈ J1,0 st

The map φ : D → X of class C 1 is holomorphic iff ψ ∗ J1,0 (X) ⊂ 1,0 (D). Let z be another coordinate function on D. We study a local problem, therefore, without loss of generality, it is possible to consider φ as mapping from (D, Jst ) to (Cz , J ),

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where the complex structure J is defined by dz = ω + μω, ω ∈ J1,0 . Therefore we have ζ → z = z(ζ ), z(0) = 0. From (4.1) we obtain ω=

dz − μdz . 1 − |μ|2

The form ω is J -holomorphic, which means that the form z∗ (dz − μdz) = (∂ζ z − μ∂ζ z)dζ + (∂ζ z − μ∂ζ z)dζ has bidegree (1, 0) on D, therefore ∂ζ z − μ∂ζ z = 0. From this, using the identity ∂ζ z we obtain ∂ζ z = μ(z)∂ζ z.

(5.1)

Obtained expression is called the equation of holomorphic disc. It is known that f satisfies this equation iff f −1 satisfies the corresponding Beltrami equation. ∂w(z) Proposition 5.1 ([1]) If ω = u + iv satisfies the equation ∂w(z) ∂ z¯ + μ(z) ∂z = 0, |μ| < 1 and a and b are holomorphic functions such that μ = a−b a+b , then W = au + ibv is holomorphic.

Indeed, ∂ ω+ω ω−ω a b (a + ib ) = (ωz + ωz ) + (ωz − ωz ) = ∂ z¯ 2 2 2 2 = ωz (

a+b a−b ) + ωz ( ), 2 2

therefore if ω is a solution of the equation ωz¯ + a−b a+b ωz = 0, then ∂z¯ W = 0. From this proposition it follows in particular that W is (a, ib)-pseudo-analytic.

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6 Relation Between Beltrami and Holomorphic Disc Equations In this section we give detailed analysis of the theory of pseudo-analytic functions in the light of Beltrami equation and holomorphic disc equation and prove the equivalence of these equations. Let (F, G) be a normalized generating pair on complex space C [3] it means that (1) F, G ∈ C p−2 , p > 2; (2)Fz , Gz ∈ Lp,2 (C) ∩ Cβ , 0 < β < 1; (3) p

I m(F (z)G(x)) ≥ K0 > 0, K0 = const, z ∈ C. As above, every function W, at every point, is uniquely represented by F (z), G(z) the following form W (z) = ϕ(z)F (z) + ψ(z)G(z),

(6.1)

where ϕ, ψ are real functions. Let W (z) be (F, G)-pseudoanalytic in C, then it is known that W (z) is the solution of the Carlemann-Bers-Vekua equation Wz = AW + BW ,

(6.2)

where A and B may be calculated by well-known formulas [3]. From the pseudo-analyticity it follows also, that there exist continuations of the partial derivatives ϕz , ϕz , ψz , ψz and F ϕz + Gψz = 0. Consider the function ω(z) = ϕ(z) + iψ(z). Then 2(F ϕz + Gψz ) = (F − iG)(ϕz + iψz ) + (F + iG)(ϕz − iψz ) = = (F − iG)(ϕ + iψ)z + (F + iG)(ϕ − iψ)z = = (F − iG)ωz + (F + iG)ωz = 0. Hence it follows ωz (F − iG) + ωz (F + iG) = 0.

(6.3)

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Lemma 6.1 F (z) − iG(z) = 0. Indeed, |F (z) − iG(z)|2 = (F (z) − iG(z)(F (z) − iG(z)) = = (F (z) − iG(z)(F (z) + iG(z)) = = |F (z)|2 + |G(z)|2 − i(F (z)G(z) − F (z)G(z)) = = |F (z)|2 + |G(z)|2 + 2I m(F (z)G(z)) ≥ 2K0 > 0,

(6.4)

when |F (z)|2 > 0, |G(z)|2 > 0, I m(F (z)G(z)) ≥ K0 for every z ∈ C. The lemma is proved. From Lemma 6.1 and (6.3) it follows ⇒ ωz + ωz

F + iG = 0. F − iG

(6.5)

Denote by q(z) = − FF (z)+iG(z) (z)−iG(z) . Lemma 6.2 |q(z)| ≤ q0 < 1, z ∈ C. The proof be divided following two steps: Step 1. |q(z)|2 =

|F (z) + iG(z)|2 (F (z) + iG(z))(F (z) + iG(z)) = ⇒ |F (z) − iG(z)|2 (F (z) − iG(z))(F (z) − iG(z)) ⇒

|F (z)|2 + |G(z)|2 − 2I m(F (z)G(z)) |F (z)|2 + |G(z)|2 + 2I m(F (z)G(z))

< 1,

(6.6)

when I m(F (z)G(z)) ≥ K0 > 0, z ∈ C. Step 2. The function F, G satisfies Carlemnan-Bers-Vekua equation Fz = aF + bF , Gz = aG + bG,

(6.7)

when F ∈ C p−1 (C), a, b ∈ Lp,2 (C) we obtain aF + bF ∈ Lp,2 (C). From (6.7) p

it follows F (z) = (z) + TC (aF + bF )(z),

(6.8)

where (z) is an entire function. From F (z), TC (aF + bF )(z) ∈ C p−2 (C) it folp

lows that (z) ∈ C p−2 (C). By the Liouville theorem we obtain (z) = const, p

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therefore (z) = C, z ∈ C. From this and (6.8) we obtain F (z) = C + TC (aF + bF )(z).

(6.9)

When TC (aF + bF )(∞) = 0, from (6.9) it follows that F (∞) = C. In a similar way we obtain G(∞) = C1 . When I m(F (z)G(z)) ≥ K0 , therefore I m(F (∞)G(∞)) ≥ K0

(6.10)

and from (6.6) and (6.10) we obtain ⇒ |q(∞)|2 =

|F (∞)|2 + |G(∞)|2 − 2I m(F (∞)G(∞)) |F (∞)|2 + |G(∞)|2 + 2I m(F (∞)G(∞))

< 1.

(6.11)

From (6.6) and (6.11) it follows |q(z)| < 1, z ∈ C, |q(∞)| < 1, therefore |q(z) ≤ q0 < 1, z ∈ C. Proposition 6.3 There exists a function q (z) such that ω is the solution of Beltrami equation with the coefficient q (z). Introduce the function q (z) : ⎧ ⎪ ⎨ q(z) ∂z ω , when ∂z ω = 0, ∂z ω q (z) = ⎪ ⎩ 0, when ∂z ω = 0.

(6.12)

and consider the equation ∂z ω − q(z)

∂z ω = 0. ∂z ω

From (6.12) it follows that ω satisfies the equation  z ω = 0. ∂z ω − q(z)∂

(6.13)

It is clear that  = |q(z) |q(z)|

∂z ω ∂z ω | = |q(z)|| | = |q(z)| ≤ q0 < 1. ∂z ω ∂z ω

(6.14)

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From (6.13) and (6.14) it follows that ω(z) is the solution of the Beltrami equation  z h = 0. ∂z h − q(z)∂

(6.15)

In the domain U ⊂ C the function ω is represented as ω(z) = (W (z)), where W (z) is a complete homeomorphism of Eq. (6.15) and (ζ ) is an analytic function on W (U ).

7 Relation Between Pseudoanalytic Functions and Quasiconformal Mapping Here we prove, that if ω : C → C quasiconformal homeorphism, (i.e. satisfies some Beltrami equation), then W (z) = A(z)ω(z) + B(z)ω(z)

(7.1)

is generalized analytic function (i.e. satisfies some Carleman-Bers-Vekua equation), where A, B have partial derivatives with respect to z and z and Δ(z) = |A|2 − |B|2 > 0. From (7.1) it follows W (z) = A(z)ω(z) + B(z)ω(z).

(7.2)

From (7.1) and (7.2) we have ω(z) =

A(z)W − B(z)W (z) . (z)

(7.3)

Indeed, multiplying both sides of expressions (7.1) and (7.2) by B and A respectively and consider their difference: B(z)W (z) − A(z)W (z) = (|B|2 − |A|2)ω = −(z)ω(z), from this it follows (7.3). Now consider partial derivative of W (z) with respect to z: Wz = Az ω + Bz ω + A(z)ωz Bωz . If we replace ωz by ωz and use expression (7.3), we obtain Wz = Az =

A(z)W − B(z)W (z) A(z)W − B(z)W (z) + Bz + Aωz + Bωz = (z) (z)

(Az A − Bz B) (−Az B + Bz A) W (z) + W (z) + A(z)ωz + Bω z .  

(7.4)

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Denote by Az A − Bz B ˜ −Az B + Bz A ˜ A(z) = , B(z) = .   Then we obtain the identity ˜ − BW ˜ = Aωz + Bωz . Wz − AW

(7.5)

Therefore we prove the following proposition: Proposition 7.1 If W (z) is a solution of the equation ˜ + BW ˜ , Wz = AW then ω is a solution of the equation ωz +

B ωz = 0 A

and visa versa, where | B A | < 1. 2 2 Remark The inequality | B A | < 1 follows from assumption |A| − |B| > 0.

Acknowledgements This work was supported by grant N FS 17-96 from the Shota Rustaveli National Science Foundation.

References 1. G. Akhalaia, G. Giorgadze, V. Jikia, N. Kaldani, N. Manjavidze, G. Makatsaria, Elliptic systems on Riemann surfaces. Lecture Notes of TICMI, vol. 13, pp. 1–147. Tbilisi University Press, Tbilisi (2012) 2. A.A. Belavin, V.G. Knizhnik, Complex geometry and the theory of quantum strings. Sov. Phys. JETP 64, 22–43 (1986) 3. L. Bers, Theory of Pseudo-analytic Functions (Courant Institute, New York, 1953) 4. B. Bojarski, On a boundary value problem for a system of elliptic equations. Dokl. Akad. Nauk SSSR (N.S.) 102, 201–204 (1955) 5. B. Bojarski, G. Giorgadze, Some analytical and geometric aspects of the stable partial indices, Proceedings of the I.Vekua Institute of Applied Mathematics, vol. 61–62 (2011–2012), pp. 14–32 6. G. Giorgadze, On monodromy of generalized analytic functions. J. Math. Sci. (N.Y.) 132, 716– 738 (2006) 7. G. Giorgadze, Moduli space of complex structures. J. Math. Sci. (N.Y.) 160, 697–716 (2009) 8. V. Kravchenko, Applied Pseudoanalytic Function Theory (Birkhauser Verlag, Basel, 2009) 9. S. Lazzarini, Flat complex vector bundles, the Beltrami differential and W-algebras. Lett. Math. Phys. 41, 207–225 (1997)

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10. G. Litvinchuk, Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift (Springer, New York, 2000) 11. G. Manjavidze, Boundary-value problems with conjugation for generalized analytic functions, in Proceedings of the I. Vekua Institute of Applied Mathematics (Tbilisi State University, Tbilisi, 1990) 12. G. Manjavidze, N. Manjavidzde, Boundary-value problems for analytic and generalized analytic functions. J. Math. Sci. 160, 745–821 (2009) 13. A. Pressley, G. Segal, Loop Groups (Clarendon Press, Oxford, 1984) 14. I.N. Vekua, Generalized Analytic Functions (Nauka, Moscow, 1988)

Dirichlet Problem for Poisson and Bi-Poisson Equations in Clifford Analysis Ümit Aksoy

Dedicated to Prof. H.G.W. Begehr on the occasion of his 80th birthday

Abstract Dirichlet problems for Poisson equation and a second order linear equation are studied in the unit ball by using an integral representation formula with respect to the Laplacian in the complex Clifford algebra Cm for m ≥ 3. Iterating the Green type kernel function, representation of the solution of the bi-Poisson equation with homogeneous Dirichlet condition is presented. Keywords Clifford analysis · Integral representations · Poisson equation Mathematics Subject Classification (2010) Primary 30G35; Secondary 31B10

1 Introduction Clifford analysis is introduced as a generalization of complex function theory to the higher dimensions. Many researchers have studied the theory of Clifford algebra valued functions, see [19, 21, 22, 25, 29, 31, 32] and references therein. As in the classical complex analysis, Cauchy integral formula is a very important tool in the Clifford analysis. Cauchy-Pompeiu type integral representation formulas expressing complex valued, quaternionic valued, and Clifford algebra valued functions have been developed in [3, 5–7, 12, 14, 15, 19, 20, 29, 31, 34–36]. In the case of complex Clifford algebra, fundamental solutions to powers of the Dirac operator and of the Laplace operator and Cauchy-type formulae are given in [6, 7, 17]. Cauchy integral representation formulas serve to solve the boundary value problems for partial differential equations, which are used in describing many problems

Ü. Aksoy () Department of Mathematics, Atilim University, Ankara, Turkey e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_2

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20

Ü. Aksoy

in science and engineering. In particular, Dirichlet problem for polyharmonic and polyanalytic functions in the complex plane is essential for studying problems in mathematical physics. In the complex plane, Dirichlet problem for Poisson and higher-order Poisson equations are investigated in different domains and explicit solutions with the respective Green functions are investigated [4, 9–11, 13]. The Dirichlet boundary value problems for linear complex partial differential equations are also discussed in the unit disc and in an annular domain [1, 2] by transforming the problems into respective singular integral equations. In recent years, Clifford analysis becomes a very important tool for the treatment of boundary value problems in domains in Rm , m ≥ 3. The Dirichlet problem for the Poisson equation was studied for the case of quaternions in [25] and for the case of Clifford-valued functions in [26] for bounded domains. In [24, 26], higher order boundary value problems were also investigated for Clifford-valued functions. Singular integral operators play an important part in boundary value problems in both complex and Clifford analysis. Stokes’ theorem provides the Theodorescu operator which is the convolution of an integrable function with the fundamental solution of the Dirac equation and it corresponds to the Pompeiu operator in classical complex analysis, see [26]. Indeed, it is a particular weak solution for the Dirac equation in Clifford analysis. Using the kernel functions of the iterated representations in terms of powers of Dirac operator and Laplacian, a hierarchy of kernel functions and integral operators providing particular solutions of higher-order Poisson and Dirac equations are given in [6, 7]. In this article, we discuss the Dirichlet problem in Clifford analysis for Poisson equation, a linear equation with Laplace operator as main part and bi-Poisson equation in the unit ball Bm in Rm using an integral representation formula in terms of the Laplacian. The paper is organized as follows: In the next section, we give basic notions about Clifford algebra and integral representations in Clifford analysis. In Sect. 3, complex Clifford algebra form of an integral representation in terms of the Laplacian for universal Clifford algebra valued functions given in [37] is presented. Then, the Dirichlet problem for Poisson equation is solved uniquely. In Sect. 4, a second order linear equation whose main part is the Laplace operator under Dirichlet conditions is studied. Using the integral operator appearing in the integral representation formula in terms of Laplacian having Green type kernel function, the problem is transformed into a singular integral equation in a manner of Vekua [33] who has treated the Beltrami equation in C. Solvability of the problem is discussed via Fredholm theory. In the last section Dirichlet problem for secondorder Poisson equation with homogeneous Dirichlet conditions is investigated and an integral representation for the solution is presented.

Dirichlet Problem for Poisson and Bi-Poisson Equations in Clifford Analysis

21

2 Preliminaries 2.1 Clifford Analysis Essentials In this part, we recall some definitions and basic concepts of Clifford analysis. For more information, see [19, 22, 25]. Let {ek : 1 ≤ k ≤ m} be an orthonormal basis of the m-dimensional real vector space Rm for m ≥ 2 with  respect to the usual scalar product so that any x ∈ Rm is represented as x = m k=1 xk ek where x = (x1 , . . . , xm ). Introducing a multiplication via ej ek + ek ej = −2δj k

1 ≤ j, k ≤ m

(2.1)

with e0 = 1 is the unit element, δj k is the Kronecker symbol, leads to a 2m -dimensional real linear, associative and non-commutative (universal) Clifford  algebra R0,m . Any a ∈ R0,m may be written as a = A aA eA in which aA ∈ R and the basis elements eA = e0 = 1 if A = ∅ eA = eα1 eα2 . . . eαk if A = {α1 , α2 , . . . αk } ⊆ {1, 2, . . . , m} with 1 ≤ α1 < α2 < · · · < αk ≤ m. The inner product a, b =



aA bA , a =

A



a A eA , b =

A



b A eA

A

gives a real Hilbert space structure in Clifford algebra. Each element x = (x0 , x1 , . . . , xm ) ∈ Rm+1 can be written as x = x 0 e0 +

m 

x k ek

k=1

 which are called as paravectors, where x0 = x0 e0 is the scalar part and m k=1 xk ek is the vector part of x. If the coefficients aA are complex rather than real, the corresponding algebra is called complex Clifford algebra which we will denote by Cm . This  algebra is m . For a = complexification of R0,m and its complex dimension is 2 A a A eA ,  aA ∈ C, denote the complex conjugate as a¯ = A a A eA where e0 = e0 = 1, ek = −ek , 1 ≤ k ≤ m, eA = eαk eαk−1 . . . eα1 and eA eB = eB eA .

22

Ü. Aksoy

A norm for an element a =



A a A eA

|a| :=

∈ Cm is defined by

 

1/2 |aA |

2

A

and in that case |a|0 = 2m/2 |a| turns into an algebra norm. A Clifford-analytic (Cm -valued) function f defined in a domain D of Rm is of the form  f (z) = fA (z)eA A

with fA (z) being complex valued function for z ∈ D. Let B(D) be a Banach space of complex valued functions defined on D. B can be the space C k (D) of k-times continuously differentiable functions in D, the space Lp (D) of all functions, whose pth power is Lebesgue integrable in D or the space W k,p of k times differentiable functions in Sobolev’s sense, whose kth derivative belongs to Lp (D). We define B(D, Cm ) := {f : D → Cm : fA ∈ B(D)}   2 1/2 . The Banach which is also a Banach space with the norm f B = A fA B spaces of these functions are denoted as C k (D, Cm ), Lp (D; Cm ) or W k,p (D; Cm ). The Dirac operator ∂ and its complex conjugate ∂ are given by ∂ :=

m 

ek ∂xk and ∂ :=

k=1

m 

ek ∂xk = ∂x1 −

k=1

m 

ek ∂xk .

k=2

In C, they correspond to Wirtinger operators (the Cauchy-Riemann and anti-Cauchy Riemann operators, respectively) ∂z¯ =

  1 1 ∂x + i∂y and ∂z = ∂x − i∂y . 2 2

∂ and ∂ are divisors of the Laplace operator  = ∂∂ = ∂∂ =

m  ∂2 . ∂xk2 k=1

Some differentiation rules are given by ∂z = z∂ = 2 − m ,

∂ z¯ = z¯ ∂ = m,

∂|z|2 = |z|2 ∂ = 2z , ∂|z|α = |z|α ∂ = α|z|α−2 z,

Dirichlet Problem for Poisson and Bi-Poisson Equations in Clifford Analysis

23

∂|z|2 = |z|2 ∂ = 2¯z , ∂|z|α = |z|α ∂ = α|z|α−2 z¯ , ∂(¯zk + zk ) = (¯zk + zk )∂ = 2k z¯ k−1,  ∂

z¯ |z|m

 =

z¯ ∂ = 0. |z|m

For further details on identities and properties, see [6, 19, 22]. The operators ∂ and ∂ act on the function f = A fA eA from the left and from the right as ∂f =

m   k=1 A

∂f =

m   k=1 A

 ∂fA ∂fA , f∂ = eA ek , ∂xk ∂xk m

ek eA

k=1 A

 ∂fA ∂fA , f∂ = eA ek . ∂xk ∂xk m

ek eA

k=1 A

A function f is called a left monogenic (or left regular) function if ∂f = 0 and a right monogenic (or right regular) function if f ∂ = 0. f is called left anti-regular if ∂f = 0 and a right anti-regular if f ∂ = 0. If f is both left and right monogenic, then it is said to be monogenic. The theory of monogenic functions for complex Clifford algebra is called as Hermitian Clifford analysis. Since  = ∂∂ = ∂∂, all components of a left and a right monogenic function satisfy the Laplace equation.

2.2 Integral Representations in Clifford Analysis This part is devoted to the presentation of integral representation formulas for Clifford analytic functions. We first start with the Gauss theorem and Cauchy-Pompeiu type representations for Cm -valued functions [6, 7, 12]. In the following, let D ⊂ Rm be a regular domain. Theorem 2.1 (Gauss Theorem) Let f, g ∈ C 1 (D; Cm ) ∩ C(D; Cm ). Then   ((f ∂)g + f (∂g))dv = f d σ g D

∂D



 ((f ∂)g + f (∂g))dv =

D

∂D

f d σ¯ g.

24

Ü. Aksoy

Here dv denotes the volume element of D, dσ the area element  of ∂D, n = (n1 , n2 , . . . , nm ) the outward directed normal vector on ∂D, n = m μ=1 nμ eμ the corresponding element in Cm , and d σ = dσ n the directed area element on ∂D, d σ¯ = dσ n¯ its complex conjugate. Theorem 2.2 (Cauchy-Pompeiu Type Representations) Any w ∈ C 1 (D; Cm ) ∩ C(D; Cm ) can be represented as w(z) =

1 ωm



ζ −z 1 d σ (ζ )w(ζ ) − m |ζ − z| ωm

∂D

w(z) =

1 ωm



ζ −z 1 d σ (ζ )w(ζ ) − m |ζ − z| ωm

∂D

 D

 D

ζ −z ∂w(ζ )dv(ζ ) |ζ − z|m

(2.2)

ζ −z ∂w(ζ )dv(ζ ). |ζ − z|m

(2.3)

The dual formulas are given by 1 w(z) = ωm

 ∂D

w(z) =

1 ωm

ζ −z 1 w(ζ )d σ (ζ ) − |ζ − z|m ωm

 w(ζ )d σ (ζ ) ∂D

ζ −z 1 − |ζ − z|m ωm

 (w(ζ )∂) D

 (w(ζ )∂) D

ζ −z dv(ζ ) |ζ − z|m

ζ −z dv(ζ ). |ζ − z|m

Here ωm denotes the area of the unit sphere in Rm . Definition 2.3 For f ∈ L1 (D; Cm ), the operator 1 Tf (z) = − wm

 D

ζ −z f (ζ )dv(ζ ) |ζ − z|m

(2.4)

is called the Clifford analytic Pompeiu operator or Teodorescu transform over D. For properties of (2.4) and some other operators arising in Clifford analysis, see [25]. For m = 2, (2.4) is the well-known Pompeiu operator T from the one-variable complex analysis, which is investigated in connection with the theory of generalized analytic functions by Vekua, see [33]. The operator (2.4) provides a particular solution to the inhomogeneous Dirac equation ∂w = f in D for any f ∈ L1 (D; Cm ).

Dirichlet Problem for Poisson and Bi-Poisson Equations in Clifford Analysis

25

Iterating Cauchy-Pompeiu type representation (2.2) leads to higher-order representation formula for kth powers of Dirac operator ∂ k and iterating (2.3) leads to a k formula for ∂ for k ≥ 2, see [6, 16]. For higher-order Dirac equation ∂kw = f for k ∈ N and f ∈ L1 (D; Cm ), the higher-order Pompeiu operator Tk f (z) =

(−1)k ωm

 D

(ζ − z)(ζ − z + ζ − z)k−1 f (ζ )dv(ζ ) 2k−1 (k − 1)!|ζ − z|m

provides a particular solution. Moreover, a representation formula related to the Laplacian  is obtained by iterating the Cauchy-Pompeiu formulas with another, see [5]. Theorem 2.4 Let D ⊂ Rm be a bounded and smooth domain and w ∈ C 2 (D; Cm ) ∩ C 1 (D; Cm ). Then for z ∈ D, w(z) =

1 ωm



1 ζ −z d σ (ζ )w(ζ ) − |ζ − z|m ωm

∂D

+

1 ωm



D

 ∂D

|ζ − z|2−m d σ (ζ )∂w(ζ ) 2−m

|ζ − z|2−m w(ζ )dv(ζ ) . 2−m

(2.5)

Above result states that, w can be represented by w = φ0 + φ1 + S1 w where S1 f (z) =

1 ωm

 D

|ζ − z|2−m f (ζ )dv(ζ ) 2−m

is the potential operator, φ0 is left regular and ∂φ1 is left anti-regular. Iterating the formula (2.5) inductively gives a higher-order representation formula in terms of kth powers of Laplacian k in different dimension cases, see [6, 8]. A hierarchy of weakly singular kernel functions are defined as the fundamental solutions in Cm for k and a class of integral operators are given providing a particular solution to higher-order Poisson equation k w = f, see [7]. Cauchy-Pompeiu type representations related to general higher-order operators l of the form ∂ k ∂ are studied with the corresponding kernel functions appearing in the domain integrals and operators having such kernels in [17, 30].

26

Ü. Aksoy

3 Dirichlet Problem for Poisson Equation In the case of functions with values in universal Clifford algebra, an integral representation formula in terms of Laplacian and the Dirichlet problem for Poisson equation in the unit ball in Rm for m ≥ 3 are investigated in [37]. In this section, we first give an integral representation formula in terms of the Laplacian for Cm -valued functions in the unit ball, which is indeed the complex Clifford form of the formula given in [37]. This representation formula is essential for the investigation of the Dirichlet problem for Poisson equation in Cm . Next, the kernel function appearing in the domain integral in this representation is discussed with some of its properties and some integral operators having this kernel and its derivatives are introduced in the context of Cm . We denote unit ball in Rm by Bm , i.e., Bm = {z : z ∈ Cm , |z| < 1} and by ∂Bm = {z : z ∈ Cm , |z| = 1}, the unit sphere. We assume m ≥ 3 in the following results. The following formula is a modified representation formula in terms of the Laplace operator in Bm to be used in the solution of Dirichlet boundary value problem for Poisson equation. Its universal Clifford algebra version was proved in [37]. Theorem 3.1 If w ∈ C 2 (Bm ; Cm ) ∩ C 1 (Bm ; Cm ), z ∈ Bm , then w(z) =

1 ωm

 ∂Bm

+

1 (2 − m)wm

 Bm

1 − |z|2 w(ζ )dσ (ζ ) |ζ − z|m

⎛ ⎜ ⎝

⎞ 1 1 ⎟ − m−2 ⎠ w(ζ )dv(ζ ). m−2 |ζ − z|  ζ  z|ζ | − |ζ | 

(3.1)

Proof For z ∈ Bm , Theorem 2.4 implies that w(z) =

1 ωm

 ∂Bm

+

1 ωm



Bm

ζ −z 1 d σ (ζ )w(ζ ) − |ζ − z|m ωm |ζ − z|2−m w(ζ )dv(ζ ) . 2−m

 ∂Bm

|ζ − z|2−m d σ (ζ )∂w(ζ ) 2−m (3.2)

Dirichlet Problem for Poisson and Bi-Poisson Equations in Clifford Analysis

27

Observe that, for z ∈ Bm with z = 0, 2−m    z  |z|2 − ζ 



2−m

Bm

w(ζ )dv(ζ )

⎡⎛  ⎤  2−m ⎞ 2−m  z  z   − ζ − ζ     2 2 |z| ⎢⎜ |z| ⎟ ⎥ = (∂∂w(ζ ))⎦ ∂ ⎠ ∂w(ζ ) + ⎣⎝ 2 − m 2 − m Bm 

⎡⎛

⎞

⎤

−ζ ⎟ −ζ ⎢⎜ ⎥ m ∂ ⎠ w(ζ ) +  m (∂w(ζ ))⎦ dv(ζ ) + ⎣⎝     z  z  |z|2 − ζ   |z|2 − ζ  z |z|2

z |z|2

holds. Using Theorem 2.1 we get  2−m  z   |z|2 − ζ 

 Bm

 2−m  z   |z|2 − ζ 

 = ∂Bm

2−m

2−m

 )∂w(ζ ) + d σ (ζ

w(ζ )dv(ζ )

 ∂Bm

z −ζ |z|2  )w(ζ ). m d σ (ζ    z  |z|2 − ζ 

(3.3)

Rewriting (3.3) gives  2−m   |z|2−m  |z|z 2 − ζ 



2−m

Bm

 2−m   |z|2−m  |z|z 2 − ζ 

 = ∂Bm

2−m

 )∂w(ζ ) + d σ (ζ

w(ζ )dv(ζ )

 ∂Bm

|z|2 |z|z 2 − ζ  )w(ζ ). m d σ (ζ    |z|m  |z|z 2 − ζ  (3.4)

Combining integral representation (3.2) and (3.4), using the facts that   k k  z    k ζ   |z|  2 − ζ  = |ζ |  2 − z , |z| |ζ | k

for any k ∈ N and n (z) = z for z ∈ ∂Bm , (3.1) follows.

28

Ü. Aksoy

In the case z = 0 we have   1 |ζ |2−m 1 ζ¯ d σ (ζ )∂w(ζ ) w(0) = d σ  (ζ )w(ζ ) − ωm |ζ |m ωm 2−m ∂Bm

−

1 ωm



Bm

∂Bm

|ζ |2−m 2−m

w(ζ )dv(ζ ),  

and applying Gauss theorem we get (3.1). Let G be the Green-type function in Cm defined by ⎛ ⎜ G(z, ζ ) = Km ⎝

⎞

1 1 ⎟ − m−2 ⎠ |ζ − z|m−2  ζ  z|ζ | − |ζ | 

(3.5)

where Km =

1 . (2 − m)ωm

It can be seen that G(z, ζ ) is a fundamental solution to the Laplace operator satisfying the following identities: • G(z, ζ ) = 0, z ∈ Bm \{ζ }, • G(z, ζ ) = G(ζ, z) for z = ζ , z, ζ ∈ Bm , • G(z, ζ ) = 0 for ζ ∈ ∂Bm , z ∈ Bm . The Poisson kernel in Cm is defined by P (z, ζ ) =

1 − |z|2 , |z| < 1, ζ ∈ ∂Bm . |ζ − z|m

If z = ζ , the function P is harmonic for |z| < 1. Then the representation (3.1) can be rewritten in the form   1 w(z) = P (z, ζ )w(ζ )dσ (ζ ) + G(z, ζ )w(ζ )dv(ζ ). ωm Bm

∂Bm

If w ∈ C 2 (Bm ; Cm ) is harmonic in Bm , then  1 w(z) = P (z, ζ )w(ζ )dσ (ζ ) ωm ∂Bm

gives the Poisson representation of harmonic functions.

Dirichlet Problem for Poisson and Bi-Poisson Equations in Clifford Analysis

29

The domain integral appearing in the above representation will be denoted as  Gf (z) := G(z, ζ )f (ζ )dv(ζ ) (3.6) Bm

where f ∈ Lp (Bm ; Cm ), p ≥ 1. Since the kernel G(z, ζ ) has weak singularity in Bm , Fubini theorem implies that Gf ∈ L1 (Bm ; Cm ) for f ∈ L1 (Bm ; Cm ). Moreover, Gf = f holds in distributional sense with Gf (z) = 0 for |z| = 1. Hence, Gf provides a particular weak solution to the Poisson equation with homogeneous Dirichlet boundary condition. Rm -analogue of this operator is defined in [27] and its various norms are estimated in Bm . The integral with Poisson kernel is defined as the operator P given by  1 Pf (z) = P (z, ζ )f (ζ )dσ (ζ ) (3.7) ωm ∂Bm

for f ∈ L1 (∂Bm ; Cm ). Thus the integral representation (3.1) can be rewritten as w(z) = Pw(z) + Gw(z).

(3.8)

The representation (3.8) serves to solve the following Dirichlet boundary value problem for the Poisson equation. This problem is considered for universal Clifford algebra valued functions in [37]. Theorem 3.2 Dirichlet problem for the Poisson equation w = f in Bm w = g on ∂Bm for f ∈ L1 (Bm ; Cm ) and g ∈ C(∂Bm ; Cm ) is uniquely solvable. The solution is given by w(z) = Pg(z) + Gf (z).

(3.9)

Proof The representation (3.1) gives (3.9), if the solution exists. Harmonicity of the Poisson kernel P and the fact that the Green function G is a fundamental solution of the Laplace operator imply that, w = f holds in the distributional sense. Moreover, on ∂Bm , Gf = 0 implies w = g. Thus the result follows.   The operators T 1 and T 2 are defined as ¯ (z) T 1 f (z) := ∂Gf = −

1 ωm

 Bm

⎛

⎞

⎜ ζ −z + ⎝  |ζ − z|m z|ζ | −

ζ¯ ⎟ m ⎠ f (ζ )dv(ζ ) ζ  |ζ | 

|ζ |2 z¯ −

(3.10)

30

Ü. Aksoy

and T 2 f (z) := ∂Gf (z) = −

1 ωm

 Bm

⎛

⎞ |ζ |2 z −

⎜ ζ −z + ⎝  |ζ − z|m z|ζ | −

ζ ⎟ m ⎠ f (ζ )dv(ζ ) ζ  |ζ | 

(3.11)

1 is a modified form of for f ∈ Lp (Bm ; Cm ), p ≥ 1. Note that, the operator T m the Teodorescu operator T given in (2.4). In R , an analogue of this operator is studied in [28]. It can be observed that T 1 and T 2 operators have weak singularities. Moreover, the operators with strong singularities 1 f (z) := ∂¯ T˜1 f (z) = ∂¯ 2 Gf (z) 

(3.12)

2 f (z) = ∂ 2 Gf (z). 2 f (z) := ∂ T 

(3.13)

and

¯ , where T is the Teodorescu may also be defined. The operator defined by f = ∂Tf operator given in (2.4) is studied previously in [23], for a generalization see [18]. In the following, boundedness and continuity properties of the weakly singular operators G, T 1 and T 2 are given. The notation C(., .) is used for generic nonnegative constant depending on the variables inside the parentheses. Lemma 3.3 Let f ∈ Lp (Bm , Cm ) with m < p < ∞, then 1. For z ∈ Bm , |Gf (z)| ≤ C(p, m)f Lp (Bm ) |T 1 f (z)| ≤ C(p, m)f Lp (Bm ) |T 2 f (z)| ≤ C(p, m)f Lp (Bm ) 2. For z1 , z2 ∈ Bm , |Gf (z1 ) − Gf (z2 )| ≤ C(p, m)f Lp (Bm ) |z1 − z2 | |T 1 f (z1 ) − T 1 f (z2 )| ≤ C(p, m)f Lp (Bm ) |z1 − z2 |(p−m)/p |T 2 f (z1 ) − T 2 f (z2 )| ≤ C(p, m)f Lp (Bm ) |z1 − z2 |(p−m)/p

Dirichlet Problem for Poisson and Bi-Poisson Equations in Clifford Analysis

Proof Let

1 p

+

1 q

31

= 1 for m < p < ∞. Then q(m − 1) < m holds.

1. Let f ∈ Lp (Bm , Cm ) , Hölder inequality implies ⎛ ⎜ |Gf (z)| ≤ 2m ⎝



⎞1/q ⎛ ⎜ ⎝

⎟ |G(z, ζ )|q dv(ζ )⎠

Bm



⎞1/p ⎟ |f (ζ )|p dv(ζ )⎠

Bm

 q ⎞1/q     m 2 1 1 ⎜  ⎟  = f Lp (Bm ) ⎝  − m−2  dv(ζ )⎠  |ζ − z|m−2 (2 − m)ωm   ζ  z|ζ | − |ζ |  Bm  ⎛

⎛ ≤

2m (2 − m)ωm

⎜ f Lp (Bm ) ⎝

+

(2 − m)ωm

1 ⎟ dv(ζ )⎠ |ζ − z|q(m−2)

Bm

⎛ 2m

⎞1/q



⎜ f Lp (Bm ) ⎝



Bm

⎞1/q 1 ⎟ dv(ζ )⎠    ζ q(m−2) z|ζ | − |ζ | 

≤ C(p, m)f Lp (Bm )

since the integrals both converge for q(m − 2) < m when the condition p > m holds. Similarly, Hölder inequality gives  ⎞1/q  q   ζ¯    |ζ | z ¯ |ζ | − m |ζ |  2 ⎜  ζ −z ⎟ m  dv(ζ )⎠ |T 1 f (z)| ≤ f Lp (Bm ) ⎝  +   ζ   |ζ − z|m  ωm z|ζ | − |ζ |   Bm  ⎛

⎛ ≤

2m ωm

⎜ f Lp (Bm ) ⎝

⎞1/q



Bm

⎛ +

2m ωm

⎜ f Lp (Bm ) ⎝

1 ⎟ dv(ζ )⎠ |ζ − z|q(m−1)

Bm

≤ C(p, m)f Lp (Bm )

⎞1/q

   z|ζ | −

⎟ q(m−1) dv(ζ )⎠ ζ  |ζ |  1

32

Ü. Aksoy

and  q ⎞1/q     2 |ζ | z − ζ  ⎜  ζ −z ⎟ m  dv(ζ )⎠ f Lp (Bm ) ⎝  + |T 2 f (z)| ≤   |ζ − z|m ζ   ωm z|ζ | − |ζ |   Bm  ⎛

2m

≤ C(p, m)f Lp (Bm ) since the integrals both converge for q(m − 1) < m. 2. For z1 , z2 ∈ Bm , applying Hölder inequality we obtain |T 1 f (z1 ) − T 1 f (z2 )| ≤

2m f Lp (Bm ) ωm q ⎛  ⎞1/q     2 2 ¯ ¯ |ζ | z¯1 − ζ |ζ | z¯2 − ζ  ζ − z2 ⎜  ζ − z1 ⎟ m − m  dv(ζ )⎠ ×⎝  + − m    |ζ − z1 |m ζ  ζ   |ζ − z | 2 z1 |ζ | − |ζ |  z2 |ζ | − |ζ |   Bm 

2m f Lp (Bm ) ωm  q ⎛ ⎞1/q  q     |ζ |2 z¯ − ζ¯  2 z¯ − ζ¯  |ζ | ζ − z ζ − z ⎜   ⎟ 1 2  1 2   − m  dv(ζ )⎠ ×⎝  −  +     |ζ | − ζ m ζ    |ζ − z1 |m |ζ − z2 |m  |ζ | − z  z   1  2 |ζ | |ζ | Bm ⎛ ⎛  m−1 2m 1 ⎜ ⎜ ≤ f Lp (Bm ) |z1 − z2 | ⎝ ⎝ q(m−1−j ) |ζ − z |q(j +1) ωm |ζ − z1 | 2 ≤

j =1

Bm

⎞⎞1/q +  z1 |ζ | −

1

q(m−1−j ) ζ  |ζ | 

  z2 |ζ | −

q(j +1) ζ  |ζ | 

⎟⎟ dv(ζ )⎠⎠

.

Using Hadamard estimate [22] we get 1 f (z2 )| ≤ C(p, m)f Lp (Bm ) |z1 − z2 ||z1 − z2 |(m−qm)/q) |T 1 f (z1 ) − T ≤ C(p, m)f Lp (Bm ) |z1 − z2 |(p−m)/p .

Dirichlet Problem for Poisson and Bi-Poisson Equations in Clifford Analysis

33

In the same way, we obtain 2 f (z1 ) − T 2 f (z2 )| |T ≤

2m f Lp (Bm ) ωm q ⎞1/q ⎛      2 2 |ζ | z1 − ζ ζ − z2 |ζ | z2 − ζ  ⎟ ⎜  ζ − z1 m − m  dv(ζ )⎠ + − ⎝  m    |ζ − z1 |m ζ  ζ   |ζ − z | 2 z1 |ζ | − |ζ |  z2 |ζ | − |ζ |   Bm 

≤ C(p, m)f Lp (Bm ) |z1 − z2 |(p−m)/p .

2 imply that Boundedness of T 1 and T |Gf (z1 ) − Gf (z2 )| ≤ C(p, m)f Lp (Bm ) |z1 − z2 |.   1 related Remark 3.4 In the case of C, the counterparts of the operators G, T 1 ,  to Poisson equation are the operators 0 , 1 , 2 which are used by Vekua [33] to solve a second order complex partial differential equation whose main part is the Laplace operator under Dirichlet condition. The explicit solution of Dirichlet problem for Poisson equation is given in the complex plane by Begehr [9–11].

4 Dirichlet Problem for a Second Order Linear Equation In this section, we consider a second order linear equation in Cm whose main part is the Laplace equation. Such an equation can be considered as a generalized Poisson equation in Clifford analysis. An integral representation for a solution of that equation is discussed under Dirichlet boundary conditions. We introduce the following Dirichlet problem for a generalized Poisson equation. We denote W 2,p (Bm , Cm ) as the Sobolev space of functions in Bm . Dirichlet Problem Find w ∈ W 2,p (Bm , Cm ) as a solution of the equation ¯ + b∂w + cw = f in Bm w + a ∂w

(4.1)

satisfying Dirichlet boundary condition w = g on ∂Bm

(4.2)

where Bm is the unit ball in Rm and a, b, c, f ∈ Lp (Bm , Cm ) and g ∈ C(∂Bm ; Cm ). We first transform the above problem into a singular integral equation via the operators discussed in the previous section.

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Ü. Aksoy

Lemma 4.1 The Dirichlet problem (4.1) and (4.2) is equivalent to the singular integral equation ˆ = fˆ , (I + K)ρ

(4.3)

ˆ = aT 2 ρ + cGρ, 1 ρ + b T Kρ

(4.4)

where w = Pg + Gρ ,

fˆ = f − a∂Pg − b∂Pg − cPg , in which G, P, T 1 and T 2 operators are given by (3.6), (3.7), (3.10) and (3.11), respectively. Proof If w = Pg + Gρ, it satisfies (4.2) and we have ∂w(z) = ∂Pg(z) + T 2 ρ(z) ∂w(z) = ∂Pg(z) + T 1 ρ(z) w(z) = ρ(z). If we put these expressions into Eq. (4.1), we get the integral equation (4.3).

 

The solvability of the Dirichlet problem is given in the following theorem. Theorem 4.2 The Dirichlet problem (4.1) and (4.2) has a solution of the form w = Pg + Gρ(z), where ρ ∈ Lp (Bm , Cm ), p > m, is a solution of the singular integral equation (4.3). Proof Lemma 3.3 implies that, the operator Kˆ is a compact operator. Thus, the operator I + Kˆ is a Fredholm operator with index zero, and Fredholm property applies for the singular integral equation (4.3). If ρ(z) is a solution of the equation (4.3), then by Lemma 4.1, w = Pg + Gρ(z) is a solution of the Dirichlet problem (4.1) and (4.2).   Remark 4.3 If the boundedness properties of the operators (3.12) and (3.13) are obtained, then the following Dirichlet problem for a more general linear equation can be handled: Find w ∈ W 2,p (Bm , Cm ) as a solution of the equation ¯ + b∂w + cw = f in Bm w + q1 ∂¯ 2 w + q2 ∂ 2 w + a ∂w

(4.5)

satisfying Dirichlet condition w = g on ∂Bm

(4.6)

Dirichlet Problem for Poisson and Bi-Poisson Equations in Clifford Analysis

35

with |q1 (z)| + |q2 (z)| ≤ q0 < 1, a, b, f ∈ Lp (Bm , Cm ) and g ∈ C(∂Bm , Cm ).

5 Dirichlet Problem for Bi-Poisson Equation In the same way in iterating the integral formula (2.5) for the Laplace operator to get a representation related to the kth power of the Laplace operator k , [5], in which the kernel functions which are the fundamental solutions to the respective powers of the Laplacian, a modified representation formula for bi-Laplacian 2 can be obtained by iterating (3.8). As it is treated in C, [10, 11], we first define a second order Green function as the convolution of Green functions as:  G2 (z, ζ ) =

G(z, ζ˜ )G(ζ˜ , ζ )dv(ζ˜ )

∂Bm

where G is defined by (3.5). The function G2 (z, ζ ) satisfies the following properties: • 2 G2 (z, ζ ) = 0 in Bm \{ζ } for any ζ ∈ Bm , • G2 (z, ζ ) = G2 (ζ, z), • G(z, ζ ) = 0 and G(z, ζ ) = 0 for ζ ∈ ∂Bm , z ∈ Bm . Now, using this Green-type function as kernel of the corresponding domain integral, we can obtain a modified form of the integral representation formula in terms of bi-Laplacian 2 by iteration procedure: If w ∈ C 4 (Bm ; Cm ) ∩ C 2 (Bm ; Cm ), z ∈ Bm , then w(z) = P2 w(z) + G2 2 w(z)

(5.1)

holds, where P2 is biharmonic, that is 2 P2 = 0 and G2 is the integral operator  G2 f (z) =

G2 (z, ζ )f (ζ )dv(ζ ).

(5.2)

Bm

The representation formula (5.1) is used to find the solution of the bi-Poisson equation under homogeneous Dirichlet conditions in the following.

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Theorem 5.1 Dirichlet problem for the bi-Poisson equation 2 w = f in Bm w = 0, w = 0 on ∂Bm for f ∈ L1 (Bm ; Cm ) is uniquely solvable. The solution is given by w(z) = G2 f (z) . Remark 5.2 Dirichlet problem with nonhomogeneous boundary conditions may also be solved using the integral representation formulas. Additionally, a fourth order linear equation with bi-Laplacian main part under Dirichlet boundary conditions in the unit ball by considering the respective integral operators with the corresponding properties can be investigated.

References 1. Ü. Aksoy, A.O. Çelebi, Dirichlet problems for generalized n-Poisson equation. Oper. Theory Adv. Appl. 205, 129–142 (2010) 2. Ü. Aksoy, A.O. Çelebi, Dirichlet problem for a generalized inhomogeneous polyharmonic equation in an annular domain. Complex Variables Elliptic Equ. 57, 229–241 (2012) 3. H. Begehr, R.P. Gilbert, Transformations, Transmutations and Kernel Functions, vol. II (Longman, Harlow, 1993) 4. H. Begehr, Complex Analytic Methods for Partial Differential Equations. An Introductory Text (World Scientific, Singapore, 1994) 5. H. Begehr, Iterations of Pompeiu operators. Mem. Differ. Equ. Math. Phys. 12, 3–21 (1997) 6. H. Begehr, Iterated integral operators in Clifford analysis. J. Anal. Appl. 18, 361–377 (1999) 7. H. Begehr, Representation formulas in Clifford analysis, in Acoustics, Mechanics, and the Related Topics of Mathematical Analysis, ed. by A. Wirgin (World Scientific, Singapore, 2002), pp. 8–13 8. H. Begehr, Integral representation in complex, hypercomplex and Clifford analysis. Integral Transf. Spec. Funct. 13, 223–241 (2002) 9. H. Begehr, Boundary value problems in complex Analysis; I. II. Bol. Asoc. Mat. Venezolana XII 65–85, 217–250 (2005) 10. H. Begehr, Biharmonic Green functions. Le Matematiche LXI, 395–405 (2006) 11. H. Begehr, Six biharmonic Dirichlet problems in complex analysis, in Function Spaces in Complex and Clifford Analysis. Proceedings of 14th International Conference Finite Infinite Dimensional Complex Analysis and Applications, ed. by H.S. Le et al. (Hue University, National University Publishers, Hanoi, 2008), pp. 243–252 12. H. Begehr, J. Dubinskii, Orthogonal decompositions of Sobolev spaces in Clifford analysis. Ann. Mat. Pura Appl. 181, 55–71 (2002) 13. H. Begehr, T. Vaitekhovich, Iterated Dirichlet Problem for the higher order Poisson equations. Le Matematiche LXIII, 139–154 (2008) 14. H. Begehr, D.Q. Dai, X. Li, Integral representation formulas in polydomains. Complex Variables Theory Appl. 47, 463–484 (2002) 15. H. Begehr, Z.X. Zhang, J. Du, On Cauchy-Pompeiu formula for functions with values in a universal Clifford algebra. Acta Math. Sci. 23, 95–103 (2003) 16. H. Begehr, J. Du, S.X. Zhang, On higher order Cauchy-Pompeiu formula in Clifford analysis and its applications. Gen. Math. 11, 5–26 (2003)

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17. H. Begehr, H. Otto, Z.X. Zhang, Differential operators, their fundamental solutions and related integral representations in Clifford analysis. Complex Variables Elliptic Equ. 51, 407–427 (2006) 18. R.A. Blaya, J.B. Reyes, A.G. Adán, U. Kähler, On the -operator in Clifford analysis. J. Math. Anal. Appl. 434, 1138–1159 (2016) 19. F. Bracks, R. Delanghe, F. Sommen, Clifford Analysis (Pitman, London, 1982) 20. J. Du, Z.X. Zhang, A Cauchy’s integral formula for functions with values in a universal Clifford algebra and its applications. Complex Variables Theory Appl. 47, 915–928 (2002) 21. J.E. Gilbert, M.A.M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis. (Cambridge University Press, Cambridge, 1991) 22. S. Huang, Y.Y. Qiao, G. Wen, Real and Complex Clifford Analysis. Series: Advances in Complex Analysis and Its Applications, vol. 5 (Springer, Berlin, 2006) 23. K. Gürlebeck, U. Kähler, On a spatial generalization of the complex P-operator. ZAA 15, 283– 297 (1996) 24. K. Gürlebeck, U. Kähler, On a boundary value problem of the biharmonic equation. Math. Meth. Appl. Sci. 20, 867–883 (1997) 25. K. Gürlebeck, W. Sprößig, Quaternionic Analysis and Elliptic Boundary Value Problems (Birkhäuser Verlag, Basel 1990) 26. K. Gürlebeck, W. Sprößig, Quaternionic and Clifford Calculus for Engineers and Physicists (Wiley, Chichester, 1997) 27. D. Kalaj, D. Vujadinovi´c, The solution operator of the inhomogeneous Dirichlet problem in the unit ball. Proc. Am. Math. Soc. 144, 623–635 (2016) 28. D. Kalaj, D. Vujadinovi´c, Gradient of solution of the Poisson equation in the unit ball and related operators (2017). arXiv:1702.00929 [math.CV] 29. E. Obolashvili, Partial Differential Equations in Clifford Analysis (Addison Wesley Longman, Harlow, 1998) 30. H. Otto, Cauchy-Pompeiusche Integraldarstellungen in der Clifford Analysis. Ph.D. thesis, FU Berlin, 2006 31. J. Ryan, Cauchy-Green type formulae in Clifford analysis. Tran. Am. Math. Soc. 347, 1331– 1341 (1995) 32. J. Ryan, Basic Clifford analysis. Cubo Math. Educ. 2, 226–256 (2000) 33. I.N. Vekua, Generalized Analytic Functions (Pergamon Press, Oxford, 1962) 34. T.N.H. Vu, Integral representations in quaternionic analysis related to Helmholtz operator. Complex Variables Theory Appl. 12, 1005–1021 (2003) 35. T.N.H. Vu, Helmholtz operator in quaternionic analysis. PhD thesis, FU Berlin, 2005 36. Z. Xu, Boundary value problems and function theory for spin-invariant differential operators. PhD thesis, Gent State University, Gent, Belgium, 1989 37. Z.X. Zhang, Integral representations and its applications in Clifford analysis. Gen. Math. 13, 81–98 (2005)

Boundary Eigenvalues of Pluriharmonic Functions for the Third Boundary Condition on the Unit Polydiscs Alip Mohammed

To the 80th birthday of Heinrich Begehr

Abstract The paper provides explicit eigenvalues and eigenfunctions of pluriharmonic functions for the third boundary condition on the unit polydiscs. It is shown that in the case of eigenvalue, for each eigenvalue, there are multiple eigenfunctions. Compatibility and solvability conditions are also studied for the case of inhomogeneous third boundary condition. Keywords The third boundary condition · Eigenvalues with multiple eigenfunctions · Pluriharmonic functions · Solvability conditions · The unit polydiscs Mathematics Subject Classification (2010) Primary 32A50, 31C10; Secondary 35J57, 65N25, 34B09

1 Introduction The Dirichlet and the Neumann problems for the inhomogeneous pluriharmonic system and pluriholomorphic system and the third boundary value problem for Cauchy–Riemann systems on the unit polydiscs were studied [4, 9–12, 18] for compatibility, solvability and explicit solutions. On the other hand, the inhomogeneous Cauchy–Riemann equations and harmonic functions (the Poisson equation) for the third boundary condition are studied [3, 6, 7, 16, 17, 19, 20] for solvability, explicit

This work was completed in spring 2018 at Khalifa University. A. Mohammed () Department of Mathematics, Khalifa University, Abu Dhabi, United Arab Emirates e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_3

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A. Mohammed

solutions and eigenvalues by means of various methods such as Fourier analysis, Green function method and variational method for the unit disc and some general domains. Moreover studies on harmonic functions with Dirichlet and Neumann boundary conditions for certain special domains are also gaining momentum[1, 2]. In this paper we study boundary eigenvalue problem of the pluriharmonic functions for the third boundary condition with constant coefficient on the unit polydiscs.

1.1 The Problem In one dimensional complex plane C, thanks to the Riemann mapping theorem, any domain with smooth boundary is biholomorphically equivalent to the unit disc. Thus solving a problem on the unit disc, at least in theory, is same as solving the underlying problem on any domain with smooth boundary in C. In higher dimensional complex space Cn (n > 1) however, the Riemann mapping theorem fails to exist [4]. Thus, for the most typical domain in C such as the unit disc, there are two essentially different domains as analogues in Cn : the unit polydiscs and the unit sphere . Their boundary functions show significantly different properties [4, 8] while the sphere divides the the entire space into two parts, inside and outside of the sphere, the distinguished (also known as characteristic or Shilov) boundary of the unit polydiscs, divides the entire space into 2n tuples [5, 9, 12]. This distinction contributes to significant challenges for study of boundary value problems of several complex variables and thus studies in Cn are conducted mainly either in polydiscs or in balls [4]. The domain we conduct our study is the unit polydiscs which is more challenging and interesting than the unit ball [5, 12–14]. Let Dn be the unit polydiscs {z : z = (z1 , · · · , zn ) ∈ Cn , |zk | < 1 , 1 ≤ k ≤ n} and ∂0 Dn be its distinguished boundary {z : z = (z1 , · · · , zn ) ∈ Cn , |z1 | = |z2 | = · · · = |zn | = 1}. Let γ0 be given function with γ0 ∈ C 1 (∂0 Dn ). Consider the following inhomogeneous system of independent equations ∂ 2u = 0, ∂zk ∂z

1 ≤ k,  ≤ n, z ∈ Dn .

(1.1)

Definition 1.1 (The Problem) For γ0 ∈ C 1 (∂0 Dn ), find a u ∈ C 2 (Dn ) ∩ C 1 (∂0 Dn ) solution u(z) of system (1.1), satisfying the third boundary condition ∂u + λu = γ0 (ζ ), ∂νζ

ζ ∈ ∂0 Dn

(1.2)

with λ being arbitrary complex parameter, i.e., λ ∈ C and ∂u /∂νζ denotes the outward normal derivative of u(ζ ) at the point ζ ∈ ∂0 Dn . Since the free parameter λ is in the boundary condition, but not in the equation, the problem is boundary eigenvalue problem for pluriharmonic functions on the unit polydiscs for the third boundary condition.

Boundary Eigenvalues of Pluriharmonic Functions for the Third BC

41

The third boundary condition (1.2) for the unit polydiscs turns out to be n   j =1

 ∂u ∂u  + zj +λu = γ (ζ ), zj ∂zj ∂zj ζ

ζ ∈ ∂0 Dn

(1.3)

√ with γ (ζ ) = γ0 (ζ ) n, see [9, 10, 12].

1.2 Pluriharmonic Functions on the Unit Polydisc It is known that any solution to (1.1) can be represented as, see page 279 [4], u(z) = ϕ(z) + φ(z)

(1.4)

where ϕ(z) and φ(z) are arbitrary holomorphic functions in Dn . Substituting (1.4) into (1.3) leads to  n   ∂ϕ ∂φ  + zj +λ(ϕ + φ) = γ (ζ ), zj ∂zj ∂zj ζ

ζ ∈ ∂0 Dn .

(1.5)

j =1

Equation (1.5) can be rewritten as n " j =1

# "  ∂φ # ∂ϕ + λϕ + ζj + λφ = γ (ζ ), ∂ζj ∂ζj n

ζj

ζ ∈ ∂0 Dn .

(1.6)

j =1

Since the left hand side of (1.6) is sum of boundary values of a holomorphic function and an anti-holomorphic function in Dn , the right hand side must be also the sum of boundary values of a holomorphic function ∂Hn and an anti-holomorphic function ∂H−n in Dn [5, 12, 15]. By the way, anti-holomorphic function in Dn can be seen as holomorphic function in D−n [11, 13, 14]. This means (1.6) to be solvable, it is necessary that   γ (ζ ) = γ (ζ )

∂ Hn

$

∂ H−n

= γ + (ζ ) + γ − (ζ ),

ζ ∈ ∂0 Dn

(1.7)

$ where the right hand side is the projection of γ (ζ ) on ∂Hn ∂H−n , γ + (ζ ) =   γ (ζ ) n and γ − (ζ ) = γ (ζ ) −n . ∂H ∂H If solvability condition (1.7) is satisfied, then Eq. (1.6) can be expressed as n  j =1

ζj

∂ϕ + λϕ = γ + (ζ ), ∂ζj

ζ ∈ ∂0 Dn

(1.8)

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and n  j =1

ζj

∂φ + λφ = γ − (ζ ), ∂ζj

ζ ∈ ∂0 Dn .

(1.9)

Notice that γ + (ζ ), γ − (ζ ) ∈ ∂Hn .

2 Fourier Series Method Theorem 2.1 Assuming the solvability condition (1.7) is satisfied for the system (1.1) with the third boundary condition (1.2). If |κ| + λ = 0, ∀κ ∈ Z+ × · · · × Z+ , then there exists a unique solution u(z) = ϕ(z) + φ(z) =

∞ "  γκ− κ # γκ+ κ z + z , |κ| + λ |κ| + λ κ=0

z ∈ Dn .

(2.1)

If there exists κ0 ∈ Z+ × · · · × Z+ such that −λ = |κ0 | = k0 > 0, k0 ∈ N, then solutions are not unique and have the following representation u(z) =

∞  κ=0 κ=κ0

γκ+ κ γκ− κ z + z + Azκ0 + Bzκ0 , |κ| + λ |κ| + λ

z ∈ Dn

(2.2)

provided additional solvability conditions γκ+0 = 0 and γκ−0 = 0 are satisfied. Arbitrary coefficients A and B remain free in the solution. Notice that the coefficients A and B can be fixed by imposing additional boundary conditions[16]. Proof We use Fourier series to solve Eqs. (1.8) and (1.9). Let ϕ(ζ ) =

∞ 

ϕκ ζ κ , γ + (ζ ) =

κ=0

φ(ζ ) =

∞  κ=0

∞ 

γκ+ ζ κ , κ = (k1 , k2 , · · · , kn ) ∈ Z+ × · · · × Z+

κ=0

φκ ζ κ , γ − (ζ ) =

∞ 

γκ∗ ζ κ , ζ = (ζ1 , ζ2 , · · · , ζn ) ∈ ∂0 Dn .

κ=0

Further denote |κ| = k1 + k2 + · · · + kn .

(2.3)

Boundary Eigenvalues of Pluriharmonic Functions for the Third BC

43

Then Eqs. (1.8) and (1.9) can be reformulated as ∞ "  

#  |κ| + λ ϕκ − γκ+ ζ κ = 0,

ζ ∈ ∂0 Dn

(2.4)

∞ " #    |κ| + λ φκ − γκ∗ ζ κ = 0,

ζ ∈ ∂0 Dn

(2.5)

  |κ| + λ ϕκ − γκ+ = 0,

κ ∈ Z+ × · · · × Z+

(2.6)

 |κ| + λ φκ − γκ∗ = 0,

κ ∈ Z+ × · · · × Z+ .

(2.7)

κ=0

and

κ=0

Equations (2.4) and (2.5) lead to

and 

Notice that throughout the paper, it is assumed that λ = 0. Otherwise the third boundary condition (1.2) degenerates to the Neumann boundary condition and it is studied by[10]. Relations (2.6) and (2.7) can be solved for the terms ϕκ and φκ as ⎧ + ⎪ if |κ| = 0 ⎪ ⎨ϕ0 = γ0 / λ, + (2.8) ϕκ = γκ / (|κ| + λ), if |κ| + λ = 0, ⎪ ⎪ ⎩ϕ = A, ∀A ∈ C & γ + = 0, if − λ = |κ | > 0. κ0

0

κ0

⎧ ∗ ⎪ ⎪ ⎨φ0 = γ0 / λ,

if |κ| = 0

γκ∗ /

(|κ| + λ), if |κ| + λ = 0, φκ = ⎪ ⎪ ⎩φ = B, ∀B ∈ C & γ ∗ = 0, if |κ | = −λ > 0. κ0 0 κ0

(2.9)

Expressions (2.8) and (2.9) completes proof of the theorem.

3 Eigenvalue with Multiple Eigenfunctions Let γ (ζ ) ≡ 0, ζ ∈ ∂0 Dn . When n = 1, for each eigenvalue −λ = |κ| = κ = k0 ∈ N, there is only one eigenfunction φ(z) = zk0 and the harmonic function is u(z) = Azk0 + Bzk0 . When n ≥ 2, for any eigenvalue −λ = |κ| = k0 ∈ N, each term z1k1 · · · znkn = zκ with |κ| = k1 + · · · + kn = k0 from the expression (z1 + z2 + · · · + zn )k0 is a linearly independent eigenfunction. When n = 2 and

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k0 = 1, there are exactly two linearly independent eigenfunctions z1 and z2 for φ(z). When n = 2 and k0 = 2, then there are exactly three linearly independent eigenfunctions z12 , z1 z2 , z22 , the same as each of the terms of (z1 + z2 )2 . If k0 = 3, there are exactly 4 eigenfunctions z13 , z12 z2 , z1 z22 , z23 . When γ (ζ ) ≡ 0, ζ ∈ ∂0 Dn , Theorem 2.1 can be reduced to the following. Lemma 3.1 Let γ (ζ ) ≡ 0, ζ ∈ ∂0 Dn . Then solvability condition (1.7) is satisfied automatically and the system (1.1) with the third boundary condition (1.2) has only trivial solution u(z) ≡ 0 for −λ = |κ|, ∀κ ∈ Z+ × · · · × Z+ . For each eigenvalue λ = −|κ0 | = k0 > 0, k0 ∈ N with κ0 ∈ Z+ × · · · × Z+ , the solutions are given by u(z) = Azκ0 + Bzκ0 ,

z ∈ Dn .

(3.1)

All the negative integers are eigenvalues and for each eigenvalue, there are multiple eigenfunction solutions, i.e., all the k0 homogeneous terms of the expression (z1 + z2 + · · · + zn )k0 are linearly independent eigenfunctions associated with the eigenvalue −λ = |κ0 | = k0 ∈ N. Each of the terms comes with the arbitrary coefficient A and its conjugate term comes with the arbitrary coefficient B in (3.1) . The coefficients A and B are finitely many different arbitrary complex numbers for each eigenvalue −λ = k0 due to κ0 being multi-index.

References 1. M. Akel, H. Begehr, Neumann function for a hyperbolic strip and a class of related plane domains. Math. Nachr. 290(4), 490–506 (2017) 2. M. Akel, S.R. Mondal, Dirichlet problems in lens and lune. Bull. Malays. Math. Sci. Soc. 41(2), 1029–1043 (2017) 3. G. Auchmuty, Steklov representations of Green’s functions for Laplacian boundary value problems. Appl. Math. Optim. https://doi.org/10.1007/s00245-016-9370-4. Article electronically published on July 25, 2016 4. H. Begehr, A. Dzhuraev, An Introduction to Several Complex Variables and Partial Differential Equations. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 88 (Addison Wesley Longman, Harlow, 1997) 5. H. Begehr, A. Mohammed, Schwarz problem for the Torus related domains. Appl. Anal. 85(9), 1079–1101 (2006) 6. H. Begehr, S. Burgumbayeva, B. Shupeyeva, Remark on Robin problem for Poisson equation. Complex Variables Elliptic Equ. https://doi.org/10.1080/17476933.2017.1303052. Published online: 19 Mar 2017 7. C. Bandle, A. Wagner, Domain perturbations for elliptic problems with Robin boundary conditions of opposite sign. St. Petersburg Math. J. 28(2), 153–170 (2017). http://dx.doi.org/ 10.1090/spmj/1443. Article electronically published on February 15, 2017. 8. S.G. Krantz, Complex Analysis: The Geometric Viewpoint (Carus Mathematical Monographs) (2004). ISBN-13:978-0883850268 9. A. Mohammed, The Neumann problem for the inhomogeneous pluriharmonic system in polydiscs, in Complex Methods for Partial Differential and Integral Equations, ed. by H. Begehr, A. Celebi, W. Tutschke (Kluwer Academic Publishers, Dordrecht, 1999), pp. 155–164

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10. A. Mohammed, The classical and the modified Neumann problem for the inhomogeneous pluriholomorphic system in polydiscs. J. Anal. Appl. 19, 539–552 (2000) 11. A. Mohammed, The classical and the modified Dirichlet problem for the inhomogeneous pluriholomorphic system in polydiscs . Complex Variables Theory Appl. 45, 213–246 (2001) 12. A. Mohammed, Boundary value problems of complex variables. PhD Thesis, Freie Universität Berlin, 2003. http://www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000000885 13. A. Mohammed, The torus related Riemann problem. J. Math. Anal. Appl. 326(1), 533–555 (2007) 14. A. Mohammed, The Riemann–Hilbert problem for certain poly domains and its connection to the Riemann problem. J. Math. Anal. Appl. 343(2), 706–723 (2008) 15. A. Mohammed, Schwarz, Riemann, Riemann-Hilbert problems and their connections in polydomains, in Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations (Birkhäuser, Basel, 2009), pp.143–166 16. A. Mohammed, A.M. Tuffaha, On boundary control of the Poisson equation with the third boundary condition. J. Math. Anal. Appl. 459(1), https://doi.org/10.1016/j.jmaa.2017.10.059. Published online October 2017 17. A. Mohammed, M.W. Wong, Solutions of the Riemann–Hilbert–Poincaré problem and the Robin problem for the inhomogeneous Cauch - Riemann equation. Proc. R. Soc. Edinb. Sect. A Math. 139(1), 157–181 (2009) 18. A. Mohammed, M.W. Wong, Solutions of Robin problems for overdetermined inhomogeneous Cauchy–Riemann systems on the unit polydisc. Complex Anal. Oper. Theory 4(1), 39–53 (2010) 19. A. Mohammed, M.W. Wong, Eigenvalues of the third boundary condition with variable coefficient for Poisson equation (to be published) 20. A. Mohammed, D. Siginer, F. Akyildiz, Eigenvalues of holomorphic functions for the third boundary condition. J. Quart. Appl. Math. 73(3), 553–574 (2015)

Survey of Some General Properties of Meromorphic Functions in a Given Domain G. Barsegian

Dedicated to Professor Heinrich G.W. Begher on the occasion of his 80th birthday

Abstract The first classical results (principles) related to arbitrary analytic (meromorphic) functions w in a given domain were obtained by Cauchy in 1814– 1831, while the next principles had arisen much later, in Ahlfors theory of covering surfaces created in 1935. In this survey we present some other (diverse type) results of the same generality which were obtained since 1970s. Previously the most attention was paid to meromorphic functions in the complex plane or in the disks, which were studied in details in the classical Nevanlinna value distribution theory. The results of this survey complement this theory by discovering some new type of phenomena or regularities and, unlike Nevanlinna theory, the result cover all meromorphic functions, including, the most important in application, functions in a given domain. Some of these results found already applications in other topics, for instance in geometry and complex equations. Meantime the results were presented earlier in some papers devoted to value distribution. Respectively many experts working in other fields are not familiar with them. The aim of this survey is to facilitate further applications of these results, by presenting them as a collection of some separate formulas or some “ready to use” tools. This will enable to apply these formulas without entering into details and interrelations of these formulas with other topics. Keywords Meromorphic functions in a domain · Nevanlinna theory · Ahlfors theory · Gamma-lines · Level sets · Estimates of derivatives · Geometric deficiencies · Universal version of value distribution

G. Barsegian () Institute of Mathematics, National Academy of Sciences, Yerevan, Armenia e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_4

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G. Barsegian

Mathematics Subject Classification (2010) 30A10, 30A99, 30C99, 30D30, 30D35, 30D99, 30F99, 30G99

1 Introduction Preceding Theories Describing Behavior of Large Classes of Meromorphic Functions We have a number of results of general nature that relate to arbitrary meromorphic functions defined in the complex plane or in the unit disk however under the additional hypothesis that they have “sufficiently large growth”. These functions we will refer them as “basic functions”.1 For similar functions we have classical results by Picard (1888), Julia, Borel, Valiron, Milloux (since 1907, [26]), classical Nevanlinna value distribution theory (1920s, [25]) and Ahlfors theory of covering surfaces (1935, [2]), Hayman’s alternative theorem (1959, [21]). All mentioned theories and results relate to studies of the number (quantities) of apoints of basic meromorphic functions w(z), that is to solutions w(z) = a. Results for Arbitrary Meromorphic Functions in a Given Domain and in the Complex Plane In contrast to the basic functions, for functions in a given domain we had just very few general results. The first key results on arbitrary analytic (meromorphic) function in a given domain were obtained in 1814–1831. The next results of the same generality arose much later in Ahlfors theory of covering surfaces [2] (1935). Some new results of similar generality were established since late 1970s. The last results mostly complement, generalize or give some alternative versions of the mentioned preceding theories related to basic functions. An interesting circumstance is that in almost all these complements we have now corresponding versions both for basic functions and for functions in a given domain. The mentioned developments describe different analytic and geometric aspects of meromorphic functions which can be attributed to three novel directions: Gammalines, proximity property of a-points and universal version of value distribution; all they will shortly be presented below. Motivation for Composing this Survey 1. Preceding presentations and publications of these results discuss in details their interrelations and contribution to Nevanlinna-Ahlfors theories. This means that those experts who do not read papers in these theories may hardly be familiar with these results. Meantime, we expect, that the results may have applications far beyond Nevanlinna-Ahlfors theories.

1 The class of basic function includes all meromorphic function in the complex plane. Meromorphic functions w in the unit disk d(1, 0) := {z : |z| < 1} are basic functions if lim supA(r)(1−r) = ∞,

where A(r) = A(r, w) is Ahlfors-Shimizu characteristic of w.

r→1

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2. Also we mention here two aspects, theoretical and applied. The basic meromorphic functions were studied largely and one can expect that the study of meromorphic functions in a given domain can attract more attention gradually. On the other hand applied scientists are interested mostly in behavior of functions in a given concrete domain. In this survey we present the results as a collection of different formulas (without “gluing” them with the classical theories). We hope, this may facilitate further applications of these formulas. In what follows, the notation D stands for a simply connected domain with smooth boundary ∂D of finite length l(∂D) and area S(D).2

2 Two Principles Related to Derivatives 2.1 Principle of Logarithmic Derivatives The next result establishes a new type inequality involving logarithmic derivatives  (k+1)  w (z)/w(k) (z) for k = 0 and any integer k ≥ 1. Theorem 2.1 (Principle of Logarithmic Derivatives, [12]) For any meromorphic function w in D¯ := D ∪ ∂D and any integer k ≥ 1,      w (z)    dσ ≤  w(z)  D

   w(k+1) (z)  kπ   l(∂D).  dσ +  (k)  w (z)  2

(2.1)

D

Sharpness For function w(z) = exp z in the disk |z| < r we have       (k+1)   w (z)  (z)  w 2    dσ = 2πr 2 and l(∂D) = 2πr  (k)  w(z)  dσ = 2πr ,   w (z) D D so that the ratio of the left and the right sides in (2.1) tends to 1 when r → ∞. This means that (2.1) is asymptotically sharp. Also we have the following result in the spirit of the second fundamental theorem in Nevanlinna value distributions theory.

2 The restriction for D we put just for simplicity. In many cases, the results are valid for larger type of domains.

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¯ any integer k ≥ 1 Theorem 2.1 ([12]) For any meromorphic function w in D, and any collection of pairwise different points aν ∈ C, ν = 1, 2, . . . , q, q     w  w − a ν=1

D

ν

    dydx ≤ 

    w(k+1)    kπ 2 q   w  dydx + kπ l(∂D),  (k)  dydx +  ρ 2 D w D

(2.2) where ρ is the minimal distance between aν . The case k = 1 was considered in a slightly modified view in [7].

2.2 Principle of Derivatives for Simple a-Points In this subsection we present a new principle related to arbitrary meromorphic function w in a given domain D, see [14]. The main component of this principle gives for the first time lower bounds for |w | for similar general class of functions. The principle can qualitatively be stated as follows: any set of simple a-points of w, contains a “large” subset of complex values zχ∗ (a) such that we have some estimates for both for |w (zχ∗ (a)| and |w(h) (zχ∗ (a)|, h > 1. Consider a given set of simple a-points zk (a) of w for k = 1, 2, . . . n, of w, where a ∈ C. For any similar set we have obviously some non-intersecting univalent neighborhoods e(zk (a)) of points zk (a) . This means that w is univalent in each e(zk (a)) and e(zk (a)) ∩ e(zj (a)) = ∅ for k = j . Notation [x] stands for the entire part of x. Theorem 2.2 (Principle of Derivatives for Simple a-Points) Let w(z) be an arbitrary meromorphic function in D¯ and let zk (a), k = 1, 2, . . . n, be a set of simple a-points of w with non-intersecting univalent neighborhoods e(z%k (a)) of & zk (a). Then the set of these a-points contains a subset zχ∗ (a), χ = 1, 2, . . . , n2 + 1 , such that for any similar zχ∗ (a) √   n  ∗  ρχ (a) , √ w (zχ (a) ≥ √ S(D) 24π

(2.3)

and for any integer h > 1, |w(h) (zχ∗ (a)| ≤

9h+2 (h − 1)! 2h+1 ρχh−1 (a)

|w (zχ∗ (a))|h .

(2.4)

  where ρχ (a) is the distance between w zχ∗ (a) and the boundary of w(e(zχ∗ (a))). The above two inequalities explicitly reflect the principle stated above qualitatively: any set of simple a-points of w , contains a subset of complex values zχ∗ (a),

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% & such that the subset is “large” (since it consists of n2 + 1 points) and for any     zχ∗ (a) we have lower bounds for w (zχ∗ (a)) (inequality (2.3) and upper bounds for |w(h) (zχ∗ (a))|, h > 1 (inequality (2.4). The following result composed of three interrelated inequalities implies the above principle. Theorem 2.3 For arbitrary meromorphic function w(z) in D¯ = D ∪ ∂D, an arbitrary set of non-intersecting simply connected domains ek ⊂ D, k = 1, 2, . . . n, where w is one-to-one and an arbitrary zk ∈ ek we have √ ' 1  ρk ≤ 6π S(D), √

|w (zk )| n n

(2.5)

k=1

where ρk is the distance w (zk ) and the boundary of w(ek ). % between & Further for some n2 + 1 indices k, respectively points zk ∈ ek , we have √   w (zk ) ≥ √ρk √ n , 24π S(D)

(2.6)

and for any integer h > 1 |w(h) (zk )| ≤

9h+2 (h − 1)! 2h+1 ρkh−1

|w (zk )|h .

(2.7)

Notice that inequalities (2.5)–(2.7) do not restrict the numbers n, domains ek and points zk so that Theorem 2.3 reflects some rather generic properties of mappings by w. Inequality (2.5) of Theorem 2.3 is new. Inequalities of type (2.6) and (2.7) were established in [8] and [10] for functions in the complex plane. The estimates were given in this case in terms of Ahlfors-Shimizu characteristic functions for w (instead of n in this paper). Later on, similar estimates were applied in [15] and [18] to complex differential equations. In this paper we “glue” the above three inequalities and formulate the inequalities for an arbitrary meromorphic function w in D. Point of View of Good a-Points of Meromorphic Functions For a given a ∈ C and disk (ρ, a) := {w : |w −a| < ρ} (ρ > 0), consider all those points zk (ρ, a) ∈ D, k = 1, 2, . . . nρ (D, a, w), for which (a) w(zk (ρ, a)) = a and (b) each zk (ρ, a) has a neighborhood σk (ρ, a) which w maps one-to-one onto (ρ, a). This means that we consider those a-points which have “good” neighborhoods so that we will refer to similar points zk (ρ, a) as good a-points.3 When we take particularly a = 0 we deal, clearly, with good zeros.

3 Notice

that the classical notion of the number n0 (D, a, w) of simple a -points is simply the limit of nρ (D, a, w) as ρ → 0.

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In fact we considered here a particular case of Theorem 2.3 when all w-images of ek coincide with (ρ, a) so that inequality (2.5) of Theorem 2.3 can be rewritten as follows. Assume w(z) is a meromorphic function in D¯ and a ∈ C. Then for an arbitrary set of good a -points zk (ρ, a) ∈ D, k = 1, 2, . . . , nρ (D, a, w) , we have ρ ' nρ (D, a, w)

nρ (D,a,w)



1

k=1

|w (zk (ρ, a))|

≤

√ ' 6π S(D).

In turn inequalities (2.6) and (2.7) imply the following assertion of a general nature (valid for an arbitrary meromorphic function). Theorem 2.3 (Principle of Derivatives for Good a-Points) Assume w(z) is a meromorphic function in D¯ and a ∈ C . Then any set %of good a-points zk (ρ, & a) ∈ D, k = 1, 2, . . . , nρ (D, a, w), contains a subset of nρ (D, a, w)/2 + 1 points zχ∗ (ρ, a) such that '   nρ (D, a, w) ρ  ∗  √ , w (zχ (ρ, a)) ≥ √ S(D) 24π

(2.3 )

and for any integer h > 1, |w(h) (zχ∗ (ρ, a))| ≤

9h+2 (h − 1)! ∗ |w (zχ (ρ, a))|h . 2h+1 ρ h−1

(2.4 )

Comment 2.1 In my opinion the above inequalities give a “ready to use” tool for geometric study of functions in a given domain, particularly for applications in complex equations. To give a hint for applications we notice that the number   ∗  nρ (D, a, w) is connected with w (zχ (ρ, a)), which can easily be estimated for different classes of functions; for instance for solutions of first order differential equations and some Painlevé equations. Respectively we can get conclusions on nρ (D, a, w) for solutions of these equations. Corresponding studies will be published elsewhere; see particularly [17].

3 Results Related to Level Sets and Gamma-Lines Solutions of equations u(x, y) = 0 (more generally u(x, y) = const) arise very often in pure and applied mathematics. Remember that the level sets u − A admit a lot of interpretations (streaming line, potential line, isobar, isoterm) in different applied fields of engineering, physics, environmental problems etc. Notice that in “non degenerating cases” these solutions are some curves. In the case u(x, y) is a polynomial in two variables P (x, y) the solutions P (x, y) = const were studied

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very largely in the frame of Hilbert’s problem 16, part 1; namely problem was to give bounds for the number of connected components of the solutions. As to other geometric aspects of the solutions P (x, y) = const they were almost non touched. Also, till end of 1970s, we have no any study related to level sets of any large class of real functions u(x, y), even harmonic functions or more generally functions Rew or |w|. Definitely this was a big gap since solutions of Rew = A ∈ (−∞, ∞), or of |w| = R ∈ (0, ∞), have important interpretations in different applied sciences. We consider now much larger concept. Let w(z) be a meromorphic function in a domain D and  be a curve in w-plane; for simplicity we assume that  is an analytic curves. Denoting by w−1 the inverse functions we consider the set w−1 () lying in D (that is preimages of  under mapping by w). We call this set w−1 () as Gamma-lines of the curve  of the function w. Now we see that the solutions of Rew = A and of |w| = R are particular cases of Gamma-lines which we obtain by taking respectively  = γ (A) := {w : Imw = A} and  = (R) := {w : |w| = R}. It is easy to see that in this case the γ (A)-lines are curves for any A and (R)-lines are curves for any R = 0. The concept of Gamma-lines (which is the set w−1 ()) is quite similar to the concept of set a-points (which is the set w−1 (a), for a given point a ∈ C). This similarity raises a question: can theory of Gamma-lines developed in analogy with Nevanlinna-Ahlfors theories describing a-points? At the end of 1970s we initiated studies in this directions and obtained an analog of the second fundamental theorems in Nevanlinna-Ahlfors theories. The topic was started in [4]; the results, later on, were enlarged in [6, 9] and summarized in the book [11]. Notice that the Gamma-lines are the sets of curves. Respectively the key aspects in their study should be their length or cardinality (like in the mentioned above Hilbert problem). In this section we present estimates for the length L(D, ) of -lines of w lying in D and give some integrated formulas for the length. This permits to get bounds for cardinalities of -lines for w in D. Also the obtained results lead to some new type inequalities for quantities of a-points. We remind that we consider here just meromorphic functions in a given domain. (The case of functions in the complex plane utilizes different characteristics and it will not be considered in this survey.)

3.1 The Tangent Variation Principle and the Second Fundamental Theorem for Gamma-Lines Denote ν() = Varz∈ α (z), where Var means variation, α (z) is the angle between the tangent to  at z ∈  and the real axis. We denote by K different absolute constants and by h = h(φ1 , . . . , φq ) constants depending only on φ1 , . . . , φq .

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Theorem 3.1 (Tangent Variation Principle) For any meromorphic function w(z) in D¯ and any smooth Jordan curve  (bounded or unbounded) with ν() < ∞ ⎫ ⎧ ⎬ ⎨   w

(z)   dσ + l(∂D) ,  L(D, ) ≤ K()  w (z)  ⎭ ⎩

(3.1)

D

where K() = 3(ν() + 1). We presented above a bit simplified version of the principle which follows from formulas (1.2.9) and (1.2.16) in [11].  ++ 

2 −1 dσ . Notation A1 (D) will stands for D |w | 1 + |w| Theorem 3.2 (The Second Fundamental Theorem for Gamma-Lines) Let w(z) be a function meromorphic in D¯ and let ν , ν = 1, 2, . . . , q , be some disjoint bounded smooth Jordan curves with ν(ν ) < ∞. Then q  i=1

L(D, ν ) ≤ K

  

 √  w (z)     w (z)  dσ + h(1 , . . . , q )A1 (D) + 2l(∂D),

(3.2)

D

where K is an absolute constant and h(1 , . . . , q ) is a constant depending on 1 , . . . , q . Further if  is an unbounded smooth Jordan curve (bounded or unbounded) with ν() < ∞, then   

 √  w (z)    dσ + h()A1 (D) + 3 2l(∂D). L(D, ) ≤ K (3.2 )  w (z)  D

The proof of this inequality follows from inequalities (1.3.1), (1.3.1 ) and (1.2.9) in [11], chapter 1. The reader familiar with Nevanlinna-Ahlfors theories [1, 25] will observe immediately a similarity between (7.2) and the second fundamental theorems in these theories. Comment 3.1 After the initial paper [4] the length of Gamma-lines were studied in Hayman and Wu’s paper [22]. They considered the case of univalent functions in the unit disk and particular , circumference and straight line merely. This study has attracted much more attention that the general case in [4]. Later on many well known complex analysts continued this study: J. Garnett, P. Jones, C. Bishop, L. Carleson, F. Gehring, J. Fernandes, M. Heinonen, O. Martio. It is interesting that the general case in [4] remained unknown for all these authors nearly 20 years (as some of them mentioned to me at personal communications).

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3.2 A Particular Case When  is a Straight Line Theorem 3.3 For any meromorphic function w in D¯ and any straight line  we have   

  w (z)  1  dσ + 1 l(∂D).  (3.3) L(D, ) < 

2 w (z)  2 D

The inequality (3.3) improves the constants in (3.1) when  is a straight line; it was proved in [16]. Notice that in the particular case we can take  = γ (A) = {w : Imw = A}; then γ (A)-lines become the level sets of Imw(z) − A . Respectively the length L(D, Imw = A) of these level sets coincides with L(D, γ (A)) and we can rewrite (3.3) as L(D, Imw = A)
0 be a continuous function for 0 ≤ R < ∞. Then ∞

L(D, (R)) dR = (R)

0



|w (z)| dσ. (|w(z)|)

(3.5)

D

Also, denoting (β) = {w : |w| > 0, arg w = β} we give below the following modification of the above identity, see [11], formula (1.1.16). Theorem 3.5 Let w(z) be a regular function in a domain D, and let (β) > 0 be a continuous function for β ∈ [0, 2π]. Then 2π 0

L(D, (β)) dβ = (β)

    w (z)  1    w(z)  (arg w(z)) dσ D

(3.6)

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3.4 On Cardinalities of the Level Sets of Functions Imw(z) Hilbert problem 16, part 1, [23] asks about cardinalities (quantities, numbers) of maximal connected components 4 of the set of level sets in the case of polynomial functions P (x, y) of two real variables x and y. Similarly we consider an analog of this Hilbert’s problem for level sets of harmonic functions or more generally for functions Imw(z) − A. In other words we consider the cardinalities of level sets of the functions Imw(z) − A; here w(z) ¯ Denote by CD (Rew = A) the number of maximal connected is defined in D. components of the set {z : Imw(z) = A} in D, which does not imply inside critical points (where w = 0) or poles (where w = ∞). We call CD (Imw = A) full Hilbert cardinality. It is pertinent to stress that the cardinalities CD (Rew = A) occurring in domain D admit many interpretations; particularly they mean the number of streaming lines and potential lines of movement of an ideal liquid in plane domain D. Further, taking a domain D ∗ ⊂ D, we consider inner Hilbert cardinality CD ∗ ,D (Imw = A) which counts only those connected components in D which have common points with D ∗ . ¯ D ∗ is a simply connected Theorem 3.6 Let w be a meromorphic function in D, domain in D and A is a real value, −∞ < A < +∞. Then   

 ,  w (z)  1

  CD∗ ,D (Imw = A) <   dσ + l(D) + 2n(D, 0, w ) + 2n(D, ∞, w ), 4ρ D w (z) (3.7) where ρ is the distance between ∂D ∗ and ∂D and n(D ∗ , 0, w ) is the number of zeros of w in D ∗ . Proof To derive the bounds of CD ∗ ,D (Imw = A) we notice first that the number of all those components which have at least one critical endpoint is less than or equal to 2n(D, 0, w ). Also notice that the number of all those components for which at least one of two endpoints contain a pole of w is less than or equal to 2n(D, ∞, w ). Thus for the the number C1 of the mentioned components we have C1 ≤ 2n(D, 0, w ) + 2n(D, ∞, w ). It remains to estimate just the number of those components which do not imply critical points or poles at both endpoints. Observe that similar component in D cannot be closed curves. Indeed then we necessarily the closed curve would have to contain a pole what contradict the definition of similar curves. Also notice that any similar component should have at least two endpoints lying on the boundary ∂D. This implies that any similar component having common points with D ∗ should have length greater or equal to 2ρ. Denoting by C2 the number of similar components and by L their total length we get C2 ≤ (1/2ρ)L ≤ (1/2ρ)L(Imw = A). Taking into account that 4 “Maximal”

here means that a given component cannot constitute a part of a larger component.

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CD ∗ ,D (Imw = A) = C1 + C2 , summing up the above two estimates and applying (3.3 ) we obtain (3.7). Comment 3.2 For the meromorphic functions f in the complex plane the number of connected components is infinite so that (following Nevanlinna theory) we can consider this number for the disks D(r) := {z : |z| < r}; then getting estimates for CD(r) (Ref = A) we can take r → ∞ what should give some conclusions on asymptotic behavior of the cardinalities. On the other hand we have CD(r) (Ref = A) ≤ CD(r),D(2r)(Ref = A) so that making use (3.7) we can give also upper bounds for CD(r) (Ref = A).

4 Three Simple Consequences Related to a-Points Definition of Blaschke Characteristic of Zeros and a-Points in a Given Domain In what follows we make use of an alternative characteristic for the zeros and apoints of functions in a given domain. Denote by zi (0) the zeros of w (in D); more generally denote by zi (a) the apoints of w, that is the solutions of w(z) = a ∈ C. Also denote by Dist(zi (a), ∂D) the distances between the a-points zi (a) of w and ∂D and consider the following sum  N (D, a, w) := Dist(zi (a), ∂D), {zi (a)∈D}

where in the sum we count each zi (a) according to its multiplicity. We can write the  last sum also as n(D,a,w) Dist(zi (a), ∂D), where n(D, a, w) is the usual counting i=1 number of a -points (multiplicities are counted). Notice that in the particular case, when D is the disk D(1) := {z : |z| < 1} and a = 0 the magnitude N (D, 0, w) coincides with the following sum characterizing the zeros zi (0): 

(1 − |zi (0)|)

{zi (0)∈D(1)}

The last sum (known as called Blaschke sum) was studied in a huge number of investigations in complex analysis. So that we can consider N (D, a, w) as a characteristic which generalizes the Blaschke sum in the case of arbitrary domains D, respectively we will refer to N (D, a, w) as Blaschke characteristic of a-points in D. Below we give three inequalities related to the above Blaschke characteristic. The first one concerns interrelations between a-points and Gamma-lines.

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Theorem 4.1 ([11] item 1.5) For any regular function w in D and any smooth Jordan curve  connecting a with ∞ N (D, a, w) ≤ L(D, , w).

(4.1)

Theorem 4.2 For any regular function w in D¯ we have N (D, a, w) ≤

1 4

  

  w (z)  1    w (z)  dσ + 4 l(∂D).

(4.2)

D

Proof of (4.2) Take an arbitrary straight line γ passing through the point a. Since γ consists of two rays connecting a with ∞ we have, due to (4.1), N (D, a, w) ≤ L(D, γ , w)/2 so that applying (3.3) with  = γ we obtain (4.2). Theorem 4.3 For any regular function w in D¯ we have N (D, a, w) ≤

1 2π

    w (z)     w(z) − a  dσ.

(4.3)

D

Proof of (4.3) Applying Theorem 3.5 to function W := w(z)−a with (β) = {w : |W | > 0, arg(W ) = β} and (β) ≡ 1 we get 2π L(D, (β), W )dβ = 0

    w (z)     w(z) − a  dσ. D

Due to the mean value theorem we have a value β ∗ such that 1 L(D, (β ), W ) = 2π ∗

    w (z)     w(z) − a  dσ.

(4.4)

D

Applying (4.1) to W and (β ∗ ) we have also N (D, 0, W ) ≤ L(D, (β ∗ ), W ) so that taking into account that N (D, 0, W ) = N (D, a, w) we obtain (4.3). Comment 4.1 We didn’t see any inequality of type (4.3) (with arbitrary D) elsewhere. For the disks D(r) inequality (4.3) follows immediately from argument principle. Indeed, due to this principle we have for the a-points zi (a) ∈ D(r) of a regular function w 1 n(D(t), a, w) ≤ 2π

 2π   w (teiϕ )     w(teiϕ ) − a  tdϕ, 0

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so that integrating we obtain the following inequality r N (D(r), a, w) :=

n(D(t), a, w)dt = 0

n(r,a,w) 

(r − |zi (a)|) ≤

i=1

1 2π

    w     w − a  dσ, D

which, obviously, is a particular case of (4.3) when D = D(r).

5 Ahlfors Fundamental Theorems in Terms of Windings and a New Interpretation of Deficient Values Introduction The results of this section fill a gap in Ahlfors theory of covering surfaces (see [2], or [25], chapter 13) and give brand new geometric interpretations of all results related to the deficient values. A Gap in Ahlfors Theory and Theorem 4.1 Nevanlinna value distribution theory deals with meromorphic functions f in the complex plane and, particularly studies a-points of w in terms of classical characteristic m(r, a) and T (r) considered in the disks D(r) := {z : |z| < r}. Ahlfors theory of covering surfaces (see [2] , or [25], chapter 13) complements Nevanlinna theory, particularly establishes some analogs of Nevanlinna fundamental theorem for functions in a given domain. Ahlfors first fundamental theorems didn’t consider the values a (like in Nevanlinna theory); instead they operate either with some domains or some curves.5 Respectively Ahlfors theory does not have corresponding analogs of the characteristics m(r, a) and the first fundamental theorem in Nevanlinna theory. In [3] an analog of m(r, a) in Ahlfors theory was introduced which is of essentially different nature. Making use it we get the first fundamental theorem for values a in Ahlfors theory. This fills the mentioned gap, see Theorem 5.1 below. Geometric Interpretations of Deficient Values According to the main qualitative corollary of Nevanlinna’s first fundamental theorem: if a function w(z) meromorphic in C takes a value a ∈ C rarely in the disks |z| ≤ r , so that a is deficient for w(z), then there are some parts on the circles |z| = r, where |w(z) − a| is “small”. In other words for the deficient values a we observe a certain closeness between w(z) and a on ∂D. It appears [3] that deficiency of a leads to another geometric behavior of the curve w(|z| = r): the curve is strongly revolved, “coiled” around the point a. This phenomena leads to new, geometric interpretation (or understanding or point of view) of any result related to the deficient values of meromorphic functions in C.

5 Ahlfors himself discussed (see [2], or [25], chapter 13) why his theory does not give corresponding analog of Nevanlinna first fundamental theorem for similar values a.

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Meantime the last phenomenon has its counterpart which is valid for arbitrary ¯ meromorphic (and some much larger classes of) functions w in D. We present this counterpart below; see [3], or [11], Section 2.2. We make use  below the classical notions: A(D)  stands for Ahlfors-Shimizu ++ |w |2 1 characteristic that is A(D) := π D dσ ; L(D) for the spherical length 2 )2 (1+|w|   + |w | of w(∂D) that is L(D) := ∂D 1+|w| 2 ds and n(D, a) for the number of a-points of w occurring in D (multiplicities are counted). Denote by ∂D(a) a subset of ∂D, where |w(z) − a| < 1; that is ∂D(a) = ∂D ∩ {z : |w(z) − a| < 1} . Theorem 5.1 Let w be a meromorphic function in D¯ and a ∈ C. Then assuming that w = a on ∂D we have  ∂ arg ∂D(a)

1 + n(D, a) = A(D) + hL(D), w−a

(5.1)

where |h| ≤ h1 (a) and h1 (a) is a finite constant depending only on a.6 Theorem 5.2 Let w be a meromorphic function in D¯ and aν , ν = 1, 2, . . . , q, be a set of pairwise different complex values. Then  q  ν=1∂D(a ) ν

 1 + n1 (D, aν ) ≤ 2A(D) + h2 L(D), w − aν q

∂ arg

(5.2)

ν=‘

where h2 is a finite constant depending only on a1 , . . . , aq . Comment 5.1 The above results are valid for much larger quasiconformal functions, see [11], Section 2.2. Comment 5.2 (On Interrelations with the Ahlfors Theory) The main application of Ahlfors theory in the theory of meromorphic function is his following second fundamental theorem: q  ν=1

[A(D) − n(D, aν )] +

q 

n1 (D, aν ) ≤ 2A(D) + h∗ L(D),

ν=‘

where h∗ is a finite constant depending only on a1 , . . . , aq .

6 Here

h1 (a) := (2π)−1 (1 + |a| + h2 (a)), where h2 (a) is .   3|w|2 |a| + 1 + |a|2 |w| + |a| inf . z∈∂D∩{z: |w(z)|>1} |w − a|2

(5.3)

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This is a direct and, in some sense more natural than, analog of Nevanlinna’s second fundamental theorem. To explain this we notice that in Nevanlinna theory (which works in the + rdisks Dr := {z : |z| < r)) we deal with the integrated magnitude N(r, a) = 0 n(r, a)t −1 dt, where n(r, a) stands, as usually, for n(Dr , a), while in Ahlfors theory we deal with more natural magnitude n(r, a). By analogy with Nevanlinna theory Ahlfors analog (5.2) leads in the case of functions in the complex plane to the Ahlfors deficiency relation: 

δ(ai ) ≤ 2, where δ(a) := 1 − lim sup r→∞

n(Dr , a) . A(r)

Similarly from (5.2) we get the following new form for this deficiency relation: ⎛ 

⎜ 1 ˜ i ) ≤ 2, where δ(a) ˜ δ(a := lim inf ⎝ r→∞ A(r)

⎞

 ∂ arg ∂d(aν )

1 ⎟ ⎠. w − aν

˜ We call δ(a) geometric deficiency of w at the point a and refer a as geometrically ˜ deficient if δ(a) > 0. An important circumstance is that the new geometric deficiency provides a brand new interpretation for deficient values. Indeed, it is well known (and widely used in Nevanlinna theory) that |w(z) − a| should be “rather close” to zero on some subset of ∂Dr provided the value a is deficient. Meantime, due+to (5.1), if a value 1 a is a geometrically deficient we conclude that the magnitude ∂D(a) ∂ arg w−a is comparatively large, what shows, (in addition to he classical interpretation), that the curve w(∂D) strongly wind (or revolve of coiling) around this value a. It deserves to be stressed that this geometric phenomenon accompany any result related to any deficient value. Now, returning statements in the above introduction, we see + to the qualitative 1 that the quantity ∂D(a) ∂ arg w−a plays in (5.1) the same role as m(r, a) in the first fundamental theorem of Nevanlinna. Respectively Theorem 5.1 is a direct analog of Nevanlinna first fundamental theorem (which now is valid for any domain unlike Nevanlinna theory working only for disks). Thus the contribution of Theorems 5.1 and 5.2 is as follows: Theorem 5.1 fills the gap in Ahlfors theory (it presents an analog of Ahlfors first fundamental theorem for values a); Theorem 5.2 in fact coincides with Ahlfors second fundamental theorem but gives in addition brand new interpretation; both theorems together give a new geometric understanding of the deficiency phenomenon.

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6 Universal Version of Value Distribution for Functions in a Given Domain The Nevanlinna-Ahlfors theories reveal value distribution type phenomena of meromorphic functions w which describe interrelations between the numbers n(r, aν ) of a-points for different values a1 , . . . , aq , q ≥ 2. Typical examples are the second fundamental theorems in these theories. The theories work well for the functions w in the complex plane while for the functions in a given disk they work only when w has “sufficiently large” growth. Meantime, there are numerous, well known classes of functions defined in the unit disk (H p , Dirichlet, Blaschke product etc.) for which the value distribution was not studied at all.7 Can there be a general value distribution type result which is valid for an arbitrary meromorphic function in an arbitrary domain D?8 Can this hold also for much larger classes of functions? In this section we show that to study value distribution type phenomena for the functions w in a given domain we need an extra characteristic function. Similar characteristic was utilized in [13] (2010) to establish an analog of the second fundamental theorems in the mentioned theories for any function w in a given domain. The result is presented below as Theorem 6.1. The next Theorem 6.2 transfers this result to much larger classes of functions. Thus we come to a universal version of value distribution (establishing a purely geometric analog of the second fundamental theorems) which is valid any ¯ meromorphic function in D¯ (as well as for much lager functions in D). On Necessity of a New Characteristic for the Meromorphic Functions in a Given Domain It is easy to show that the mentioned classical theories do not contain enough characteristics to describe the numbers of zeros (or a-points) in general case. Let w(z) be a meromorphic function in the closure D¯ of a given domain D with smooth boundary ∂D. Consider the case of the Ahlfors theory which works with the spherical area πA(D) of w(D), with the spherical length L(D) of w(∂D) and with the number n(D, a) of a-points of w (taken with counting multiplicities) in D. Taking w = n−2 zn in the disk d(1, 0) := {z| |z| < 1} we observe that for “sufficiently large” n the magnitudes A(d(1, 0)) and L(d(1, 0)) are as small as possible and the magnitude n(d(1, 0), 0) is as large as possible. This means that that A, L and n can not be evaluated in terms of each other.

similar classes of functions only the case q = 1 and a1 = 0 was studied. other words can there be an analog of the second fundamental theorems in the classical Nevanlinna and Ahlfors theories which is valid also in general case? As we mentioned above the classical Nevanlinna and Ahlfors theories work properly only for the basic functions and they do not touch the general case; particularly these theories do not work for the mentioned numerous classes of functions. 7 For 8 In

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Thus, to describe value distribution of analytic (meromorphic) functions in the given domains we should deal with another set of characteristic functions or we should make use an additional characteristic along with A and L. The New Characteristic and Main Result Denote by K(D) the absolute integral curvature of the curve w(∂D), that is  1 K(D) := |k(s)|ds, 2π w(∂D) where k(s) is the curvature of the curve w(∂D). We show first that the value distribution of an arbitrary meromorphic function in the given domain can be described in terms of A(D), L(D) and K(D). (More general case is given at the end of this item). Theorem 6.1 ([13]) For any set of pairwise different complex values a1 , . . . , aq ∈ ¯ , q > 1, and an arbitrary meromorphic function w in D¯ with w = a1 , . . . , aq C and w = 0 on ∂D and we have q 

|n(D, aν ) − A(D))| ≤ K(D) + hL(D),

(6.1)

ν=1

where h := h(a1 , . . . , aq ) < ∞ is a constant depending on a1 , . . . , aq . The value of h(a1 , . . . , aq ) is given in the proof, see [13] . Sharpness in the Case of Functions in a Given Domain Consider the simplest functions w = zn , n ≥ 1, in a small disk DR = {z : |z| < r}, where R < 1. Clearly when R → 0 we have A(DR )) → 0 and L(DR ) → 0. Taking two values aν in Theorem 6.1, a1 = 0 and a2 = 2, we notice that for any R , 0 < R < 1, we have n(DR , a1 ) = n and n(DR , a2 ) = 0 and K(DR ) = n . Thus we obtain that the ratio of both sides in (5.1) tends to 1 when n → ∞; this means that (6.1) is asymptotically sharp. If we wish to deal with a fixed domain we consider the functions w = n−2 zn in the disk D1 . In this case we have A(D1 )) → 0 and L(D1 ) → 0 when n → ∞. Considering the same a1 and a2 we obtain again that the ratio of both sides in (6.1) tends to 1 when n → ∞. (Moreover, one can give a hint how to construct some “large” classes of “more complicated” function w in the unit disk D(1) for which (6.1) is asymptotically sharp.) Sharpness in the Case of Entire Functions and Meromorphic Functions in the Complex Plane Consider Theorem 6.1 for the entire function exp z with two values a1 = 0 and a2 = ∞. It is easy to check that in the domains Dr we have K(Dr )A−1 (Dr ) → 2, L(Dr )A−1 (Dr ) → 0. On the other hand we have n(D, 0) = n(D, ∞) = 0. Thus we obtain that (6.1) is asymptotically sharp for exp z when r → ∞.

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Now we consider the case of the meromorphic functions in the complex plane. For the Weierstrass doubly periodic function P we take arbitrary values ¯ q > 1. Notice thatA(Dr )r −2 → c1 , 0 < c1 = const < ∞, a1 , . . . , aq ∈ C, −1 K(Dr )r → c2 , 0 < c2 = const < ∞, and L(Dr )r −1 → c3 , 0 < c2 = const < ∞, for r → ∞ so that we have K(Dr )A−1 (Dr ) → 0 and L(Dr )A−1 (Dr ) → 0. On the other hand for an arbitrary aν we have |n(Dr , aν ) − A(Dr ))| A−1 (Dr ) → 0. From here we obtain 1  1 |n(Dr , aν ) − A(Dr ))| = lim K(Dr ) + hL(Dr ) = 0; r→∞ A(Dr ) r→∞ A(Dr ) q

lim

ν=1

what means that (6.1) is asymptotically sharp for P when r → ∞. Comment 6.1 (The Magnitude K(D) as the New Characteristic) The sharpness of (6.1) in all cases (the domains and the complex plane) shows that the magnitude K(D) should be considered as an additional (to A(D) and L(D)) characteristic: as we mentioned above something new is needed to handle the case of arbitrary domains. It is interesting that all these three characteristics are of geometric nature. Comparison with the Classical Results and Some Discussions The second fundamental theorem of Ahlfors, in a bit shorted version, asserts: for any set of pairwise different complex values a1 , . . . , aq , q ≥ 3, we have q 

[A(D) − n(D, aν )] ≤ 2A(D) + h1 (a1 , . . . , aq )L(D),

(6.2)

ν=1

where h1 (a1 , . . . , aq ) < ∞ is a constant depending on a1 , . . . , aq . Let’s compare (6.1) and (6.2). First we pay attention to the very important circumstance that in (6.1) we deal with the modules |n(D, aν ) − A(D))| meantime in (6.2) with the difference A(D) − n(D, aν ). Due to this circumstance (6.1) is ¯ On the other meaningful and describes distribution of the a-points for any w in D. hand, it is well known that (6.2) describes distribution of the a-points only when L(D) is essentially less that A(D) (see [2, 25], chapter 13). As it was mentioned above this is so for only those classes of functions that have “equidistribution” : for instance for meromorphic functions in the complex plane as well as in the disks D(r) but provided that corresponding characteristic function grows rather rapidly. Ahlfors’ theorem does not work when we have enough powerful set of vales a1 , . . . , aq such that n(D, aν ) are essentially larger that A(D). But this is quite common and important in application case (remember the simplest example n−2 zn ) when we deal with the functions in arbitrary domains. Corresponding Riemann surfaces have a very interesting geometry. They have one or several algebraic branch points whose neighborhoods look like some thin gimlets with many coils (“spins” in the above terminology). In this case the boundary the Riemann surface of q make many coils; the number of coils is comparable with ν=1 n(D, aν ) since we assumed in this case that A(D) is comparatively small. It is obvious geometrically

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that K(D) should be large enough in similar case. Theorem 6.1 reflects namely this circumstance. In the case when w is meromorphic in the complex plane Miles showed [24] that q 

|A(D) − n(D, aν )| ≤ CA(D) + h1 (a1 , . . . , aq )L(D),

ν=1

where C is an absolute constant. The same result followed also from our Theorem 1 in [6]. Universal Version for Larger Functions Let W (z) be a complex functions in a given simply connected domain D¯ with the topology of analytic (meromorphic) function. This means that W maps D¯ onto a Riemann surface R := {z ∈ D, W (z)}.9 In addition we assume that the curve γW = W (∂D) is an analytic curve and belongs to F (see Sect. 2). The class of ¯ complex functions satisfying the above restrictions we will denote by W˜ (D). Due to the main theorem of conformal mappings (see [25], chapter 1, item 2) there is a meromorphic function w in D¯ which maps D¯ onto the Riemann surface R. Notice that all magnitudes n, A, K and L in Theorem 6.1 are the same for both functions w and W . In addition assumption that γW ⊂ F leads to w = 0 on ∂D. So that Theorem 6.1 implies immediately to the following result. Theorem 6.2 Theorem 6.1 remains true if we substitute the meromorphic function ¯ with W = a1 , . . . , aq on ∂D. by any complex function W ∈ W˜ (D) Acknowledgements This work was supported by Marie Curie (IIF) award. The author thanks the referee for careful checking.

References 1. L. Ahlfors, Untezuchungen zur Theorie der Konformen Abbildungen und ganze Funkzionen. Acta Soc. Sci. Fenn. 1(9), 1–40 (1930) 2. L. Ahlfors, Zur Theorie der Überlagerungsflachen. Acta Math. 65, 157–194 (1935) 3. G. Barsegian, Deficient values and the structure of covering surfaces. Izvestia Acad. Nauk Armenii 12, 46–53 (1977) (in Russian) 4. G. Barsegian, New results in the theory of meromorphic functions. Dokl. Acad. Nauk SSSR 238(4), 777–780 (1978) (in Russian, translated in Soviet Math. Dokl.) 5. G. Barsegian, On geometric structure of image of disks under mappings by meromorphic functions. Math. Sbornik 106(148)(1), 35–43 (1978) 6. G. Barsegian, On the geometry of meromorphic functions. Mathem. sbornic 114(156)(2), 179–226 (1981) (in Russian, translated in Math. USSR Sbornic)

9 Notice that this is so for any enough smooth quasiconformal function (particularly any meromorphic function).

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7. G. Barsegian, Exceptional values associated with logarithmic derivatives of meromorphic functions. Izvestia Acad. Nauk Armenii 16(5), 408–423 (1981) 8. G. Barsegian, Estimates of derivatives of meromorphic functions on sets of a-points. J. Lond. Math. Soc. 34(3), 534–540 (1986) 9. G. Barsegian, Tangent variation principle in complex analysis. Izvestia Acad.Nauk Armenii 27(3), 37–60 (1992) (in Russian, translated in Journal Cont. Math. Anal., Allerton Press) 10. G. Barsegian, Estimates of higher derivatives of meromorphic functions on sets of its a-points. Bull. Honk Kong Math. Soc. 2(2), 341–346 (1999) 11. G. Barsegian, Gamma-Lines: On the Geometry of Real and Complex Functions (Taylor and Francis, London, New York, 2002) 12. G. Barsegian, Some interrelated results in different branches of geometry and analysis, in Further Progress in Analysis (World Scientific Publishing, Hackensack, NJ, 2009), pp. 3–32 13. G. Barsegian, An universal value distribution for arbitrary meromorphic functions in a given domain, in Progress in Analysis and Its Applications (World Scientific Publishing, Hackensack, NJ, 2010), pp. 123–128 14. G. Barsegian, A new principle for arbitrary meromorphic functions in a given domain. Georgian Math. J. 25(2), 181–186 (2018). https://doi.org/10.1515/gmj-2018-0032 15. G. Barsegian, D.T. Lê, On a topological property of multi-valued solutions of some general classes of complex differential equations. Complex Variables Theory Appl. 50(5), 307–318 (2005) 16. G. Barsegian, G. Sukiasyan, Methods for study level sets of enough smooth functions, analysis and applications, in “Topics in Analysis and Applications”: the “NATO Advanced Research Workshop, August 2002, Yerevan”, Series, ed. by G. Barsegian, H. Begehr (NATO Science Publications, Kluwer, 2004) 17. G. Barsegian, W. Yuan, On some generalized Painlevé and Hayman type equations with meromorphic solutions in a bounded domain. Georgian Math. J. 25(2), 187–194 (2018). https:// doi.org/10.1515/gmj-2018-0033 18. G. Barsegian, I. Laine, D.T. Lê, On topological behavior of solutions of some algebraic differential equations. Complex Variables Elliptic Equ. 53, 411–421 (2008) 19. G. Barsegyan, A proximity property of the a-points of meromorphic functions. Mat. Sb. 120(162), 42–67 (1983); Math. USSR-Sb. 48, 41–63 (1984) 20. G. Barsegyan, The property of closeness of a-points of meromorphic functions and the structure of univalent domains of Riemann surfaces. Izv. Akad. Nauk Armyan. SSR Ser. Mat. 20(6), 407–425 (1985) (Russian, with English and Armenian summaries) 21. W. Hayman, Picard values of meromorphic functions and their derivatives. Ann. Math. 70(2), 9–42 (1959) 22. W. Hayman, J.M. Wu, Level sets of univalent functions. Comment. Math. Helf. 56(3), 366–403 (1981) 23. D. Hilbert, Mathematische Probleme, Vortrag, gehalten auf dem internationalen MathematikerKongress zu Paris 1900; Archiv der Mathematik und Physik, 3rd series, 1, 44–63, 213–237 24. J. Miles, Bounds on the ratio sup n(r, a)/A(r) for meromorphic functions. Trans. Am. Math. Soc. 162, 383–393 (1971) 25. R. Nevanlinna, Eindeutige analytische Funktionen (Springer, New York, 1936) 26. G. Valiron, Fonctions entières d’ordre fini et fonctions méromorphes. Enseignement Mathématique”, No. 8, Institut de Mathématiques, Université, Genève, 1960, 150 pp.

Boundary Value Problems in Polydomains Ahmet Okay Çelebi

Dedicated to Prof. H.G.W. Begehr on the occasion of his 80th birthday

Abstract In this paper we give a short survey of the boundary value problems in polydomains in the last decades. Firstly we develop an alternative method to derive integral representations for functions in Cn . This unified method provides representations which are suitable to be employed in discussions for all linear boundary value problems. In the rest of the article we have improved some results obtained for Schwarz and Dirichlet type problems. Keywords Polydisc · Schwarz problem · Riquier problem · Complex partial differential equations Mathematics Subject Classification (2010) Primary 32W10, 32W50; Secondary 31A10

1 Introduction The investigation on several complex variables has started in 19th century. But only in the last several decades the researchers are attracted by the partial differential equations in several complex variables: see for example [9, 14, 19, 23] and references therein for discussions in Cn . The unit disc D := {z ∈ C : |z| < 1} of C is extended in two different approaches: (i) as a ball Bn = {z ∈ C : |z1 |2 + · · · + |zn |2 < 1}; (ii) as a polydisc Dn = {z ∈ C : |z1 | < 1, . . . , |zn | < 1}

A. O. Çelebi () Yeditepe University, Istanbul, Turkey e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_5

69

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which also states that Dn = D1 × D2 × · · · × Dn where Dj := {zj ∈ C : |zj | < 1}. We can find many articles on the boundary value problems defined in balls or polydomains. The geometrical structures of polydiscs are comparatively complicated than balls. That is why we can find fewer articles investigating the solutions of boundary value problems on polydiscs. In order to find the solutions of boundary value problems in a domain in C, it is very important to derive integral representations of the functions in the domains considered. Since the Riemann mapping theorem is valid in C, the investigations have started with the unit disc. Afterwards the discussions have been extended to domains different from unit disc. Many researchers have been interested in the solutions of the model equations for Riemann, Riemann-Hilbert, Dirichlet, Schwarz, Neumann, Robin and mixed type problems [10–13, 15, 17, 18, 20, 21, 30, 31]. In some of the cases solutions are also obtained for linear complex partial differential equations, [4, 5]. Because of the complex nature of the Pompeiu operator over the unit ball in Cn , the iterations of integral representations are not given explicitly yet [14]. On the other hand for the problems in polydiscs, the higher-dimensional differential equations are converted into a particular system of equations in Cn with the leading term involving either a pluriharmonic or pluriholomorphic operators. A detailed investigation is given by Begehr and Dzhuraev [14]. Using this technique many articles have appeared [24–26, 28, 29]. Lately, we have given the integral representations for functions w : Cn → Cn by iterations which is used to derive solutions of boundary value problems in polydomains for differential equations involving either a polyharmonic or polyholomorphic leading terms [6, 8, 22]. In this paper, we review and improve the results in our recent studies on Schwarz and Riquier problems [6, 22] for linear elliptic equations in Cn on polydomains. We start with the preliminaries in Sect. 2. Section 3 is devoted to the derivations of unified integral representations for the functions in Cn . In this section the investigations will be carried by the iterations of the integral representations which are given for the functions in C. In Sects. 4 and 5 we have discussed the derivation of the solutions for Schwarz and Riquier problems in Cn which extends the results obtained in [6, 22].

2 Preliminaries We start with reminding the Schwarz and Dirichlet type problems in C. These boundary value problems have attracted many researchers in the several decades [1, 3, 4, 10–13, 15–18, 20, 21]. Firstly we consider the existence and uniqueness result for Schwarz problem in a disc in C.

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Theorem 2.1 ([16]) The Schwarz problem for the inhomogeneous polyanalytic equation in the unit disc D defined by ∂zm w = f in D , Re ∂zl w = γl on ∂D , Im ∂zl w(0) = cl , 0 ≤ l ≤ m − 1 , is uniquely solvable in the distributional sense for f ∈ L1 (D), γl ∈ C(∂D; R), cl ∈ R, 0 ≤ l ≤ m − 1. The solution is w(z) = i

m−1  l=0

(−1)m + 2π(m − 1)!

 (−1)l cl (z + z)l + l! 2πil! m−1 l=0

 (ζ −z+ζ − z)m−1 D

 γl (ζ ) ∂D

dζ ζ +z (ζ − z + ζ − z)l ζ −z ζ

 f (ζ ) ζ + z f (ζ ) 1 + zζ  + dξ dη ζ ζ −z ζ 1 − zζ

(2.1)

where ζ = ξ + iη. For m = 1, the solution is given by the so-called Cauchy-Schwarz-PoissonPompeiu formula as w(z) =

1 2πi

 Rew(ζ ) ∂D

 " f (ζ ) ζ + z f (ζ ) 1 + zζ # dξ dη . + ζ ζ −z ζ 1 − zζ

ζ + z dζ 1 +iImw(0)− ζ −z ζ 2π

D

The domain integral appearing in the solution (2.1) is a particular solution of the inhomogeneous polyanalytic equation given in Theorem 2.1 having homogeneous boundary conditions. It is introduced by Begehr (see [7]) as an integral operator: Definition 2.2 ([7]) Let m ∈ N, f ∈ Lp (D), 1 ≤ p. Then for z ∈ D, m m f (z) := (−1) T 2π(m − 1)!

/

 (ζ − z + ζ − z)

m−1

D

f (ζ ) ζ + z f (ζ ) 1 + zζ¯ + ζ ζ −z ζ¯ 1 − zζ¯

0 dξ dη .

Moreover, T 0 f = f . We employ the notation m w(z) = i ∂T

m−1  l=0

 (−1)l cl (z + z)l + l! 2πil! m−1 l=0

 γl (ζ ) ∂D

dζ ζ +z (ζ − z + ζ − z)l ζ −z ζ

in the sequel. This simplifies the representation of the solution as m w(z) + T m ∂ m w w(z) = ∂ T z¯

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or m w(z) + T m (L1,m w) w(z) = ∂ T

(2.2)

where L1,m = ∂z¯m is the operator occurring in the model equation. It is easy to observe that 1 (∂ T k−1 f (z)) = ∂ T k (f (z)) ∂T and T 1 (T k−1 f (z)) = T k (f (z)). Secondly we consider the Dirichlet type problems in the unit disc D ⊂ C, with harmonic operators as leading terms. In this domain the harmonic Green’s function is defined as    1 − zζ¯ 2  . G1 (z, ζ ) = log  ζ −z  The properties of the harmonic Green function are well-known. Since the relevant differential operator is self-adjoint, Green’s function is symmetric, i.e., G1 (z, ζ ) = G1 (ζ, z) holds. G1 (z, ζ ) is related to the Dirichlet problem for Poisson equation. Theorem 2.3 ([10]) The Dirichlet problem ∂z ∂z¯ w = f (z) in D, w = γ on ∂D with f ∈ Lp (D) ∩ C(D), p ≥ 1, γ ∈ C(∂D) is uniquely solvable. The solution is  w(z) = −

∂νζ G1 (z, ζ )γ (ζ )

1 dζ − ζ π

∂D

 D

G1 (z, ζ )f (ζ )dξ dη.

A polyharmonic Green function Gm is defined iteratively by 1 Gm (z, ζ ) = − π

 D

G1 (z, ζ˜ )Gm−1 (ζ˜ , ζ )d ξ˜ d η˜

(2.3)

for m ≥ 2. Gm are related to the m-Dirichlet or Riquier problem for higher order Poisson equation in the unit disc D. Theorem 2.4 ([17]) The Riquier problem (∂z ∂z¯ )m w = f in D, (∂z ∂z¯ )μ w = γμ , 0 ≤ μ ≤ m − 1 on ∂D

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with f ∈ L1 (D) ∩ C(D), γμ ∈ C(∂D), 0 ≤ μ ≤ m − 1 is uniquely solvable. The solution is   m  1 1 dζ − w(z) = − ∂νζ Gμ (z, ζ )γμ−1 (ζ ) Gm (z, ζ )f (ζ )dξ dη. 4πi ζ π D μ=1

∂D

(2.4) Definition 2.5 For k, l ∈ N0 , n ∈ N, k + l ≤ 2m, we define  1 f (z) = − ∂zk ∂z¯l Gm (z, ζ )f (ζ )dξ dη Gk,l m π D

where ζ = ξ + iη, for a suitable complex valued function f given in D. Previously, we have given [4] a set of properties in C, which are stated below: Lemma 2.6 For f ∈ Lp (D) where p > 2 (i)     k,0 G1 f (z) ≤ C(k, p)f Lp (D)

(2.5)

for k = 0, 1, (ii)    |z1 − z2 |(p−2)/p if k = 1  k,0  k,l p f (z ) − G f (z ) ≤ C(k, p)f  G1 1 2  L (D) 1 |z1 − z2 | if k = 0 (2.6) for z1 , z2 ∈ D and (iii) G12,0 f Lp (D) ≤ C(p)f Lp (D)

(2.7)

G12,0 f L2 (D) ≤ f L2 (D)

(2.8)

for p > 1. Moreover

holds. Theorem 2.7 If  f ∈

Lp (D) , 1 ≤ k ≤ 2m − 1 W k+1−2m,p (D) , k ≥ 2m

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then 1,0 k−1 0,0 Gk,0 Gm−1 f (z)) m f (z) = G1 ((D − D∗ )

(2.9)

p and Gk,0 m f ∈ L (D) hold.

Theorem 2.8 Let f ∈ Lp (D), p > 2 and k + l < 2m. Then,    k,l  Gm f (z) ≤ Cf Lp (D)

(2.10)

for z ∈ D . Theorem 2.9 Let f ∈ Lp (D), p > 2 and k + l < 2m. Then for z1 , z2 ∈ D,    |z1 − z2 |(p−2)/p if k + l = 2m − 1   k,l p f (z ) ≤ Cf  .  Gm f (z1 ) − Gk,l 2 L (D) m |z1 − z2 | otherwise (2.11) p Theorem 2.10 If k + l = 2m, then Gk,l m f ∈ L (D) and

Gk,l m f Lp (D) ≤ Cp f Lp (D)

(2.12)

for f ∈ Lp (D) with p > 1. For discussion of the operators for Schwarz and Riquier problems together with their properties, see [2–4]. Let us note that the representation of the solution given by (2.4) may be decomposed into m (L2,m w) = − 1 G π

 D

Gm (z, ζ )L2,m w(ζ )dξ dη

and m w(z) = − ∂G

 m  1 dζ ∂νζ Gμ (z, ζ )L2,μ−1 w(ζ ) 4πi ζ

μ=1

∂D

where L2,k w(z) = (∂z ∂z¯ )k w(z). Hence (2.4) can be written as m (L2,m w(z)). m w(z) + G w(z) = ∂ G

(2.13)

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75

The integral representations (2.2) and (2.13) may be unified observing that they are the compositions of boundary integral terms and a domain integral. Now let us assume that D ⊂ C is a domain and m,D w(z) + K m,D (Pm,D w(z)). w(z) = ∂ K

(2.14)

is the relevant integral representation for a complex valued function. We should keep in mind that Pm,D is the operator of the model equation that we want to deal m , ∂ K m which are generated depending on the corresponding Green’s with and K function.

3 Integral Representations for Functions in Cn In this section we give a unified derivation of the integral representations of functions in Cn , which leads to the solutions of boundary value problems. To simplify the computations we take the domain D as the unit disc D. We denote the polydisc in Cn as Dn := {z = (z1 , . . . , zn ) : |zk | < 1, k = 1, . . . , n} = D1 × D 2 × · · · × D n with the distinguished boundary ∂Dn := {z = (z1 , . . . , zn ) : |zk | = 1, k = 1, . . . , n} = ∂D1 × ∂D2 × · · · × ∂Dn where Dk = {zk : |zk | < 1} and ∂Dk = {zk : |zk | = 1}, k = 1, . . . , n. We employ the notation w(ˆzj ) := w(z1 , . . . , zj −1 , zˆj , zj +1 , . . . , zn ) So equation (2.14) may be written as m,Dj w(ˆzj ) + K m,Dj (Pm,Dj w(ˆzj )). w(ˆzj ) = ∂ K

(3.1)

To derive the integral representation we use an induction technique, given previously [6, 22]. Let us start with the case of n = 2: m,D1 w(ˆz1 , z2 ) + K m,D1 (Pm,D1 w(ˆz1 .z2 )) w(z1 , z2 ) = ∂ K m,D2 w(z1 , zˆ 2 ) + K m,D2 (Pm,D2 w(z1 , zˆ 2 )) w(z1 , z2 ) = ∂ K

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substituting the second equation in the first one, we get in C2 m,D1 [∂ K m,D2 w(ˆz1 , zˆ 2 ) + K m,D2 (Pm,D2 w(ˆz1 , zˆ 2 ))] w(z1 , z2 ) = ∂ K m,D2 w(ˆz1 , zˆ 2 ) + K m,D2 (Pm,D2 w(ˆz1 , zˆ 2 )))] m,D1 [Pm,D1 (∂ K +K m,D2 w(ˆz1 , zˆ 2 )] + ∂ K m,D1 [K m,D2 (Pm,D2 w(ˆz1 , zˆ 2 ))] m,D1 [∂ K = ∂K m,D2 w(ˆz1 , zˆ 2 )] + K m,D1 [Pm,D1 K m,D1 [Pm,D1 (∂ K m,D2 (Pm,D2 w(ˆz1 , zˆ 2 )))] +K m,D1 [∂ K m,D2 w(ˆz1 , zˆ 2 )] + ∂ K m,D1 [w(ˆz1 , zˆ 2 ) − ∂ K m,D2 w(ˆz1 , zˆ 2 ))] = ∂K m,D1 Pm,D1 w(ˆz1 , zˆ 2 )] + K m,D2 (Pm,D1 Pm,D2 w(ˆz1 , zˆ 2 )) m,D2 [K +∂ K m,D2 w(ˆz1 , zˆ 2 ) + ∂ K m,D1 w(z1 , zˆ 2 ) − ∂ K m,D1 ∂ K m,D2 w(ˆz1 , zˆ 2 ) m,D1 ∂ K = ∂K m,D1 w(ˆz1 , zˆ 2 )] + K m,D2 (Pm,D1 Pm,D2 w(ˆz1 , zˆ 2 )) m,D2 [w(z1 , zˆ 2 ) − ∂ K +∂ K m,D2 w(ˆz1 , z2 ) − ∂ K m,D1 ∂ K m,D2 w(ˆz1 , zˆ 2 ) m,D1 w(z1 , zˆ 2 ) + ∂ K = ∂K m,D2 (Pm,D1 Pm,D2 w(ˆz1 , zˆ 2 )). +K

For the functions in C3 , using a similar technique we get m,D1 w(z) + ∂ K m,D2 w(z) + ∂ K m,D3 w(z) − ∂ K m,D1 ∂ K m,D2 w(z) w(z) = ∂ K m,D1 ∂ K m,D2 w(z) − ∂ K m,D2 ∂ K m,D3 w(z) −∂ K m,D1 ∂ K m,D2 ∂ K m,D3 w(z) + K m,D3 (Pm,D1 Pm,D2 Pm,D3 w(z)). +∂ K In Cn , the representation we are looking for, assumes the form w(z) =

n  r=1

(−1)r+1

 j1 +···+jn =r

m,Dj ∂ K m,Dj . . . ∂ K m,Dj w(z) ∂K n 1 2

m,Dn (Pm,D1 Pm,D2 . . . Pm,Dn w(z)), +K

(3.2)

by induction. Equation (3.2) is a unified integral representation for functions in Cn . It enables us to handle some of the boundary value problems in Cn for model equations by the m,Dj . choice of the differential operator Pm,Dj and the relevant integral operator K In the sequel we state some particular cases of (3.2) as examples. For a derivation of the integral representation of a function w ∈ C m (Dn ) suitable to discuss Schwarz type problems, we take m,Dj (Pm,Dj ) = T m,Dj (∂ m w) Pm,Dj = ∂z¯mj , K z¯ j and m,Dj w m,Dj w = ∂ T ∂K

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77

m,Dj w are defined by (2.2). Hence we get where T m,Dj ∂z¯mj w and ∂ T w(z) =

n  (−1)r+1

 j1 +···+jn =r

r=1

m,Dj ∂ T m,Dj . . . ∂ T m,Dj w(z) ∂T n 1 2

+T m,D1 T m,D2 . . . T m,Dn ∂z¯m1 ∂z¯m2 . . . ∂z¯mn w(z).

(3.3)

Let us note that this type of representation was given previously [6] for the case of n = 2. Remark 3.1 In fact the representation (3.2) may be employed for all linear boundary m,Dj and ∂ K m,Dj . value problems in Cn , by the choice of suitable operators P m,Dj , K Secondly, let us take a Dirichlet-type problem (∂z1 ∂z¯ 1 )m (∂z2 ∂z¯ 2 )m . . . (∂zn ∂z¯ n )m w(z) = f (z) and the relevant boundary conditions. Thus we choose Pm,Dj = (∂zj ∂z¯ j )m as the differential operator. The corresponding integral operator is generated by the Green’s function given in (2.3) as m,Dj f (z) = K

 Dj

Gm,Dj (zj , ζj )f (z)dξj dηj .

Thus (3.2) is converted into w(z) =

n 

(−1)s+1

 j1 +j2 +···+jr =s

s=1

˜ Dj ×···×Dj ,n w + G ˜ Dm ,n [ ∂G r 1

n 1

(∂zk ∂z¯k )m w]

k=1

(3.4) where     1 n ˜ m GD ,n w = − ... GDn ,m (z1 , . . . , zn ; ζ1 , . . . , ζn )w(ζ )dξ dη π D1 Dn in which G

Dn ,m

(z1 , . . . , zn ; ζ1 , . . . , ζn ) =

n 1

GDj ,m (zj , ζj )

j =1

˜ Dj ×···×Dj ,m w = ∂ G ˜ Dj ×···×Dj ,m (∂ G ˜ Dj ,m w) ∂G r r 1 1 r−1

(3.5)

and dξ dη = dξ1 dη1 . . . dξn dηn . This representation is obtained previously in [22].

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4 Schwarz Boundary Value Problem This section is devoted to the review and extend the study of the higher-order model and linear equations under Schwarz-type boundary conditions in unit polydisc D of C. Definition 4.1 The Schwarz boundary value problem in Cn is to find a function w ∈ W nm,p (Dn ) satisfying ∂z¯m1 ∂z¯m2 . . . ∂z¯mn w(z) = f (z) in Dn

(4.1)

subject to the Schwarz boundary conditions μj

μj

μj

1

2

k

μj

μj

μj

1

2

k

Re ∂z¯ j 1 ∂z¯ j 2 . . . ∂z¯ j k w(z) = γμj1 μj2 ...μjk (z) on ∂Dj1 ×∂Dj2 ×· · ·×∂Djk

(4.2)

Im ∂z¯ j 1 ∂z¯ j 2 . . . ∂z¯ j k w(z) = cμj1 μj2 ...μjk (z) with zjs = 0 if js = {j1 , j2 , . . . , jk } (4.3) where cμj1 μj2 ...μjk (z) are pluriharmonic, f ∈ Lp (Dn ) for p > 1, and γμj1 μj2 ...μjk (z) are restrictions of the continuous functions γμ (z) defined on ∂Dn , to the part of the boundary ∂Dj1 × ∂Dj2 × · · · × ∂Djk . Previously we have given the solutions of Schwarz boundary value problem in C2 [6]. In this section we will extend the results obtained there to problems in Cn under two subsections.

4.1 Schwarz Problem for Model Equation We take the model equation in Cn subject to the homogeneous Schwarz boundary conditions, and prove the next theorem. Theorem 4.2 The Schwarz problem (4.1)–(4.3) with homogeneous boundary conditions is uniquely solvable and the solution is given by w(z) = T (m,m,...,m),Dn f (z), c ∈ Cn where f ∈ Lp (Dn ), p ≥ 2 and m,Dn f (z). (m,m,...,m),Dn f (z) = T m,D1 T m,D2 . . . T T

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79

k f (z) = Proof Firstly we remind some properties of the integral operators, [3, 16]: T 1 (T k−1 f (z)) is given in C and has the properties T Re ∂z¯l T k f (z) = 0 on ∂D, 0 ≤ l ≤ k − 1;

(4.4)

Im ∂z¯l T k f (0) = 0, 0 ≤ l ≤ k − 1.

(4.5)

Thus it is easy to prove that the Schwarz problem for ∂z¯m w(z) = f (z) with homogeneous boundary conditions has the unique solution w(z) = T m f (z), z ∈ C. If we return back to the problems in Cn , it is trivial by (3.3) that, the function m,D1 T m,D2 . . . T m,Dn f (z), z ∈ Cn w(z) = T

(4.6)

satisfies the differential equation ∂z¯m1 ∂z¯m2 . . . ∂z¯mn w(z) = f (z). Now let us prove that (4.6) satisfies also the boundary conditions. First of all notice that m,D2 . . . T m,Dn f (z) = T m−α1 ,D1 T m−α2 ,D2 . . . T m−αn ,Dn f (z) ∂z¯α11 ∂z¯α22 . . . ∂z¯αnn T m,D1 T for 0 ≤ αj ≤ m, j = 1, 2, . . . , n. Thus Re T m−αj1 ,Dj1 T m−αj2 ,Dj2 . . . T m−αjk ,Djk f (z) # " = Re T m−αj1 ,Dj1 T m−αj2 ,Dj2 . . . T m−αjk ,Djk f (z) = 0, 0 ≤ αjs ≤ m − 1 for 0 ≤ αjs ≤ m − 1 by (4.4) and Im T m−αj1 ,Dj1 T m−αj2 ,Dj2 . . . T m−αjk ,Djk f (z) " # = Im T m−αj1 ,Dj1 T m−αj2 ,Dj2 . . . T m−αjk ,Djk f (z) = 0,

0 ≤ αjs ≤ m − 1

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and zjs = 0 in Dj1 , . . . , Djk . So (4.6) is the solution of the Schwarz problem satisfying the homogeneous conditions.   Note We know that the integral operators in this representation are commutative and T m−αj g ∈ Lp if g ∈ Lp . Remark 4.3 1) The solution of the homogeneous equation with inhomogeneous Schwarz conditions is given as w(z) =

n 



(−1)r+1

j1 +···+jn =r

r=1

m,Dj ∂ T m,Dj . . . ∂ T m,Dj w(z) ∂T n 1 2

(4.7)

by (3.3) for z ∈ Cn . This solution is also unique. 2) The solution of the inhomogeneous equation subject to the inhomogeneous conditions is the combination of the solutions (4.6) and (4.7).

4.2 Schwarz Problem for Linear Equations in Cn In the sequel, we consider a linear differential equation in Cn of nm-th order, subject to the homogeneous Schwarz conditions. We need the notations ∂ := (∂z1 , . . . , ∂zn ), ∂ := (∂z¯ 1 , . . . , ∂z¯ n ) as differential operators and α = (α1 , . . . , αn ), β = (β1 , . . . , βn ) as multi-indices where αi , βi ∈ N, i = 1, 2, . . . , n to simplify the equations. Thus a linear nm-th order partial differential equation in Cn may be written as ∂

m·1

w+



β

Aαβ ∂ α ∂ w = f (z)

(4.8)

|α|≤nm,|β|≤nm

where 1 = (1, 1, . . . , 1), |α| + |β| ≤ nm with β = (m, m, . . . , m) and Aαβ , f ∈ Lp (Dn ) for p > 1. The Schwarz problem is defined as finding the solution of (4.8) subject to Schwarz conditions β

Re ∂ w = γβ (z) on ∂Dn , |β| < nm β

Im ∂ w = cβ (z) on Dj , zj = 0 for βj = 0 where γβ (z) are continuous, cβ (z) are polyanalytic.

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81

In this subsection we discuss the solutions of (4.8) under homogeneous Schwarz conditions. To find the solution of linear equation, we will convert the differential m·1 equation into a singular integral equation. Assuming ∂ w(z) = g(z) we get w(z) = T m·1,Dn g(z). Now we compute the derivative m−βn ,Dn g m−β1 ,D1 T m−β2 ,D2 . . . T ∂ α ∂ w(z) = ∂ α T ⎛ ⎞ n 1 α =⎝ ∂zjj T m−βj ,Dj ⎠ g β

(4.9)

j =1

for |α| + |β| ≤ nm. Now let us concentrate on the operators ∂zjj T m−βj ,Dj for 0 ≤ αj ≤ m, 0 ≤ βj ≤ m. α

We have two cases to be discussed: (1) In the case of αj + βj = m, we use the notation m−βj ,Dj g = ∂zαjj T m−βj ,Dj g.  These are strongly singular integral operators of Calderon-Zygmund type. m−βj ,Dj are weakly singular. (2) If αj + βj ≤ m − 1, the operators ∂zjj T α

Thus, (4.9) assumes the form β

∂ α ∂ w(z) =

s 1

⎛ α m−βj ,Dj ⎝ ∂zjj T

j =1

n 1

⎞ αj ,Dj g ⎠ 

j =s+1

where |α| + |β| ≤ nm with |β| < nm. Previously, for the strongly singular integral operators in C, we have shown that [2] αj ,Dj gLp (Dj ) ≤ C(p, αj )gLp (Dj )  αj ,Dj are Lp where C(p, αj ) > 0 are constants, and this inequality state that  bounded. Then it is easy to show that n 1

αj ,Dj g 

j =s+1

is Lp -bounded when αj + βj = m for g ∈ Lp (Dn ).

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For the weakly singular operators the following two properties have been obtained previously [3, 7]: (i)    αj  ∂zj Tm−βj ,Dj g  ≤ gLp (Dj ) for zj ∈ Dj where c > 0 is a constant. Thus these operators are bounded. So the operators defined by s 1

∂zjj T m−βj ,Dj g α

j =1

with g ∈ Lp (Dn ) are bounded if αj + βj ≤ m − 1. (ii)   αj  αj m−βj ,Dj g(ˆzj∗∗ ) ≤ CgLp (Dj ) |ˆzj∗ − zˆ j∗∗ | ∂zj Tm−βj ,Dj g(ˆzj∗ ) − ∂zj T for 0 ≤ αj ≤ m − βj − 1, zˆ j∗ , zˆ j∗∗ ∈ Dj . So ∂zjj T m−βj ,Dj is uniformly continuous. α

Using (i) and (ii) we deduce by Arzela-Ascoli theorem that the operators α m−βj ,Dj are compact if αj + βj ≤ m − 1. Hence it is easy to see that ∂zjj T 2s αj p n j =1 ∂zj Tm−βj ,Dj g is also compact if αj + βj ≤ m − 1 for g ∈ L (D ). Thus Eq. (4.7) is converted into the integral equation ⎛ ⎞ s n  1 1 α αj ,Dj g ⎠ = f˜. g+ Aαβ ∂zjj T m−βj ,Dj ⎝  |α|≤nm,|β|≤nm

j =1

j =s+1

This singular integral equation may be written as + K)g = f˜ (I + 

(4.10)

where 

= Πg

aαβ

|α|+|β|≤nm ∀j,αj +βj =m

and = Kg

 |α|+|β|≤nm ∃j,αj +βj 2 is a solution of (4.10).

5 Dirichlet-Type Boundary Value Problems in Polydiscs In this section we review and extend the study of higher-order model equations and relevant linear differential equations having a polyharmonic leading term. The integral representation for w ∈ C 2mn (Dn ) given by (3.4) may be written as  n   i n  w(z) = 4π r

|J |=r∂D ×···×∂D j1 jr

r=1



r 1







(∂ζs ∂ζ¯s )

s=1

μs −1

w(ζ )

r 1

 ∂νjs GDjs ,μjs (zjs , ζjs )

s=1

dζr dζ1 ... ζr ζ1

⎤ ⎡    n 1 1 n + − GDn ,m (z1 , . . . , zn ; ζ1 , . . . , ζn ) ⎣ (∂ζs ∂ζ¯s )m ⎦ w(ζ )dξ dη π Dn j =1

(5.1)

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A. O. Çelebi

where J r = [j1 , j2 , . . . , jn ], |js | ∈ {0, 1}, s = 1, 2, . . . , n in which the entries js may commute and |J r | ≤ n. Let us note that |js | = r represents the entries js , which have the property that |js | = 0. Now let us state the Riquier problem in Dn ⊂ Cn for higher order polyharmonic equation: Definition 5.1 For any f ∈ Lp (Dn ), find a function w ∈ W 2mn,p (Dn ) satisfying ∂ m·1 ∂¯ m·1 w(z) = f (z) in Dn

(5.2)

∂ α ∂¯ α w(z) = γα (z) on ∂Dn

(5.3)

for all 0 ≤ αj < m, 1 ≤ j ≤ n where the functions γα are restrictions of a polyharmonic function on the distinguished boundary ∂Dn for each j , is called a Riquier problem on Dn . The above definition for the model equation has been given in Cn and its solution in C2 is obtained previously [22]. The solution of Riquier problem in Cn is given in the following statement. Theorem 5.2 The Riquier problem (5.2)–(5.3) is uniquely solvable for f Lp (Dn ), p > 2 by  n   i n  w(z) = 4π



|J |=r∂D ×···×∂D j1 jr

r=1



r 1

∈

 ∂νJs GDJs ,αJs (zJs , ζJs )

s=1

γα1 α2 ...αr (ζ1 , . . . , ζr , zr+1 , . . . , zn )

dζr dζ1 ... ζr ζ1

   1 n + − GDn ,m (z1 , . . . , zn ; ζ1 , . . . , ζn )f (ζ1 , . . . , ζn )dξ dη. π Dn

(5.4)

Proof may be obtained by direct computation since the boundary integrals yield polyharmonic functions. In the case of homogeneous boundary conditions the solution in Cn is    1 n w(z) = − GDn ,m (z; ζ )f (ζ )dξ dη π Dn which will be denoted as ˜ Dn ,m f (z). w(z) = G

(5.5)

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85

˜ Dn ,m 5.1 The Properties of Integral Operators Related to G One of the ways to discuss the existence and uniqueness of the solutions for linear higher order differential equations in Cn is to transform the given problem into integral equations. ˜ Dn ,m f (z). Let us use the notation ∂ Thus we need the derivatives of w(z) = G ¯ and ∂ given in Sect. 4.2. Definition 5.3 For m ∈ N and |α| ≤ nm, |β| ≤ nm, we define    1 n β ˜ ∂ ∂ GDn ,m f (z) := − ∂ α ∂ GDn ,m (z; ζ )f (ζ1 , ζ2 )dξ dη π α ¯β

D2

with dξ dη = dξ1 dη1 . . . dξn dηn for a suitable complex valued function f in Dn . Remark 5.4 We should observe that ˜ Dn ,m f (z) = ∂ α ∂¯ β ∂ α ∂¯ β G  =

n 1 Dn j =1

⎛

 =

⎝ Dn

α



n 1 Dn j =1

GDj ,m (zj , ζj )f (ζ )dξ dη

β

∂zjj ∂z¯ jj GDj ,m (zj , ζj )f (ζ )dξ dη ⎞⎛

1

α β ∂zjj ∂z¯ jj GDj ,m (zj , ζj )⎠ ⎝

αj ≥βj

⎞

1

α β ∂zjj ∂z¯ jj GDj ,m (zj , ζj )⎠ f (ζ )dξ dη.

αj 1 and at least for one j, αj = 2. Then ˜ Dn ,1 f Lp (Dn ) ≤ Cf Lp (Dn ) ∂ α G holds. ˜ Dn ,1 f ∈ Lp (Dn ). On the other hand Proof It is trivial that ∂ α G ⎡ ⎤  n 1 α ˜ Dn ,1 f (z) = ⎣ ∂zjj GDj (zj , ζj )⎦ f (ζ )dξ dη. ∂αG Dn

j =1

If α1 = 2 we have ˜ Dn ,1 f (z) = ∂z21 G



⎡

 D1

∂z21 GD1 (z1 , ζ1 )

⎣ D2 ×...Dn

n 1

⎤ α ∂zjj GDj (zj , ζj )⎦ f (ζ )dξ dη.

j =2

Thus ˜ Dn ,1 f Lp (Dn ) ≤ Cf Lp (Dn ) ∂ α G ˜ Dn ,1 is Lp -bounded. that is the operator ∂ α G Lemma 5.8 For f ∈

Lp (Dn )

with p > 2, the following hold:

˜ Dn ,1 f (z)| ≤ Cf Lp (Dn ) (i) |∂ α G α ˜ Dn ,1 f (z∗ ) − ∂ α G ˜ Dn ,1 f (z∗∗ )| ≤ Cf Lp (Dn ) |z∗ − z∗∗ | (ii) |∂ G for z∗ , z∗∗ ∈ Dn .

 

88

A. O. Çelebi

Proof (i) Let us assume that ∂zα11 is a factor in ∂ α . ⎤ ⎡  n 1 α ⎣ ˜ Dn ,1 f (z) = ∂αG ∂zjj GDj (zj , ζj )⎦ f (ζ )dξ dη. Dn

 =

j =1

⎡

 D1

∂z21 GD1 (z1 , ζ1 )

⎣ D2 ×...Dn

n 1

⎤ ∂zjj GDj (zj , ζj )⎦ f (ζ )dξ dη. α

j =2

By Lemma 2.6(i) we get 5 5 ⎤ ⎡ 5 5 n 1 5 5 α j α ˜ n 5 ⎦ ⎣ ∂zj GDj (zj , ζj ) f (ζ )dξ dη5 |∂ GD ,1 f (z)| ≤ C 5 5 5 5 D2 ×...Dn j =2

Lp (Dn )

≤ Cf Lp (Dn ) . (ii) Let (zj∗ ) = (z1∗∗ , . . . , zj∗∗ , zj∗+1 , . . . , zn∗ ) with (z0∗ ) = z∗ be the set of points in Dn . Then     n−1   α ˜ n ∗ α ˜ n ∗  ˜ Dn ,1 f (z∗ ) − ∂ α G ˜ Dn ,1 f (z∗∗ )| =  ∂ f (z ) − ∂ f (z ) |∂ α G G G D ,1 D ,1 j j +1   j =p  ≤ Cf Lp (Dn )

n−1  j =p

|zj∗ − zj∗+1 | ≤ Cf Lp (Dn ) |z∗ − z∗∗ |.  

Using Remark 5.4, we may improve the statements of Lemmas 5.7 and 5.8 as in the following. Lemma 5.9 Let us assume that f ∈ Lp (Dn ), p > 1 and least for one j , αj + βj = 2m. Then ˜ Dn ,m f  ≤ Cf Lp (Dn ) ∂ α ∂¯ β G holds. Lemma 5.10 For f ∈ Lp (Dn ) with p > 2, the following hold: ˜ Dn ,m f (z)| ≤ Cf Lp (Dn ) (i) |∂ α ∂¯ β G α β ˜ Dn ,m f (z∗ ) − ∂ α ∂¯ β G ˜ Dn ,m f (z∗∗ )| ≤ Cf Lp (Dn ) |z∗ − z∗∗ | (ii) |∂ ∂¯ G for z∗ , z∗∗ ∈ Dn .

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89

5.2 Riquier Problem for Linear Higher-Order Equations in Cn Now we define the Riquier problem for a linear 2nm-th order equation ∂ m˙1 ∂

m˙1



w+

β

Aαβ ∂ α ∂ w = f (z) in Dn

(5.8)

|α|≤nm |β|≤nm

subject to the conditions α

∂ α ∂ w(z) = γα (z) on ∂Dn

(5.9)

where |α| + |β| < 2nm and Aαβ , f ∈ Lp (Dn ), γα are continuous functions. We know that the solution of the Riquier problem for higher-order polyharmonic equation is given by (5.5) ˜ Dn ,m f (z) w(z) = G having zero boundary conditions. In order to find the solution of (5.8) subject to homogeneous boundary conditions, we will convert the problem into an integral equation. We assume that ˜ Dn ,m g(z) = w(z) = G



n 1 Dn j =1

GDj ,m (zj , ζj )g(ζ )dξ dη

(5.10)

provides the solution of (5.8) subject to the homogeneous conditions for some function g ∈ Lp (Dn ). In order to determine g we substitute (5.10) in (5.8) which gives g(z) +



α β

Aαβ ∂ ∂

|α|≤nm |β|≤nm,|α|+|β| 0, κ > 0. For the physical meaning of these parameters, we refer the reader to [24]. It is known that the Dirichlet external problem for (1.1) admits unique solution for any value of the parameter ω2 , provided that the thermoelastic radiation conditions are satisfied at infinity. This is not the case for the internal Dirichlet problem: there is an eigenfrequency spectrum, i.e. a discrete set of values of ω2 , which depends on the domain and we denote by ", for which the uniqueness theorem does not hold. In the particular case ω = 0 we have the equations of thermoelasto-static state μu + (λ + μ)∇ div u − γ ∇u4 = 0, u4 = 0.

(1.2)

The main result we obtain in the present paper is that, if ω2 ∈ / ", the system of exponential polynomials which are solutions of system (1.1) is complete in [Lp (∂)]4, where  is a bounded domain in R3 with a Lyapunov boundary such that R3 \  is connected. An exponential polynomial is a vector Q(x)eiζ ·x , where Q = (Q1 , Q2 , Q3 , Q4 ) has polynomial components and ζ is a constant vector in C3 . In the particular case of the thermoelasto-static state we prove the completeness in [Lp (∂)]4 of the polynomial solutions of system (1.2). The paper is organized as follows. Section 2 is devoted to a description of the concept of completeness in the sense of Picone and to review some known results. In particular, we describe necessary and sufficient conditions for the completeness of polynomial solutions in the case of scalar partial differential operators with constant coefficients.

Completeness Theorems

95

The thermoelastostatics (1.2) is considered in Sect. 3. Here all the elastothermostatics polynomials are determined and their completeness in [Lp (∂)]4 is established. In Sect. 4 it is proved that the class of exponential polynomials solutions of (1.1) is complete in [Lp (∂)]4 , provided that ω2 is not an eigenfrequency. Finally in Sect. 5 we consider multiple connected domains. In this case the system of elastothermostatics polynomials is not complete in [Lp (∂)]4 and we determine the closure (in Lp norm) of the linear space generated by this system.

2 The Completeness in the Sense of Picone Everybody knows that Theorem 2.1 The trigonometric system {cos kϑ, sin kϑ}

(k = 0, 1, 2, . . . .)

is complete in the space Lp (−π, π) (1 ≤ p < ∞). If we consider the usual polar coordinates in the plane z = !eiϑ and denote the unit disk {z ∈ C |z| < 1} by D, Theorem 2.1 can be rephrased as follows Theorem 2.2 The system {Re zk , I m zk }

(k = 0, 1, 2, . . .)

(2.1)

is complete in the space Lp (∂D) (1 ≤ p < ∞). We note that the system (2.1) is nothing but the system of harmonic polynomials. This means that every harmonic polynomial in two real independent variables can be written as a finite linear combination of elements of system (2.1). Theorem 2.2 suggests immediately the following more general question: Let  be a bounded domain in R2 ; is the system of harmonic polynomials (2.1) complete in the space Lp (∂) ? The very first person who considered such a problem was the physicist Marcel Brillouin [4]. He proposed some approximation methods for very particular domains, hinging on completeness properties on the boundary. He did not give any proof. He wrote: “J’espére que quelques mathématiciens attaqueront cette question délicate”. The first one who posed the problem in a precise and more general way was Mauro Picone [32]. Let E be a partial differential operator Eu =

 |α|≤2m

aα (x)D α u

96

A. Cialdea

defined in Rn and and let B1 , . . . , Bs some partial differential operators defined on the boundary # of a bounded domain . Let us suppose that there exists a solution of the problem Eu = 0

in 

Bh u = fh

(2.2)

on # (h = 1, . . . , s)

if and only if (f1 , . . . , fs ) satisfies a finite number of compatibility conditions s   h=1 #

fh ψh(k) dσ = 0,

k = 1, . . . , μ,

(ψ (k) = (ψ1(k) , . . . , ψn(k) ) being μ linearly independent vectors, depending on the operators E, Bh and on the domain ) that is to say that problem (2.2) is an index problem. Let us denote by {ωk } a particular sequence of solutions of the equation Eu = 0 in A, where A is a domain such that  ⊂ A. The problem posed by Picone is to find under which conditions the system {(B1 ωk , . . . , Bs ωk )} is complete in the space -

. s     (k) (v1 , . . . , vs ) ∈ [Lp (#)]s  vh ψh dσ = 0, k = 1, . . . , μ . h=1 #

I have considered the Lp -norm, but of course the problem of completeness can be stated with respect to different norms as well. It is clear that in the particular case   n = 2,  = D, A = R2 , E = , s = 1, Bu = u

∂D

{ω} = {Re zk , I m zk | k = 0, 1, 2, . . .} the completeness in the sense of Picone is given by Theorem 2.2. Analogously, if we take B = {x ∈ Rn | |x| < 1},   n ≥ 3,  = B, A = Rn , E = , s = 1, Bu = u

∂B

and {ω} the system of harmonic polynomials in n variables:  |x|h Yhs

x |x|

 (s = 1, . . . ,

(2h + n − 2)(h + n − 3)! , h = 0, 1, . . .) (n − 2)! h! (2.3)

Completeness Theorems

97

{Yhs } being the system of spherical harmonics, the completeness in the sense of Picone coincides with the known completeness of spherical harmonics. It must be said that the completeness is usually considered in a different way. Let us consider a general operator with complex-valued coefficients Eu =

m 

aα (x) D α u,

aα ∈ C ∞ (Rn )

(2.4)

|α|=0

which we suppose to be elliptic (in the sense of Petrowski)  α(x) ξ α = 0 ∀ x ∈ Rn , ξ ∈ Rn \ {0}. |α|=m

Let K be a compact subset of Rn . Denote by (K) the space of functions which are continuous on K and are C ∞ in its interior, where they satisfy the equation Eu = 0, i.e. (K) = {f ∈ C(K) ∩ C ∞ (K \ ∂K) | Ef = 0 in K \ ∂K}. We equip this space with the uniform norm f  = max |f (x)|. x∈K

Suppose now that A is a domain containing K and let S ⊂ C ∞ (A) be a sequence of particular solutions of the equation Eu = 0 in the domain A. The problem is to see when S is complete in (K). In the very particular case n = 2, E = 1/2(∂x + i∂y ), S given by the complex polynomials, the celebrated Mergelyan Theorem gives a complete answer: Theorem 2.3 (Mergelyan) The system S is complete in (K) if and only if R2 \ K is connected. Even if a definite answer like the Mergelyan one is not known for the partial differential operators (2.4), several general results are available. We mention, in particular, the important contributions of Lax [25], Malgrange [27] and Browder [5]. These results are connected to another classical theorem due to Runge (see, e.g., [33]). The completeness in the sense of Picone is much more sophisticated than the completeness in the sense of Mergelyan. Let us consider a simple, but significant example. Consider the Dirichlet problem for Laplace equation: u = 0 in  (2.5) u=ϕ on # where ϕ is a given function belonging to C 0 (#) (Lp (#)).

98

A. Cialdea

If we know that the system of harmonic polynomials {ω} is complete in C 0 (#) we can find a sequence {ωn } of harmonic polynomials such that ωn − ϕC 0 (#) → 0 (ωn − ϕLp (#) → 0). Because of well known results, the sequence {ωn } will be convergent also in C 0 () (in Lp ()). It is clear that the limit of the sequence {ωn } is the solution of the Dirichlet problem (2.5). This shows that the completeness in the sense of Picone is a deeper result than the Mergelyan type theorems. As Fichera notes in [23], the completeness in the sense of Picone is equivalent to the completeness in the sense of Mergelyan plus existence theorem. Moreover, as Fichera writes in [23], the completeness in the sense of Picone is, in the case p = 2, particularly useful for applications. There are two methods for the numerical solutions of the BVP’s for harmonic functions which, in fact, are founded on the completeness properties of the harmonic polynomials in the case p = 2. We refer to [23, pp. 304–305] for a description of these numerical methods. The first completeness theorem in the sense of Picone was proved by Fichera [20]. He proved the following results concerning the Dirichlet, the Neumann and the mixed problem for Laplace equation (Lp (#)),

Theorem 2.4 (Fichera [20]) Let  is a bounded domain of Rn with a C 2 boundary # and such that Rn \ is connected. Denoting by {ωk } the sequence of homogeneous harmonic polynomials we have: (i) {ωk } is complete in L2 (#); + (ii) {∂ν ωk } is complete in {v ∈ L2 (#) | # v dσ = 0};    (iii) {(ωk  , ∂ν ωk  )} is complete in L2 (#1 )×L2 (#2 ), where #1 and #2 are two #1

#2

measurable susbsets of # with positive hypersurface measure, #1 ∩ #2 = ∅ and #1 ∪ #2 = #. The proof given by Fichera can be extended to the case of Lp -norms (1 ≤ p < ∞) without difficulty in the cases (i) and (ii). As far as the mixed problem is concerned (case (iii)), the situation is different. It is clear that the completeness for p = 2 implies the completeness for 1 ≤ p < 2, but for p > 2 this is still an open problem. After Fichera’s results, several completeness theorems have been obtained for particular partial differential equations. We mention the biharmonic equation [3, 18, 30], the elasticity system [10, 19, 21], the heat equation [26], the Helmholtz equation [29], the Stokes system [15] and general 2nd order elliptic equations [2, 8, 22, 31]. We also mention [9] where the Laplacian in any number of variables is considered and completeness theorems for the oblique derivative problem are proved. All of these results are proved on smooth boundary, namely on Lyapunov or C 1 boundaries. Very few results are known on non smooth boundaries (see [6, 7, 16, 17]).

Completeness Theorems

99

Quite recently completeness theorems in the sense of Picone have been obtained for the Dirichlet problem for general elliptic (scalar) equations of any order with constant coefficients. Let us consider a scalar operator 

Eu =

aα D α u,

aα ∈ R

(2.6)

|α|≤2m

which we suppose to be elliptic Q(ξ ) =



aα ξ α > 0,

∀ ξ ∈ Rn \ {0}.

|α|=2m

Moreover we require a(0,...,0) = 0. This is the necessary and sufficient condition for the existence of polynomial solutions of the equation Eu = 0. Denote by {ωk } a system of polynomial solutions of Eu = 0 (i.e. a sequence of polynomial solutions of the equation Eu = 0 such that any polynomial solution of the same equation can be written as a finite linear combination of ωk ). Let  be a bounded domain of Rn such that Rn \  is connected. Consider the Dirichlet problem in  for the operator E: Eu = 0 ∂νh u

in 

= fh

on #

(h = 0, . . . , m − 1)

Consider at first an elliptic operator with no lower order terms: Eu =



aα D α u,

aα ∈ R.

(2.7)

|α|=2m

Theorem 2.5 ([11]) Let E be the elliptic operator (2.7) and {ωk } be a system of polynomial solutions of the equation Eu = 0. Let  be a bounded domain of Rn such that its boundary # is C 1 and Rn \  is connected. The system {(ωk , ∂ν ωk , . . . , ∂νm−1 ωk )} is complete in [Lp (#)]m (1 ≤ p < ∞). In the particular case of the polyharmonic operator E = m , this result was previously proved in [14]. Theorem 2.5 shows that the completeness of the Dirichlet data of a system of polynomial solutions of an elliptic equation with constant coefficients always holds, provided that there are no lower order terms. If lower order terms are present, the

100

A. Cialdea

question is much more delicate. In order to describe the result in this situation, let us rewrite operator (2.6) in the form Eu =

0,m 

(−1)|p| apq D p D q u,

apq ∈ R, a00 = 0.

(2.8)

|p|,|q|

As usual, Q denotes the characteristic polynomial Q(ξ ) =

0,m 

(−1)|p| apq ξ p+q

(2.9)

|p|,|q|

and B(u, v) is the bilinear form associated to the operator E: B(u, v) =

0,m 

 D q u D p v dx.

apq

|p|,|q|



We suppose that the following Gårding inequality holds B(u, u) ≥ Cu2H m () ,

˚∞ (), ∀u∈C

(2.10)

˚∞ () denotes the space of compactly supported functions of C ∞ (). where C It is well known that (2.10) implies ellipticity, but it is not equivalent.1 Under condition (2.10) existence and uniqueness results for the Dirichlet problem hold (see, e.g., [28, p. 503]). Theorem 2.6 ([12]) Let E be the operator (2.8) satisfying Gårding inequality (2.10). Denote by {ωk } a system of polynomial solutions of the equation Eu = 0. Let  be a bounded domain of Rn such that its boundary # is C 1 and Rn \  is connected. The system {(ωk , ∂ν ωk , . . . , ∂νm−1 ωk )} is complete in [Lp (#)]m (1 ≤ p < ∞) if and only if all the irreducible factors over C of the characteristic polynomial (2.9) vanish at the origin. This Theorem implies that there are elliptic operators with real constant coefficients having polynomial solutions for which an existence and uniqueness theorem holds for the Dirichlet problem, but the corresponding completeness property in the sense of Picone does not. A simple example of such operators is E = 2 − . Theorem 2.6 was extended to the uniform norm in [13].

1 The

Gårding inequality which is equivalent to ellipticity is the following B(u, u) ≥ Cu2H m () − cu2L2 () ,

˚∞ (). ∀u∈C

Completeness Theorems

101

3 Elastothermostatics In this section we consider the three-dimensional system μu + (λ + μ)∇ div u − γ ∇u4 = 0, u4 = 0.

(3.1)

First, let us construct a system of polynomial solutions of this system, i.e. of vectors whose components are polynomials. We shall call them elastothermostatics polynomials. Lemma 3.1 If U = (u, u4 ) is a solution of (3.1), then div(u) = 0,

curl(u) = 0.

(3.2)

Proof Taking the divergence of the vector μu + (λ + μ)∇ div u − γ ∇u4 we get (λ + 2μ)(div u) − γ u4 = 0. Since u4 is a harmonic function, we get (λ + 2μ)(div u) = 0 and the first equation in (3.2) is proved. If we derive with respect to xj the i-th component of μu + (λ + μ)∇ div u − γ ∇u4 we find μ∂j ui + (λ + μ)∂j i div u − γ ∂j i u4 = 0. This implies μ(∂j ui − ∂i uj ) = 0 and this ends the proof.

 

It is clear that a constant vector satisfies (3.1). As far as homogeneous polynomials of higher degree, we have the following result Lemma 3.2 Let u = (u1 , u2 , u3 ) and u4 be homogeneous polynomials of degree k and k − 1 respectively (k ≥ 1). The vector U = (u, u4 ) satisfies system (3.1) if, and only if, u4 = 0 and there exist a harmonic polynomial α, homogeneous of degree k, such that u(x) = α(x) −

|x|2 ∇[(λ + μ) div α − γ u4 ] . 2[μ(2k − 1) + (λ + μ)(k − 1)]

(3.3)

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A. Cialdea

Proof Let us suppose that U = (u, u4 ) satisfies system (3.1). Obviously u4 = 0. Moreover Lemma 3.1 implies that 2 u = 0. Because of the classical Almansi theorem [1], there exist two harmonic vector functions α, β such that u(x) = α(x) + |x|2 β(x).

(3.4)

Since U is homogeneous of degree k, α and β will be homogeneous of degree k and k−2 respectively. In view of Euler theorem on homogeneous functions, we have ui = 6βi + 4xj ∂j βi = 2(2k − 1)βi .

(3.5)

Lemma 3.1 shows that div β = 0. Moreover ∂j ui = 2(2k − 1)∂j βi and then, in view of (3.2), curl β = 0. From div β = 0 we get div u = div α + 2xi βi . Therefore ∂j (div u) = ∂j (div α) + 2βj + 2xi ∂j βi = ∂j (div α) + 2(k − 1)βj ,

(3.6)

since curl β = 0 and Euler theorem imply xi ∂j βi = xi ∂i βj = (k − 2)βj . From (3.5) and (3.6) it follows μu + (λ + μ)∇ div u − γ ∇u4 = 2μ(2k − 1)β + (λ + μ)[∇(div α) + 2(k − 1)β] − γ ∇u4 = 2[μ(2k − 1) + (λ + μ)(k − 1)]β + (λ + μ)∇(div α) − γ ∇u4 . We note that the coefficient [μ(2k − 1) + (λ + μ)(k − 1)] cannot be zero. Indeed, if it were zero, then k=

2μ + λ 3μ + λ

and this is impossible, because k is an integer.

Completeness Theorems

103

The vector U satisfying system (3.1), we find β=

1 ∇[−(λ + μ) div α + γ u4 )] . 2[μ(2k − 1) + (λ + μ)(k − 1)]

This together with (3.4) gives (3.3). The vice-versa can be proved by a straightforward calculation.

 

Let us denote by {ωhs } (s = 1, . . . , 2h + 1, h = 0, 1, . . .) a complete system of harmonic polynomials (see (2.3)). The next Theorem determines all the elastothermostatics polynomials. Theorem 3.3 Let u = (u1 , u2 , u3 ) and u4 be homogeneous polynomials of degree k and k − 1 respectively (k ≥ 1). Denote by τk the constant 2[μ(2k − 1) + (λ + μ)(k − 1)]. The vector U = (u, u4 ) satisfies system (3.1) if, and only if, U is a linear combination of the following 2(4k + 1) polynomials:  (λ + μ) 2 (λ + μ) 2 (λ + μ) 2 |x| ∂11 ωks , − |x| ∂21 ωks , − |x| ∂31 ωks , ωks − τk τk τk  (λ + μ) 2 (λ + μ) 2 (λ + μ) 2 |x| ∂12 ωks , ωks − |x| ∂22 ωks , − |x| ∂32 ωks , − τk τk τk  (λ + μ) 2 (λ + μ) 2 (λ + μ) 2 − |x| ∂13 ωks , − |x| ∂23 ωks , ωks − |x| ∂33 ωks , τk τk τk   γ γ γ 2 2 2 |x| ∂1 ωk−1,p , |x| ∂2 ωk−1,p , |x| ∂3 ωk−1,p , ωk−1,p τk τk τk

 0 ,  0 ,  0 ,

(3.7) (s = 1, . . . , 2k + 1; p = 1, . . . , 2k − 1). Proof The result follows immediately from Lemma 3.2. We have just to remark that, since in R3 there are 2k + 1 homogeneous harmonic polynomials of degree k, the number of polynomials given by (3.7) is 3(2k + 1) + (2k − 1) = 8k + 2.   Let us denote by {j } the system of elastothermostatics polynomials, i.e. the system constituted by the vectors (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) and all the polynomials given by (3.7) (s = 1, . . . , 2k + 1; p = 1, . . . , 2k − 1; k = 1, 2, . . .), ordered in one sequence. We remark that the polynomials written in the first three lines in (3.7) are just the polynomial solution of the linear elasticity system μu + (λ + μ)∇ div u = 0 (see [10, 21]). From now on  is a bounded domain in R3 with a Lyapunov boundary C 1+h and such that R3 \  is connected. Theorem 3.4 Let 1 ≤ p < ∞. The system {j } is complete in [Lp (#)]4 .

104

A. Cialdea

Proof Let F = (f1 , f2 , f3 , f4 ) ∈ [Lq (#)]4 (q = p/(p − 1), if p > 1; q = ∞, if p = 1) such that 4  

fh hj dσ = 0,

j = 1, 2, . . . ,

(3.8)

h=1 #

where hj is the h-th component of j . In particular we have 3  

fh hj dσ = 0

h=1 #

for all the polynomials written in the first three lines of (3.7). Since these polynomials are complete in [Lp (#)]3 (see [21, Th.XXII]), we get f1 = f2 = f3 = 0. Thus  f4 4j dσ = 0 #

for any j . In particular (see (3.7)) we have  f4 ωdσ = 0 #

for any harmonic polynomial ω. The system of harmonic polynomial being complete (see [20]), we get also f4 = 0. We have proved that conditions (3.8) imply F = 0. In view of the Hahn-Banach theorem, the system {j } is complete in [Lp (#)]4 .   We note that this completeness Theorems implies a kind of Runge property for system (3.1). Indeed we have: Theorem 3.5 Let A be a domain such that R3 \ A is connected. Let U = (u, u4 ) ∈ [C ∞ (A)]4 be a solution of system (3.1) in A. For any compact set K ⊂ A there exists a sequence Vn of elastothermostatics polynomials such that Vn → U uniformly in K. Proof Let  be a bounded domain with smooth boundary such that K ⊂ ,  ⊂ A and R3 \  is connected. Because of the completeness property proved in Theorem 3.4 there exists a sequence Vn of polynomial solutions of (3.1) such that Vn → U in [L2 (∂)]4. In view of classical results concerning elliptic systems, this implies that Vn → U uniformly in K.  

Completeness Theorems

105

4 The Steady Oscillation Equation Consider now the three-dimensional steady oscillation equation ⎧ ⎨μu + (λ + μ)∇ div u − γ ∇u4 + ! ω2 u = 0, ⎩u4 + i ω u4 + i ω η div u = 0. κ

(4.1)

It is easy to check that if ω = 0 there are no polynomial solutions. Let us denote by % the class of exponential polynomial solutions of (4.1) The aim of this section is to prove that % is complete in [Lp (#)]4 , provided that 2 ω ∈ / ". Let us denote by (x, ω) = {kj (x, ω)} the fundamental solution of (4.1) given by   3 6  δkj ∂2 (1 − δk4 )(1 − δj 4 ) δ3l − αl 2πμ ∂xk ∂xj l=1 3 4 ∂ ∂ − γ δj 4 (1 − δk4 ) +βl iωηδk4(1 − δj 4 ) ∂xj ∂xk 7 eiλl |x| , (k, j = 1, . . . 4) +δk4 δj 4 γl |x|

kj (x, ω) =

(4.2)

where αl = βl =

(−1)l (1 − iωk −1 λ−2 l )(δ1l + δ2l ) 2π(λ +

(−1)l (δ1l + δ2l ) , 2π(λ + 2μ)(λ22 − λ21 )

2μ)(λ22

γl =

−

λ21 )

−

δ3l , 2π!ω2

(−1)l (λ2l − kl2 )(δ1l + δ2l ) 2π(λ22 − λ21 )

,

(l = 1, 2, 3)

and λ21 , λ22 and λ23 are determined by λ21 + λ22 =

! ω2 iωηγ iω + + , κ λ + 2μ λ + 2μ

λ21 λ22 =

iω ! ω2 , κ λ + 2μ

λ23 =

! ω2 μ (4.3)

(see [24, pp. 94–97]). We remark that (4.2) makes sense if λ21 = λ22 . If λ21 = λ22 the fundamental solution can be obtained by passing to the limit. We omit the details. If γ = 0 the solutions of (4.3) are complex numbers. In this case we choose them in such a way I m λj > 0 (j = 1, 2).

106

A. Cialdea

The matrix (x, ω) is not symmetric and each of its columns satisfies system (4.1) for any x = 0. The rows do not satisfies such a system, but the column of the matrix ∗kj (x, ω) = j k (−x, ω),

(k, j = 1, 2, 3, 4)

(4.4)

satisfy the associated system ⎧ ⎨μu + (λ + μ)∇ div u − i ω η∇u4 + ! ω2 u = 0, ⎩u4 + i ω u4 + γ div u = 0 κ

(4.5)

for any x = 0. We recall here the following theorem proved in [24, p. 138] Theorem 4.1 Let U = (u, u4 ) be a solution of system (4.1). It admits the representation U = (u(1) + u(2), u4 ) where ( + λ21 )( + λ22 )u(1) = 0, ( + λ23 )u(2) = 0,

curl u(1) = 0,

div u(2) = 0,

( + λ21 )( + λ22 )u4 = 0, the constants λ2j (j = 1, 2, 3) being determined by (4.3). We also recall the thermoelastic radiation conditions for system (4.1). The vector U , solution of (4.1) in the exterior of a bounded domain, is said to satisfy the radiation conditions at infinity if ⎧ (1) −1 ⎪ ⎪ ⎨u (x) = o(R ), u(2) (x) = O(R −1 ), ⎪ ⎪ ⎩u (x) = o(R −1 ), 4

∂u(1) (x) = O(R −2 ), k = 1, 2, 3 ∂xk ∂u(2) (x) − i λ3 u(2) = o(R −1 ), ∂R ∂u4 (x) −2 ∂xk = O(R ), k = 1, 2, 3,

where u(1) and u(2) are given by Theorem 4.1.

(4.6)

Completeness Theorems

107

is a regular solution2 of system (4.1) (of system (4.5)) in Theorem 4.2 If U (U) 3 |# = 0) and satisfies the thermoelastic radiation R \ , such that U |# = 0 (U ≡ 0) in R3 \ . conditions (4.6) at infinity, then U ≡ 0 (U Proof For the proof of uniqueness for system (4.1), we refer to [24, p. 143–144]. The same proof works for the system (4.5), after we observe that the radiation conditions for system (4.5) coincide with (4.6), since the constants λj are still given by (4.3).   Let 1 be a ball containing  in its interior. Let U(1 ) be the class of potentials U = (u1 , u2 , u3 , u4 ),  uh (x) = hj (x − y, ω)ϕj (y) dσy (h = 1, 2, 3, 4) #1

with ϕj varying in C 0 (#1 ). If U ∈ U(1 ), then U ∈ C ∞ (1 ) ∩ C 0 (1 ) and U satisfies system (4.1). The next result shows that the restrictions on # of elements of U(1 ) are complete in [Lp (#)]4 . / " and 1 ≤ p < ∞. The system Theorem 4.3 Let ω2 ∈  {U # | U ∈ U(1 )}

(4.7)

is complete in [Lp (#)]4 . Proof We have to show that, if β ∈ [Lq (#)]4 (q = p/(p − 1), if p > 1; q = ∞, if p = 1) and  β U dσ = 0 (4.8) #

for any U ∈ U(1 ), then β = 0. Conditions (4.8) mean 



hj (x − y, ω)ϕj (y) dσy = 0

βh (x) dσx #

#1

for any ϕj ∈ C 0 (#1 ), i.e. 

 βh (x) hj (x − y, ω) dσx = 0 .

ϕj (y) dσy #1

2

#

By a regular solution we mean a vector to which one can apply Green formulas. For example, a simple layer potential with densities in Lp (#) is regular in this sense.

108

A. Cialdea

The arbitrariness of ϕj implies  βh (x) hj (x − y, ω) dσx = 0,

(j = 1, 2, 3, 4)

#

for any y ∈ #1 . In view of (4.4), we may write  #

βh (x) ∗j h (y − x, ω) dσx = 0,

(j = 1, 2, 3, 4)

for any y ∈ #1 . Let  vj (y) = #

βh (x) ∗j h (y − x, ω) dσx .

(4.9)

The vector V = (v1 , v2 , v3 , v4 ) satisfies the associated system (4.5). Since the fundamental solution , and then ∗ , satisfies the radiation conditions (4.6), the uniqueness theorem 4.2 shows that V ≡ 0 in R3 \ 1 (see also footnote 2). The vector V being solution of the system (4.5) in R3 \ , V is there analytic. As V vanishes in R3 \ 1 and R3 \  is connected, V ≡ 0 in R3 \ . Because of well known properties of the simple layer potentials, we have also  #

βh (x) ∗j h (y − x, ω) dσx = 0,

(j = 1, 2, 3, 4)

for almost any y ∈ #. Therefore the vector V is solution of the Dirichlet problem for the associated system (4.5) in  with zero data on the boundary. Since ω2 ∈ /" we know that we have the uniqueness for the Dirichlet problem for system (4.1). On the other hand the eigenfrequencies of systems (4.1) and (4.5) are the same (see [24, p. 548]) and then V = 0 in . We have thus proved that  #

βh (x) ∗j h (y − x, ω) dσx = 0,

(j = 1, 2, 3, 4)

for all y ∈ R3 \ # (j = 1, 2, 3, 4). Let us consider the matrix operator R = {Rij }, where {Rij } for i, j = 1, 2, 3 coincides with the stress operator of linear elasticity and R4j = Rj 4 = δj 4 ∂/∂ν (ν being the external unit normal vector on #). Since ∗ (x, ω) − ∗ (x, 0) = O(1) and keeping in mind the expression of ∗ (x, 0) (see [24, p. 97]) we get RV+ − RV− = 2β . This implies β = 0 and the theorem is proved.

 

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Theorem 4.4 Let ω2 ∈ / " and 1 ≤ p < ∞. The class % of exponential polynomial solutions of (4.1) is complete in [Lp (#)]4 . Proof Let F be a given vector in [Lp (#)]4 . Theorem 4.3 shows that, for any ε > 0, there exists a U ∈ U(1 ) F − U [Lp (#)]4 < ε .

(4.10)

By a Malgrange result [27, Prop. 7, p. 299], we can find a sequence of exponential polynomials solutions of (4.1) converging to U in C ∞ (1 ). Therefore, there exists an exponential polynomial P satisfying system (4.1) such that U − P [Lp (#)]4 < ε .

(4.11)  

Inequalities (4.10) and (4.11) imply the result.

∈ " the class It is clear from the proof of Theorems 4.3 and 4.4 that if % is not complete in [Lp (#)]4 . More precisely, as in the first part of the proof of Theorem 4.3, we have that β satisfies conditions (4.8) if, and only if, the potential (4.9) is an eigensolution of the associated system (4.5) in  with zero data on the boundary. Since there are only a finite number of eigensolutions of this problem, we have that the set of such vectors β is a finite dimensional space spanned by a finite numbers of vectors β1 , . . . , βm . This means that the closure of the system (4.7) is given by the vectors V ∈ [Lp (#)]4 such that ω2

 V · βj dσ = 0,

j = 1, . . . , m.

(4.12)

#

As in Theorem 4.4, this implies that the closure of % is given by the same space of vectors V ∈ [Lp (#)]4 satisfying conditions (4.12). In order to construct the exponential polynomials satisfying (4.1), it could be useful the following lemma (see [24, Lemma 5.1, p. 870]) Lemma 4.5 Let us suppose that the vectors v (k) and the scalar functions vk , k = 1, 2, are related by v (1) =

γ ω2

− λ21 (λ + 2μ)

∇v1 ,

v (2) =

γ ω2

−

λ22 (λ + 2μ)

∇v2 ,

(4.13)

and they satisfy the equations ( + λ2k )v (k) = 0,

(4.14)

( + λ2k )vk = 0

(4.15)

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A. Cialdea

(k = 1, 2). Let u(2) be a solution of the equations ( + λ23 )u(2) = 0,

div u(2) = 0.

(4.16)

Then the vector U = (u(1) + u(2), u4 )

(4.17)

v (1) − v (2) , λ22 − λ21

(4.18)

where u(1) =

u4 =

v1 − v2 λ22 − λ21

is a solution of system (4.1). Thanks to this decomposition, the problem of finding exponential polynomials solutions of (4.1) is reduced to the problem of finding exponential polynomials solutions of the scalar Helmholtz equations (4.15) and divergence free exponential polynomials solutions of the vector Helmholtz equations (4.16). Indeed suppose that v1 , v2 and u(2) satisfy (4.15) and (4.16); define v (1) , v (2) , u(1) and u4 by (4.13) and (4.18). Note that (4.14) are satisfied. In view of Lemma 4.5 the vector U (4.17) is solution of system (4.1). Finally we remark that using the potential theory results contained in [12] one can prove the completeness results 3.4, 4.3 and 4.4 for a domain  with a C 1 boundary.

5 Multiple Connected Domains Saying that  is a multiple connected domain, or - more precisely - an (m + 1)connected domain, we mean that  is an open connected set of the form m 8

 = 0 \

j ,

(5.1)

j =1

where each j (j = 0, . . . , m) is a bounded domain of Rn with connected boundaries #j ∈ C 1,λ (λ ∈ (0, 1], j = 0, . . . , m) and such that j ⊂ 0 and j ∩ k = ∅,

j, k = 1, . . . , m, j = k.

For such domains the sets of solutions we have previously considered are not complete. In this section we determine the closure of the space generated by such systems. What comes out is that such a closure can be described by means of the boundary values of the solutions of some transmission problems. The next results,

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which we consider for simplicity only in the case of Elastothermostatics (3.1), clarify this statement. Let us introduce some notations. Denote by B the matrix differential operator the associated operator given by the left hand side of (3.1) and by B = (μu + (λ + μ)∇ div u, v4 + γ div v) , BU by R the boundary operator RU = (T u − γ νu4 , ∂u4 /∂ν), (T being the stress operator of linear elasticity) and finally = (T u, ∂u4 /∂ν) . RU It is known that the following Gauss–Green formula holds 

− V · BU )dx = (U · BV 



− V · RU )dσ (U · RV

(5.2)

#

for any smooth vectors U , V (see [24, p. 532]). Theorem 5.1 Let 1 ≤ p < ∞. Let  be the (m + 1)-connected domain (5.1). Suppose F = (f, f4 ) = (f1 , f2 , f3 , f4 ) ∈ [Lq (#)]4 (q = p/(p − 1), if p > 1; q = ∞, if p = 1) is such that  fh hj dσ = 0,

j = 1, 2, . . . .

(5.3)

#

Then there exists V = (v, v4 ) solution of the associated system3 -

μv + (λ + μ)∇ div v = 0, v4 + γ div v = 0

(5.4)

in  satisfying the boundary condition V = 0,

on #0 ,

(5.5)

3 This and the other BVPs are considered in the spaces of vectors which can be represented as simple layer potentials with Lp densities. See [24] for more details.

112

A. Cialdea (k)

such that, denoting by W (k) = (w(k) , w4 ) the solution of the BVP ⎧ (k) (k) ⎪ ⎪ ⎨μw + (λ + μ)∇ div w = 0, in k w4(k) + γ div w4(k) = 0, ⎪ ⎪ ⎩W (k) = V ,

in k

(5.6)

on #k ,

(k = 1, . . . , m), we have + V F =R − Wk + V − R F =R

on #0 , on #k , k = 1, . . . , m,

(5.7)

Conversely, if V is a solution of the system (5.4) satisfying condition (5.5) and Wk (k = 1, . . . , m) are solutions of the BVPs (5.6), then the vector F given by (5.7) is orthogonal to the system {j }, i.e. (5.3) hold. Proof Let us denote by (x) = {kj (x)} the fundamental solution of system (5.4). It is given by kj (x) = (1 − δk4 )(1 − δj 4 )kj (x) +

δk4 δj 4 1 γ δj 4 xk (1 − δk4 ) + , 4π |x| 2π |x|

where {kj (x)} is the Kelvin matrix (see [24, p. 96–97]). The matrix (x) can be seen as the limit lim (x, ω) .

ω→0

Let ∗ (x) = {∗kj (x)}, where ∗kj (x) = j k (−x). The matrix ∗ (x) is the fundamental solution of the operator B. Let us show that conditions (5.3) imply that  #

∗hj (x − y) fj (y) dσy = 0,

∀x ∈ / 0 .

(5.8)

/ 0 and 1 ≤ h ≤ 4. Let A be a domain such that 0 ⊂ A, x0 ∈ / A and Fix x0 ∈ R3 \ A is connected. The vector (∗h1 (x0 − y), ∗h2 (x0 − y), ∗h3 (x0 − y), ∗h4 (x0 − y)) = (1h (y − x0 ), 2h (y − x0 ), 3h (y − x0 ), 4h (y − x0 )) is, as a function of y, a smooth solution of system (3.1) in A. Theorem 3.5 shows that there exists a sequence of elastothermostatics polynomials Vn such that Vn (y) → (1h (y − x0 ), 2h (y − x0 ), 3h (y − x0 ), 4h (y − x0 ))

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uniformly for y ∈ 0 . Therefore  #

∗hj (x0 − y) fj (y) dσy = lim



n→∞ #

Vn · F dσ = 0

because of (5.3), and (5.8) is proved. Set  1 vh (x) = ∗ (x − y) fj (y) dσy , 2 # hj

x ∈ .

The vector V = (v1 , v2 , v3 , v4 ) is solution of the adjoint system (5.4). In view of (5.8), V satisfies condition (5.5). The first equation in (5.7) follows from the known jump relations for the thermoelastopotentials (see [24, formulas (2.33)– (2.34), p.541]). On the other hand, for any 1 ≤ k ≤ m fixed, the potential (k) wh (x)

1 = 2

 #

∗hj (x − y) fj (y) dσy ,

x ∈ k ,

is solution of the BVP (5.6) in k and F satisfies the condition (5.7) on #k . Conversely, suppose (5.4)–(5.7) hold. Keeping in mind the Gauss-Green formula (5.2), we have 

 fh hj dσ = #

 #

 #

F · j dσ + #0

V · R+ j dσ −

k=1 #k

W

(k)

F · j dσ =

k=1 #k

+ V dσ − j · R m  

m  

m   k=1 #k

− W (k) dσ = j · R

· R− j dσ =

m   k=1 #k

V · (R+ j − R− j )dσ.

Since j are smooth vectors on R3 , we have R+ j = R− j on each #k and (5.3) follows.   As a corollary we have that the linear space generated by the system {j } can be described as the orthogonal complement of a certain linear space. Indeed we have Theorem 5.2 The closure in [Lp (#)]4 of the linear space generated by the system {j } is constituted by the vectors G ∈ [Lp (#)]4 such that  F ·G=0 #

114

A. Cialdea

for any F ∈ [Lq (#)]4 given by (5.7), with V and W (k) (k = 1, . . . , m) satisfying (5.4)–(5.6). Proof It follows immediately from Theorem 5.1.

 

Finally we mention that a complete system in the (m+1)-connected domain (5.1) can be obtained following the order of ideas introduced in [19, 20]. We omit the details.

References 1. E. Almansi, Sull’integrazione dell’equazione differenziale 2m u = 0. Ann. Mat. (3) 2, 1–51 (1899) 2. L. Amerio, Sul calcolo delle soluzioni dei problemi al contorno per le equazioni lineari del secondo ordine di tipo ellittico. Am. J. Math. 69, 447–489 (1947) 3. R.B. Ancora, Problemi analitici connessi alla teoria della piastra elastica appoggiata. Rend. Sem. Mat. Univ. Padova 20, 99–134 (1951) 4. M. Brillouin, La mèthode des moindres carrès et les èquations aux dèrivèes de la Physique Mathèmatique. Ann. de Physique 6, 137–223 (1916) 5. F.E. Browder, Approximation by solutions of partial differential equations. Am. J. Math. 84, 134–160 (1962) 6. A. Cialdea, Un teorema di completezza per i polinomi biarmonici in un campo con contorno angoloso. Rend. Mat. Appl. (7) 5, 327–344 (1985) 7. A. Cialdea, Teoremi di completezza connessi con equazioni ellittiche di ordine superiore in due variabili in un campo con contorno angoloso. Rend. Circ. Mat. Palermo (2) 34, 32–49 (1985) ∂u 8. A. Cialdea, L’equazione 2 u + a10 (x, y) ∂u ∂x + a01 (x, y) ∂y + a00 (x, y)u = F (x, y). Teoremi di completezza Atti Accad. Naz. Lincei, VIII. Ser., Rend. Cl. Sci. Fis. Mat. Nat. 81, 245–257 (1987) 9. A. Cialdea, Sul problema della derivata obliqua per le funzioni armoniche e questioni connesse. Rend. Accad. Naz. Sci. XL 12, 181–200 (1988) 10. A. Cialdea, Formule di maggiorazione e teoremi di completezza relativi al sistema dell’elasticità tridimensionale. Riv. Mat. Univ. Parma (4) 14, 283–302 (1988) 11. A. Cialdea, Completeness Theorems: Fichera’s fundamental results and some new contributions. Matematiche (Catania) LXII, 147–162 (2007) 12. A. Cialdea, Completeness theorems for elliptic equations of higher order with constant coefficients. Georgian Math. J. 14, 81–97 (2007) 13. A. Cialdea, Completeness theorems in the uniform norm connected to elliptic equations of higher order with constant coefficients. Anal. Appl. (Sing.) 10, 1–20 (2012) 14. A. Cialdea, A. Malaspina, Completeness theorems for the Dirichlet problem for the polyharmonic equation. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 29, 153–174 (2005) 15. A. Cialdea, G. Nino, Completeness theorems for the Stokes system. Preprint 16. M.P. Colautti, Sul problema di Neumann per l’equazione 2 − λcu = f in un dominio piano a contorno angoloso. Mem. Accad. Sci. Torino (3) 4, 1–83 (1959) 17. M.P. Colautti, Teoremi di completezza in spazi hilbertiani connessi con l’equazione di Laplace in due variabili. Rend. Sem. Mat. Univ. Padova 31, 114–164 (1961) 18. G. Fichera, Teoremi di completezza connessi all’integrazione dell’equazione 4 u = f . Giorn. Mat. Battaglini (4) 77, 184–199 (1947) 19. G. Fichera, Sui problemi analitici dell’elasticità piana. Rend. Sem. Fac. Sci. Univ. Cagliari 18, 1–22 (1948)

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20. G. Fichera, Teoremi di completezza sulla frontiera di un dominio per taluni sistemi di funzioni. Ann. Mat. Pura Appl. (4) 27, 1–28 (1948) 21. G. Fichera, Sull’esistenza e sul calcolo delle soluzioni dei problemi al contorno, relativi all’equilibrio di un corpo elastico. Ann. Sc. Norm. Super. Pisa 248, 35–99 (1950) 22. G. Fichera, Alcuni recenti sviluppi della teoria dei problemi al contorno per le equazioni alle derivate parziali lineari. Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali, Trieste, 1954 (Ed. Cremonese, Roma, 1955), pp. 174–227 23. G. Fichera, The problem of the completeness of systems of particular solutions of partial differential equations, in Numerical Mathematics. Symposium on Occasion of the Retirement L. Collatz, Hamburg 1979, vol. 49. International Series of Numerical Mathematics, ed. by R. Ansorge, K. Glashoff, B. Werner (1979), pp. 25–41 24. V. Kupradze, T.G. Gegelia, M.O. Basheleishvili, T.V. Burchuladze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity (North-Holland Publishing Company, Amsterdam, 1979) 25. P. Lax, A stability theory of abstract differential equations and its applications to the study of local behaviours of solutions of elliptic equations. Commun. Pure Appl. Math. 8, 747–766 (1956) 26. E. Magenes, Sull’equazione del calore: teoremi di unicità e teoremi di completezza connessi col metodo di integrazione di M. Picone., I, II. Rend. Sem. Mat. Univ. Padova 21, 99–123, 136–170 (1952) 27. B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution. Ann. Inst. Fourier Grenoble 6, 271–355 (1955–1956) 28. V.G. Maz’ya, T.O. Shaposhnikova, Theory of Sobolev Multipliers. Grundlehren Math. Wiss., vol. 337 (Springer, Berlin, 2009) 29. R.F. Millar, On the completeness of sets of solutions to the Helmholtz equation. IMA J. Appl. Math. 30, 27–38 (1983) 30. C. Miranda, Formule di maggiorazione e teorema di esistenza per le funzioni biarmoniche de due variabili. Giorn. Mat. Battaglini (4) 2(78), 97–118 (1948) 31. C. Miranda, Partial Differential Equations of Elliptic Type. Second revised edition. Ergeb. Math. Grenzgeb, vol. 2 (Springer, New York/Berlin, 1970) 32. M. Picone, Nuovo metodo di approssimazione per la soluzione del problema di Dirichlet. Rend. R. Acc. Lincei (5) 31, 357–359 (1922) 33. E.M. Stein, R. Shakarchi, Complex Analysis. Princeton Lectures in Analysis, II (Princeton University Press, Princeton, 2003)

A Circle Pattern Algorithm via Combinatorial Ricci Flows Dong-Meng Xi, Shi-Yi Lan, and Dao-Qing Dai

Dedicated to Professor Heinrich G.W. Begher on the occasion of his 80th birthday

Abstract A circle pattern is a configuration of circles with a prescribed combinatoric and prescribed intersection angles. Based on the idea of combinatorial Ricci flows, we present an iterative process which converges exponentially fast to radii of circle patterns in the Euclidean and hyperbolic planes. This provides a new and effective method to find the radii of circle patterns. Keywords Triangulation · Circle pattern · Ricci flow · Discrete Dirichlet problem Mathematics Subject Classification (2010) 65N10, 65N30, 52C15, 30G62

1 Introduction A circle pattern in the complex plane or hyperbolic plane is a configuration of circles with a prescribed combinatoric and prescribed intersection angles. In particular, a circle pattern is also called a circle packing when all its intersection angles are equal to zero. The theory of circle patterns is a fast developing field of research on the border of complex analysis and discrete differential geometry (see[3–6, 17, 26]). It has played crucial roles in the construction of polyhedra in hyperbolic 3-space

D.-M. Xi · S.-Y. Lan School of Sciences, Guangxi University for Nationalities, Nanning, People’s Republic of China e-mail: [email protected] D.-Q. Dai () Department of Mathematics, Sun Yat-Sen University, Guangzhou, People’s Republic of China e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_7

117

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[14, 24, 25], in the approximation of analytic functions [8, 12, 16], in development of a discrete analytic function theory [1, 2, 26] and in medical imaging [18, 22], etc. The algorithm of circle pattern is an important aspect of the theory of circle patterns, since applications of circle patterns involve their computations. Thurston [28] proposed an algorithm to find the radii of circle packing in his original proof of the existence of zero curvature circle packing metric. The idea of the algorithm is to adjust iteratively the radius of each circle so that the neighboring circles fit around. A change of any single radius affects most strongly the curvature at that vertex, so the process converges reasonable well. Mohar [23] showed that Thurston’s algorithm converges in polynomial time to the radii of circle packing. According to Thurston’s idea, Collins and Stephenson [11] described an iterative process to approximate numerically the radii of circle packings. Stephenson’s program circlepack [27] constructs circle packings by terms of Thurston’s scheme. Hamilton [15] introduced the 2-dimensional Ricci flow on a compact surface df with a Riemannian metric (#, fij ), which is given by the equation dtij = −2Kfij where K is the Gaussian curvature of the surface #. It is proved by Hamilton [15] and Chow [9] that for any closed surface with any initial Riemannian metric, the solution of the Ricci flow exists for all time, and after normalizing the solution to have a fixed area, the solution converges to a constant curvature metric conformal to the initial metric as time goes to infinity. Chow and Luo [10] investigated the analog of Hamilton’s Ricci flows in the combinatorial setting, which are defined by dri dri dt = −Ki ri in the Euclidean metric and by dt = −Ki sinh ri in the hyperbolic metric respectively, where Ki denotes the curvature of combinatorial surface at vertex vi . They showed that these combinatorial Ricci flows have solutions for all time for any initial metric, and converge exponentially fast to the radii of Euclidean circle pattern and hyperbolic circle pattern, respectively. In the meantime, it was pointed out that this may produce a faster algorithm to find circle packing metric, but such an algorithm was not given in [10]. In 2007 Jin, Luo and Gu [19] employed discrete variational Ricci flow to compute geometric structures of surfaces. Recent developments of discrete Ricci flows can be found in [29–31]. In this paper, we shall use the idea of combinatorial Ricci flows above to describe an algorithm which approximates numerically the radii of circle patterns. Consider a weighted triangulation (T , ") of a closed topological disc with a weight function " : T (1) → [0, π/2], where T (1) denotes the set of edges of T . It is well known that there is uniquely a circle pattern P in the complex plane (respectively, in the hyperbolic plane) which realizes (T , "), when the radii of boundary vertices of T are given in the Euclidean metric (respectively, in the hyperbolic metric). Our gaol here is to establish an iterative process based on the idea of combinatorial Ricci flows and prove that this process converges exponentially fast to the radii of circle pattern P . To the end, consider any label (putative radii) vector R = (r1 , r2 , . . . , rZ ) of (T , ") with a fixed boundary label vector in terms of Euclidean metric and hyperbolic one, where Z denotes the number of vertices in T . Then we may realize each face vi , vj , vk  in (T , ") by a Euclidean triangle and a hyperbolic triangle, respectively. This allows us to compute a curvature K(vi )

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at each interior vertex vi in T . Set ui = ln ri in the Euclidean geometry and ui = ln tanh(ri /2) in the hyperbolic one, then the combinatorial Ricci flows suggest that we define a iterative process with a parameter η > 0 (see (3.5)). Next, we −−→ show that the Jacobian matrix of K(n) = (K1 (n), K2 (n), . . . , KN (n)) in terms −−→ of u(n) = (u1 (n), u2 (n), . . . , uN (n)) is positive definite and its eigenvalues are bounded, where N denotes the number of interior vertices in T . Hence we conclude that the iterative process (3.5) converges exponentially fast to the natural radius → vector − u of P by taking a suitable η > 0. Finally, the details of implementation are described and the efficiency of our algorithm is demonstrated with some examples. This gives a new and effective approach to seek the radii of circle patterns. Compared with Thurston’s algorithm described in [11], the main differences are first that the convergence rates of iterative processes are different. The iterative process in [11] is local linear convergence, while our iterative process is global exponential one. Secondly, the algorithm in [11] is performed by adjusting the radius of only a vertex one at a time, whereas our algorithm generates simultaneously the approximating radii of all vertices one at a time. In addition, our work is also different from one of [19] although the ideas of discrete Ricci flows are both utilized. First, we apply directly combinatorial Ricci flow to construct an iterative algorithm; instead, discrete variational Ricci flow, i.e., minimizing the discrete Ricci energy by Newton’s method, is used in [19]. Thus our algorithm is simpler than one of [19]. Next, the Ricci flow approach in [19] is employed to compute geometric structures of surfaces, while we seek the radii of circle patterns using the method of combinatorial Ricci flow. The sphere is the most rigid and difficult classical setting. It is worth to point out that our algorithm does not work in the spherical geometry, since the evolution of curvature in the this setting does not satisfy the maximum principle, i.e., the curvature has not monotonicity in time t. To the authors’ knowledge, no circle packing algorithm intrinsic to the spherical geometry has been found; spherical circle packings are typically obtained by stereographically projecting from the disc. This paper is organized as follows. In Sect. 2 we shall give briefly some definitions and results related to circle patterns. In Sect. 3, we will use the idea of combinatorial Ricci flows to establish an iterative process for finding radii of circle patterns and give the definition of exponential convergence for this process. Using the techniques of eigenvalues of matrixes we prove that the iterative process (3.5) converges exponentially fast to the radii of circle pattern realizing (T , ") for the special case that " ≡ 0 and the convergence of algorithm for general case that " ∈ [0, π/2] is addressed briefly in Sect. 4. In Sect. 5 we will describe details of implementation and give some examples to show the the efficiency of our algorithm.

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2 Preliminaries In this section, we will give briefly some definitions and results related to circle patterns; see [7, 17, 20, 21, 26] for more details. Triangulation Let T be a finite triangulation of a closed topological disk X. (0) (0) Denote by T (0) , TI , T∂ and T (1) the sets of vertices, interior vertices, boundary vertices and edges of T , respectively. If " : T (1) → [0, π/2] is a function defined on T (1), then we call (T , ") a weighted triangulation of X. Assume further that (T , ") satisfies the following two conditions: (A) for 3 any three edges e1 , e2 , e3 forming a null homotopic loop in X, if i=1 "(ei ) ≥ π, then these three edges form the boundary of a triangle of T ; (B)  for any four edges e1 , e2 , e3 , e4 forming a null homotopic loop in X, if 4 i=1 "(ei ) ≥ 2π, then these four edges form the boundary of union of two adjacent triangles of T . (0)

A set β = {(v1 , k1 ), (v2 , k2 ), . . . , (vm , km )} ⊂ TI × N is called a branch structure for (T , ") if the following condition is satisfied: for each simple closed edge path  = {e1 , e2 , . . . , en } in T which surrounds at least one vertex of T , the inequality n  [π − "(ej )] > (2l() + 2)π j =1

holds, where l() is the number of vertices of β inside , counting repetitions. We define the degree d(v) of a vertex v in T as the number of vertices in T adjacent to v and the degree d of T as the least upper bound on the degree d(v) of any vertex v in T . Circle Pattern A configuration of circles P in the Euclidean or hyperbolic plane is called a circle pattern realizing (T , ") if (a) there exists a one-to-one correspondence between the vertices v ∈ T (0) and the circles Pv ∈ P such that Pv and Pw intersect at overlap angle "([v, w]), where [v, w] ∈ T (1) is the edge joining v and w; (b) P is orientation preserving, that is, if v1 , v2 , v3 are three vertices of a face in T taken in the positive order, then circles Pv1 , Pv2 , Pv3 form positively directed triple of circles in the Euclidean or hyperbolic plane. We remark that if " ≡ 0 in the above definition, then a circle pattern P realizing (T , ") is usually said to be a circle packing for T . In other words, a circle packing P for T may be viewed as a special case of circle pattern realizing (T , ") when " ≡ 0. Label Vector Consider any sub-complex T˜ ⊂ T with vertex number m. If to each vertex vi ∈ T˜ (0) one assigns a positive number ri for i = 1, 2, . . . , m, where 0 < ri < ∞ in the Euclidean geometry and 0 < ri ≤ ∞ in the hyperbolic one, then the

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vector RT˜ (0) = (r1 , r2 , . . . , rm ) is called a label vector for T˜ . In particular, if a label vector RT (0) = (r1 , r2 , . . . , rZ ) equals the radii of a circle pattern P realizing (T , ") where Z denotes the cardinal number of T (0) , then RT (0) is also called a radius vector of P . Furthermore, if we take natural variables ui = ln ri in the Euclidean geometry → and ui = ln tanh(ri /2) in the hyperbolic one, then the vector − u = (u1 , u2 , . . . , uZ ) is said to be a natural radius vector of P corresponding to the Euclidean geometry or the hyperbolic one, respectively. Curvature If a label vector RT (0) = RT (0) ∪ RT (0) is given for a weighted I ∂ triangulation (T , ") with a branch structure β, then we can realize each face ij k vi , vj , vk  in T by a Euclidean triangle TE of edge lengths lij , lj k and lki , where 2 lmn = (rm + rn2 + 2rm rn cos "(emn ))1/2

for m, n = i, j, k and enm denotes the edge joining vn and vm . Namely, the triangle ij k TE is formed by the centers of three circles of Euclidean radii ri , rj and rk intersecting at angles "(eij ), "(ej k ) and "(eki ). Likely, we may also realize each ij k triangle vi , vj , vk  in T by a hyperbolic triangle TH of edge lengths lij , lj k and lki where lmn = cosh−1 [cosh rm cosh rn + sinh rm sinh rn cos "(emn )] ij k

for m, n = i, j, k, that is, TH is formed by the centers of three circles of hyperbolic radii ri , rj and rk intersecting at angles "(eij ), "(ej k ) and "(eki ). Let α(R(vi ); R(vj ), R(vk )) denote the inner angle at vertex vi in the Euclidean ij k ij k triangle TE or hyperbolic one TH corresponding to the face vi , vj , vk . Then one has α(R(vi ); R(vj ), R(vk )) = arccos(

2 − l2 lij2 + lik jk

2lij lik

)

(2.1)

in the Euclidean geometry and α(R(vi ); R(vj ), R(vk )) = arccos(

cosh lij cosh lik − cosh lj k ) sinh lij sinh lik

(2.2)

in the hyperbolic geometry. Thus, for each vertex v ∈ TI(0) and its flower Fv = {v; v1 , . . . , vk } which is a sub-complex formed by v and its neighbors in T , the angle sum θ (v; R) at v for R may be expressed by θ (v; R) =

 v,vi ,vi+1 

α(R(v); R(vi ), R(vi+1 )),

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where the summation is over faces v, vi , vi+1  ∈ Fv . We define the curvature K(v) at v by K(v) = 2β(v)π − θ (v; R),

(2.3)

where β(v) = k + 1 and k is the number of times v occurs in β. A label vector R is said to be a β-pattern vector for (T , ") if K(v) = 0 for each vertex v ∈ T (0) . Obviously, if R is a β-pattern vector for (T , "), then we obtain a branched circle pattern with branch set β which realizes (T , "). In particular, this results in a univalent or univalent locally circle pattern for (T , ") when β = ∅. The circle patterns we intend to compute are guaranteed by the following fundamental existence and uniqueness result (see [7, 21]), which is also called the discrete Dirichlet problem. Proposition 1 Let (T , ") be a weighted triangulation of a closed topological disk with a weight function " : T (1) → [0, π/2] which satisfies the conditions (A) and (B), and let β be a branch structure for (T , "). If we are given a function g : T∂(0) → (0, ∞) (respectively(0, ∞]) defined on the boundary vertices of T . Then there exists a unique Euclidean (respectively hyperbolic) β-pattern vector R for (T , ") such that R(w) = g(w) for each boundary vertex w of T . Our aim is to describe an efficient algorithm which approximates numerically β-pattern vector R in Proposition 1 based on the idea of combinatorial Ricci flows.

3 The Iterative Process In this section we first recall the concepts of combinatorial Ricci flows and some related results (also see [10, 20]). Next we use the former’s idea to establish an iterative process for finding radii of circle patterns, and give the definition of exponential convergence for this process. Consider a weighted triangulation (T , ")(" ∈ [0, π/2]) of a closed topological disk X with a label vector RT (0) = RT (0) ∪RT (0) = {r1 , r2 , . . . , rZ }, where Z denotes I

∂

the cardinal number of T (0) . Note that the boundary label vector RT (0) is fixed in our ∂ algorithm discussed below, so we concentrate only on the interior label vector RT (0) . I

For each vertex vi ∈ TI(0) , let Ki = K(vi ) denote the curvature at vi which is given by (2.3). Then the combinatorial Ricci flow associated with (T , ") in the Euclidean metric is defined by dri = −Ki ri , dt

(3.1)

and the corresponding combinatorial Ricci flow in the hyperbolic one is given by dri = −Ki sinh ri . dt

(3.2)

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It is proved in [10, 20] that given any initial label vector R 0 (0) for the weighted TI

triangulation (T , ") satisfying the conditions (A) and (B), then (i) the solution to the combinatorial Ricci flow (3.1) in the Euclidean geometry with R 0 (0) exists for TI

all time and converges exponentially fast to a Euclidean metric β-pattern vector for (T , "); (ii) the solution to the combinatorial Ricci flow (3.2) in the hyperbolic geometry with R 0 (0) exists for all time and converges exponentially fast to a TI

hyperbolic metric β-pattern vector for (T , "). It is easy to see that (3.1) and (3.2) can be rewritten as d ln ri = −Ki dt

(3.3)

d ln tanh(ri /2) = −Ki , dt

(3.4)

and

respectively. As before, we take natural variables ui = ln ri in the Euclidean setting and ui = ln tanh(ri /2) in the hyperbolic one. Then the interior label vector RT (0) = I → − → {r1 , r2 , . . . , rN } becomes − u = (u1 , u2 , . . . , uN ) and the curvature vector K = → (K , K , . . . , K ) may be viewed as a function of vector − u , where N denotes the 1

2

N

(0)

cardinal number of TI . Thus the combinatorial Ricci flows (3.3) and (3.4) suggest that we can define an iterative process for finding radii of a circle pattern realizing (T , ") by −−−−−→ −−→ −−→ u(n + 1) = −ηK(n) + u(n),

(3.5)

−−→ −−→ where u(n) = (u1 (n), u2 (n), . . . , uN (n)), K(n) = (K1 (n), K2 (n), . . . , KN (n)) and η > 0 is a constant to be determined later on. A solution to (3.5) is called convergent if (a) lim Ki (n) = 0 exists for each n→∞

1 ≤ i ≤ N; and (b) lim ui (n) = ui exists for all 1 ≤ i ≤ N. n→∞ A convergent solution of (3.5) is called convergent exponentially fast if there are positive constants c1 , c2 so that for all n ≥ 0, |Ki (n)| ≤ c1 e−c2 n , and |ui (n) − ui | ≤ c1 e−c2 n .

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It is clear that the latter is equivalent to the following condition: there are positive constants σ1 , σ2 so that for all n ≥ 0, −−→ K(n) ≤ σ1 e−σ2 n , and −−→ → u(n) − − u  ≤ σ1 e−σ2 n , where  ·  denotes the Euclidean norm of RN . In the next section, we will prove that the solution to iterative process (3.5) exists and converges exponentially fast to radii of a circle pattern realizing (T , ")(" ≡ 0), as n → ∞.

4 The Convergence of Algorithm In this section, the convergence of (3.5) will be proven in the Euclidean metric and the hyperbolic one, respectively. More concretely, Proposition 1 gives that there exists a unique circle pattern P realizing (T , ")(" ∈ [0, π/2]) for a given boundary label vector RT (0) . Assume that RT (0) is the radius vector of P , then it is clear that the ∂ → − → − → curvature K associated with − u is equal to 0 . On the other hand, if an initial vector −−→ −−→ u(0) = (u1 (0), u2 (0), . . . , uN (0)) is given, then K(0) follows from (2.3). Hence −−→ label vector u(n)(n = 1, 2, · · · ) can be obtained from the iterative process (3.5). We shall show that as n → ∞, the solution to (3.5) converges exponentially fast → to the natural radius vector − u of P in the Euclidean geometry and the hyperbolic one, respectively. For simplicity, we shall restrict ourselves to the special case that " ≡ 0. To be more precise, we have the following.

4.1

Convergence in Euclidean Geometry

Theorem 1 Let T be a triangulation of a closed topological disk with a given Euclidean boundary label vector RT (0) , and let P be a unique circle pattern for ∂ (T , ")(" ≡ 0) in the Euclidean plane such that its boundary radius vector is equal to RT (0) . Then the iterative process (3.5) converges exponentially fast to the natural ∂ → radius vector − u of P when η is taken such that η < 1/(d' ), where d denotes the degree of T and ' =

 √

5−1

√ √ 2( 5 − 1) . √ 5+1

(4.1)

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In order to prove this theorem, we need the following lemmas. Lemma 1 Let M123 be a 3 × 3 positive semi-definite matrix ⎛ ⎞ b1 + b2 −b2 −b1 M123 = ⎝ −b2 b2 + b3 −b3 ⎠ , −b1 −b3 b3 + b1

(4.2)

where bi > 0 and bi + bj < κ for i = j ∈ {1, 2, 3}; and its leading principal 2 × 2 minor is denoted by   b1 + b2 −b2 M˜ 123 = . −b2 b2 + b3 Then (i) eigenvalues νi of M123 satisfy νi < 2κ for i = 1, 2, 3; (ii) eigenvalues μj of M˜ 123 satisfy μj < 2κ for j = 1, 2. Proof We first prove (ii). It is easy to see that μ1 + μ2 = b1 + 2b2 + b3 for matrix M˜ 123 , which implies (ii) holds using the condition bi + bj < κ for i = j ∈ {1, 2, 3}. Next, note that the eigenvalues ν1 , ν2 , ν3 of M123 are roots of the following equation ν[ν 2 − 2(b1 + b2 + b3 ) + 3(b1b2 + b2 b3 + b3 b1 )] = 0, which implies ν1 = 0, ν2 = b1 + b2 + b3 + (b12 + b22 + b32 − b1 b2 − b2 b3 − b3 b1 )1/2 , ν3 = b1 + b2 + b3 − (b12 + b22 + b32 − b1 b2 − b2 b3 − b3 b1 )1/2 . Thus, to prove (i) holds, it is sufficient to show that b12 + b22 + b32 − b1 b2 − b2 b3 − b3 b1 < (2κ − b1 − b2 − b3 )2 . The latter is equivalent to the following inequality 3(b1 b2 + b2 b3 + b3 b1 ) − 4κ(b1 + b2 + b3 ) + 4κ 2 > 0.

(4.3)

Indeed, since that bi > 0 and bi + bj < κ for i = j ∈ {1, 2, 3}, we have κ 2 − κ(b1 + b2 + b3 ) + b1 b3 + b2 b3 = (κ − b1 − b2 )(κ − b3 ) > 0; κ 2 − κ(b1 + b2 + b3 ) + b1 b2 + b3 b2 = (κ − b1 − b3 )(κ − b2 ) > 0; κ 2 − κ(b1 + b2 + b3 ) + b2 b1 + b3 b1 = (κ − b2 − b3 )(κ − b1 ) > 0; κ 2 − κ(b1 + b2 + b3 ) + b1 b2 + b1 b3 + b2 b3 > (κ − b1 − b2 )(κ − b3 ) > 0.

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2

v2 α2

1

v1

3

α1

α3

v3

Fig. 1 A topological triangle #123 and its Euclidean realizing one #v1 v2 v3

Adding up to the above inequalities yields (4.3). This completes the proof of the lemma.  Lemma 2 Let #123 be a topological triangle in T with Euclidean label vector (r1 , r2 , r3 ), and let #v1 v2 v3 denote the Euclidean triangle realizing #123 and αi the inner angle at vi in #v1 v2 v3 for i = 1, 2, 3 (see Fig. 1) where αi = α(R(vi ); R(vj ), R(vk )) is expressed by (2.1). Set li = rj +rk for {i, j, k} = {1, 2, 3} and ui = ln ri for i = 1, 2, 3. Then the Jacobian matrix M123 of functions −α1 , −α2 , −α3 in terms of u1 , u2 , u3 is positive semi-definite and can be expressed as the form (4.2), where b1 = [l12 + l22 − l32 + 2(r22 − r32 ) + (l12 + l32 − l22 )(r12 − r22 )/ l32 ]/(4li lj sin αk ),

(4.4)

b2 = [l12 + l32 − l22 + 2(r12 − r22 ) + (l22 + l32 − l12 )(r32 − r12 )/ l22 ]/(4li lj sin αk ),

(4.5)

b3 = [l22 + l32 − l12 + 2(r32 − r12 ) + (l12 + l22 − l32 )(r22 − r32 )/ l12 ]/(4li lj sin αk ),

(4.6)

which satisfy bi > 0, and bi + bj ≤ '

(4.7)

for i = j ∈ {1, 2, 3} and ' is expressed by (4.1). Proof First, from [13, Lemma 6] we obtain that the matrix M123 is positive semi-definite and can be written as the form of (4.2), where b1 , b2 and b3 are given by (4.4),(4.5) and (4.6), respectively. Moreover, we easily deduce from the expressions of bi that bi > 0 for i = 1, 2, 3. Thus, the remainder is to prove that (4.7) holds. Notice that b1 +b2 = [2l12 +(l12 +l22 −l32 )

r12 − r32 l22

+(l12 +l32 −l22 )

r12 − r22 l32

]/(4l1 l2 sin α3 ),

(4.8)

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So simplifying and applying transformation of trigonometric function to (4.8) combined with li = rj + rk for {i, j, k} = {1, 2, 3}, we deduce that b1 + b2 = (C3 + C2 )(S2 + S3 − S1 )/(2S2 S3 ),

(4.9)

where Si = sin αi and Ci = cos αi for i = 1, 2, 3. The identity α1 + α2 + α3 = π gives S1 = S2 C3 + S3 C2 . Hence using the trigonometric identity we get from (4.9) that b1 + b2 = (C3 + C2 )(c2 s3 + c3 s2 )/(2c2 c3 ),

(4.10)

where si = sin(αi /2) and ci = cos(αi /2) for i = 1, 2, 3. Since that c1 = c2 s3 +c3 s2 and Ci = 2ci2 − 1, we obtain from (4.10) that b 1 + b 2 = c1

c22 + c32 − 1 . c2 c3

(4.11)

c2 +c2 −1

Fix α1 , it is easy to see that 2 c2 c33 arrives at the maximum value if and only if α2 = α3 . Hence it follows from (4.11) that √ 2 cos2 θ − 1 2 1 − x 2 (2x 2 − 1) π b1 + b2 ≤ cos( − 2θ ) = , 2 cos2 θ x where 0 < θ =

α2 2

=

α3 2


2' d and a eigenvector − y corresponding to λj such that − y M− y = → − → − 2 2 λ y  > 2' d y  , which contradicts with the inequality (4.13). Therefore, we finish the proof of the lemma.  −−→ − − → Lemma 4 Let u(0) ∈ RN be any given initial vector, and let u(n) be the label −−→ vectors given by the iterative process (3.5). Then as η < 1/(' d), (i) u(n) are bounded for all n ∈ N; and (ii) there is a constant χ > 0 such that each eigenvalue → − − −−→ −−−−−→ −−→ − → K → ( ξ ) satisfies λξ > χ for all ξ = u(n) + θ (u(n + 1) − u(n)), θ ∈ λξ of ∂∂ − → u (0, 1), n ∈ N. −−−−−→ Proof It is clear that the iterative process (3.5) can be written as u(n + 1) = → −−→ − → − → − f (u(n)), where f is a vector function in RN . From the definition of K we deduce − → easily that f is differentiable in RN . On the other hand, Proposition 1 gives that there exists a circle pattern label R for (T , ")(" ≡ 0), which implies that there

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→→ − → → exists − u such that − u = f (− u ). Thus from the mean value theorem and (3.5) we deduce that → − ∂f − −−−−−→ → → −−→ − − →→ → −−→ → u(n + 1) − − u  =  f (u(n) − f (− u )) ≤  − ( ξ ) · (u(n) − − u ) ∂→ u → − ∂K − −−→ → −−→ → → ≤ η − ( ξ ) − I · u(n) − − u  ≤ ρu(n) − − u , → ∂u

(4.14)

→ −−→ − − → −−→ where ξ = u(n) + θ ( uˆ − u(n))(0 < θ < 1), and I denotes the unit matrix and ρ := max {|ηλi − 1|}. 1≤i≤N

(4.15)

Lemma 3 gives that 0 < λi ≤ 2d' . Hence when taking η < 1/(' d), we deduce → → −−→ − −−→ − from (4.15) that ρ ≤ 1. Thus we obtain from (4.14) that u(n) − uˆ  ≤ u(0)− uˆ  → − −−→ for any n ∈ N. Note that uˆ and u(0) are both bounded. So it is easy to see that −−→ there exits a constant L > 0 such that u(n) ≤ L for each n ∈ N, which implies (i) holds. −−→ → − It follows from (i) that there exists a constant L > 0 such that  ξ  < 2u(n)+ −−−−−→ u(n + 1) ≤ 3L. If the conclusion of (ii) doesn’t hold, then for each k ∈ N there is → − − → − K → ( ξk ) has an eigenvalue λξk < 21k . Let U = [−3L, 3L]N , then U is a ξk such that ∂∂ − → u → − bounded closed subset in RN . Hence U ⊂ RN is a compact subset. Since { ξk } ⊂ U , → − → − → − → − there exists must be a subsequence {ξkl } of { ξk } such that {ξkl } converges to ξ ∈ U → − − K → ( ξ )] = 0, which leads to a contradiction with the as l → ∞. This implies det[ ∂∂ − → u → − − → K positive definite of ∂∂ − ( ξ ), by Lemma 3. Therefore we conclude (ii) holds.  → u Proof of Theorem 1 First of all, we get from the iterative formula (3.5) that 1 −−−−−→ −−→ −−−−−→ → −−−−−→ − → −−→ − K(n + 1) =  K (u(n + 1)) − K (u(n)) − (u(n + 1) − u(n)) η → − ∂K − 1 −−−−−→ −−→ → ≤ − ( ξ ) − I · u(n + 1) − u(n) → η ∂u → − ∂K − −−→ → = η − ( ξ ) − I · K(n) ∂→ u −−→ ≤ ρK(n), where ρ is given by (4.15) and I denotes the N × N unit matrix. Take η < 1/(d' ), then we get from Lemmas 3 and 4(ii) that χ < λi < 2d' . Thus we deduce

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−−→ −−→ from (4.15) that 0 < ρ < 1. Hence we conclude that K(n) ≤ ρ n K(0) for all n ∈ N. → Next, let − u denote the natural radius vector of circle pattern P for (T , ")(" ≡ − → 0). Then similar to the proof of Lemma 4 and combining with the definition of f , we obtain −−−−−→ → → −−→ − − →→ u(n + 1) − − u  =  f (u(n) − f (− u )) → − ∂f − −−→ → → ( ξ ) · u(n) − − u ≤ − ∂→ u → − ∂K − −−→ → → = η − ( ξ ) − I · u(n) − − u → ∂u −−→ → ≤ ρu(n) − − u , where ρ is given by (4.15) and satisfies 0 < ρ < 1 as taking η < 1/(d' ). This −−→ → −−→ → u  for all n ∈ N. So we finish the proof of the yields u(n) − − u  ≤ ρ n u(0) − − theorem. 

4.2 Convergence in Hyperbolic Geometry Theorem 2 Suppose that T is a triangulation of a closed topological disk with a given hyperbolic boundary label vector RT (0) . Let P be a unique circle pattern for ∂ (T , ")(" ≡ 0) in the hyperbolic plane whose boundary radius vector of P is equal to RT (0) . Then the iterative formula (3.5) converges exponentially fast to the natural ∂ → radius vector − u of P when η is chosen such that 0 < η < 1/(3d), where d is the degree of T . Similar to the situation of Euclidean geometry , we need the following lemmas in order to prove Theorem 2. Lemma 5 Let H123 be a 3 × 3 positive definite matrix ⎞ H1 h3 h2 = ⎝ h3 H2 h1 ⎠ h2 h1 H3 ⎛

H123

(4.16)

satisfying the following conditions hi < 0, − hj − hk < Hi < γ

(4.17)

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131

for {i, j, k} = {1, 2, 3}, where γ > 0 is some constant, and let H˜ 123 =



H1 h3 h3 H2



denote the leading principal 2×2 minor of H123 . Then (i) the eigenvalues νj of H123 satisfy νj < 3γ for j = 1, 2, 3; (ii) the eigenvalues μi of H˜ 123 satisfy μi < 2γ for i = 1, 2. Proof From the property of eigenvalue for matrix and the assumptions of lemma, it is easy to see that (ii) holds. Next we will show that (ii) is valid. Note that the matrix H123 can be written as H123 = H (1) + H (2), where ⎞ h3 h2 −h3 − h2 ⎠ =⎝ h3 −h3 − h1 h1 h2 h1 −h1 − h2 ⎛

H (1)

and ⎛

H (2)

⎞ H1 + h2 + h3 0 0 ⎠. =⎝ 0 H2 + h1 + h3 0 0 0 H3 + h1 + h2

So we get from Lemma 1 that → → → → → → − → x =− x +− x < 3γ − x H (1)− x H (2) − x 2 x H123− → for any − x = (x1 , x2 , x3 ) ∈ R3 , which implies that the eigenvalues νj of H123 satisfy νj < 3γ for j = 1, 2, 3. Therefore the proof of this lemma is completed.  Lemma 6 Let #123 ∈ T be a topological triangle with hyperbolic label vector (r1 , r2 , r3 ), and let #v1 v2 v3 be the hyperbolic triangle realizing #123 and αi the inner angle at vertex vi in #v1 v2 v3 for i = 1, 2, 3 (see Fig. 2) where αi = α(R(vi ); R(vj ), R(vk )) is expressed by (2.2). Write li = rj + rk for {i, j, k} = {1, 2, 3} and ui = ln tanh(ri /2) for i = 1, 2, 3. Then the Jacobian matrix H123 of functions −α1 , −α2 , −α3 in terms of (u1 , u2 , u3 ) is positive definite and can be expressed as the form (4.16), where h1 =hij k [cosh r1 −

cosh l2 cosh l1 − cosh l3 cosh2 l1 − 1

cosh r2 −

cosh l3 cosh l1 − cosh l2 cosh2 l1 − 1

cosh r3 ],

(4.18) h2 =hij k [cosh r2 −

cosh l1 cosh l2 − cosh l3 cosh2 l2 − 1

cosh r1 −

cosh l3 cosh l2 − cosh l1 cosh2 l2 − 1

cosh r3 ],

(4.19)

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2

v2 α2 v1

1

α1

α3

3

v3

Fig. 2 A topological triangle #123 and its hyperbolic realizing one #v1 v2 v3

h3 =hij k [cosh r3 −

cosh l1 cosh l3 − cosh l2 cosh2 l3 − 1

cosh r1 −

cosh l2 cosh l3 − cosh l1 cosh2 l3 − 1

cosh r2 ],

(4.20) H1 =hij k [

(cosh l1 cosh l2 − cosh l3 )(cosh l2 cosh r1 − cosh r3 ) sinh2 l2 +

H2 =hij k [

sinh2 l3

],

(4.21)

],

(4.22)

],

(4.23)

(cosh l2 cosh l3 − cosh l1 )(cosh l3 cosh r2 − cosh r1 ) sinh2 l3 +

H3 =hij k [

(cosh l1 cosh l3 − cosh l2 )(cosh l3 cosh r1 − cosh r2 )

(cosh l2 cosh l1 − cosh l3 )(cosh l1 cosh r2 − cosh r3 ) sinh2 l1

(cosh l3 cosh l1 − cosh l2 )(cosh l1 cosh r3 − cosh r2 ) sinh2 l1 +

(cosh l3 cosh l2 − cosh l1 )(cosh l2 cosh r3 − cosh r1 ) sinh2 l2

and hij k = 1/(sin αi sinh lj sinh lk ), which satisfies the conditions (4.17) where γ is chosen as γ = 2. Proof It follows from [13, Lemma 12 and Lemma 11] that H123 is positive definite and can be written as the form (4.16), where h1 , h2 , h3 , H1 , H2 and H3 are given by (4.18),(4.19), (4.20),(4.21),(4.22) and (4.23), respectively. Thus, it suffices to verify that (4.17) holds. It is easy to see from the expressions of hi and Hi that hi < 0 and −hi − hj < Hk for {i, j, k} = {1, 2, 3}. The remainder is to seek a constant γ > 0 such that Hj < γ for j = 1, 2, 3. Consider first H1 . Notice that li = rj + rk for {i, j, k} = {1, 2, 3}. In the light of Cosine Law and Sine Law in the hyperbolic geometry we obtain from (4.21) that H1 =

(C2 + C3 ) sinh r1 , S3 sinh l2

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where C2 = cos α2 , C3 = cos α3 and S3 = sin α3 . Similar to the proof of Lemma 2, we may deduce that H1 < 2. With the same arguments as above, we may conclude that Hi < 2 for i = 2, 3. Thus (4.17) holds when γ is taken as γ = 2. This completes the proof of the lemma.  Lemma 7 For a fixed boundary label vector RT (0) and any interior label vector ∂ RT (0) = (r1 , r2 , . . . , rN ) with 0 < ri ≤ ∞ in the hyperbolic setting, let I → − K = (K1 , K2 , . . . , KN ) denote the curvature vector corresponding to the interior (0) vertices TI which is viewed as a vector function of (r1 , r2 , . . . , rN ). Then the → − → Jacobian matrix H of K in terms of − u = (u , u , . . . , u )(u = ln tanh(r /2)) 1

2

N

i

i

is positive definite and its eigenvalues λi satisfy 0 < λi ≤ 6d for i = 1, 2, . . . , N, where d denotes the degree of T . → − Proof The proof is similar to the proof of Lemma 3. By the definition of K and combined with Lemma 6, we may deduce that λi > 0 for i = 1, 2, . . . , N. Next, we group the faces of T into three sets F1 , F2 and F3 , where F1 denotes the set of all triangles consisted of one interior vertex and two boundary vertices, F2 the set of all triangles consisted of two interior vertices and one boundary vertex and F3 the set of all triangles consisted of three interior vertices. Then for each vector → − x = (x1 , x2 , . . . , xN ) ∈ RN , we get from Lemmas 6 and 5 that − → → x H− x =



(xi , xj , xk )Hij k (xi , xj , xk ) +

#ij k∈F3

+



Hij k xi2



6(xi2 + xj2 + xk2 ) +

#ij k∈F3

≤ 6d

N 

(xi , xj )Hij k (xi , xj )

#ij k∈F2

#ij k∈F1

≤



 #ij k∈F2

4(xi2 + xj2 ) +



2xi2

#ij k∈F1

→ xi2 = 6d− x 2 ,

i=1

where Hij k is the Jacobian matrix of functions (−αi , −αj , −αk ) in terms of (ui , uj , uk ) which depends only on two variables if #ij k ∈ F2 and depends only on one variable if #ij k ∈ F1 , since that the boundary label vector RT (0) is fixed. ∂ This implies that eigenvalues λi of H satisfy 0 < λi ≤ 6d for i = 1, 2, . . . , N. So we finish the proof of the lemma.  Proof of Theorem 2 It is clear that that Lemma 4 also holds in the hyperbolic setting, where natural variables ui = ln tanh(ri /2). So combined with Lemma 7, identical with the proof of Theorem 1 we can conclude that the Theorem 2 holds.  Remark Theorems 1 and 2 give that for the special case that " ≡ 0, the iterative process (3.5) converge exponentially fast to the radii of circle pattern P realizing

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(T , ") in the Euclidean geometry and the hyperbolic one, respectively. However, the convergence of (3.5) also holds for general case that " ∈ [0, π/2]. In fact, from the proofs of Theorems 1 and 2 we see that the key step is how to choose η, which reduces to seek the upper bound of the eigenvalues of matrix associated with any #ij k in T . From the proof of Lemmas 2 and 6 we may deduce that there exist constants η1 > 0 and η2 > 0 such that the eigenvalues λi of the Jacobian matrix of functions (−α1 , −α2 , −α3 ) in terms of (u1 , u2 , u3 ) satisfy λi < η1 in the Euclidean setting and λi < η2 in the hyperbolic one. Thus the remainder is similar to the proof of Theorems 1 and 2, we can deduce that (3.5) converge exponentially fast to the radii of circle pattern P realizing (T , ")(" ∈ [0, π/2]) by choosing a suitable η > 0.

5 Implementation and Examples In this section we will describe the details of implementation for the iterative process (3.5) and show the efficiency of our algorithm by giving some examples. Given a triangulation T of a closed topological disk and a boundary label vector (0) RT (0) = {!1 , !2 , . . . , !N∂ } where N∂ denotes the cardinal number of T∂ , our task ∂ is to compute the radii of circle pattern P realizing (T , ")(" ≡ 0) whose boundary radius vector is equal to RT (0) , as guaranteed by Proposition 1. More concretely, ∂ index first the vertices of the triangulation T by {w1 , w2 , . . . , wN∂ ; v1 , v2 , . . . , vN }, with wi denoting boundary vertices and vj , interior vertices. Next we need to find values {r1 , r2 , . . . , rN } corresponding to {v1 , v2 , . . . , vN } so that the label → − → − vector R satisfies K (R) = (K(v1 ), K(v2 ), . . . , K(vN )) = 0 , where R = {!1 , !2 , . . . , !N∂ ; r1 , r2 , . . . , rN } and K(vj ) is given by (2.3) for j = 1, 2, . . . , N. Thus our algorithm is described as follows. Algorithm Given a triangulation T , a boundary label vector RT (0) and a tolerance ∂ *0 > 0. → 1. Set initial label vector to − u in terms of u = ln r and take η = 1/(0.601d) in i

i

the Euclidean setting; and by terms of ui = ln tanh(ri /2) and η = 1/(3d) in the hyperbolic setting. → − 2. Compute the curvature vector K . → → − → 3. Compute − u = −η K + − u. → − 4. Compute the error * =  K /(3N), if * ≤ *0 , then stop; if * > *0 , then goto to step 2.

Test Environment The foregoing algorithm are coded in MATLAB2007 and implemented on Intel CoreT M i5 CPU M540 2.53 GHz with 4G RAM under Windows 7 operation system.

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Fig. 3 A triangulation T163 with 127 interior vertices

Results We will use directly the iterative process (3.5). The error will be computed → − as  K /(3N), where N denotes the number of interior vertices of T . In order to demonstrate the efficiency of our algorithm, we perform three tests in the Euclidean geometry and the hyperbolic one respectively, which involves triangulations with hundreds of interior vertices. The number of iteration, the time of computation and the error of the final value are shown in Table 1, where Tn denotes a triangulation with n vertices. In particular, a triangulation T163 with 127 interior vertices is indicated in Fig. 3 and the corresponding circle packing realizing T163 in the Euclidean plane is shown in Fig. 4, which is computed by our algorithm with boundary radii all equal to 2. From the proofs of Theorems 1 and 2 in Sect. 4 we see that the choice of η depends on the eigenvalues of matrixes associated with triangles and the degrees of vertices in T , which determines the convergence rate of the sequence of label vectors. In our tests, we take η = 0.2 with boundary radii all equal to 2 and the initial interior radii all equal to 1 in the Euclidean setting; and η = 0.24 with boundary radii all equal to 15 and the initial interior radii all equal to 1 in the hyperbolic setting. This is because that for each hyperbolic triangle as described in Lemma 6, the eigenvalue of the associated Jacobian matrix Hij k is tested to be in a range (0, 1.05), and the maximum of degrees of vertices in these triangulations is no more than 8. Moreover, it is easy to see from our tests (see Table 1) that when the number of interior vertices in T increases, the number of iteration does increase only slightly. So our algorithm may be more appropriate to compute substantial circle patterns.

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Fig. 4 A circle packing realizing T163 in the Euclidean plane

Table 1 Results for three samples in Euclidean and hyperbolic geometry Triangulation T163 T317 T619 T163 T317 T619

Euc/Hyp Euc Euc Euc Hyp Hyp Hyp

Num of Int 127 261 535 127 261 535

Num of Iter 191 198 211 87 94 96

Time (s) 0.88 1.83 3.98 0.54 1.27 2.64

Error 9.8649E-5 9.9962E-5 9.8438E-5 9.8760E-5 9.4914E-5 9.5238E-5

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Acknowledgements This work is supported in part by NSF of China (11661011, 11631015) and NSF of Guangxi (2016GXNSFAA380099).

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Strong Asymptotic Analysis of OLPs on the Unit Circle by Riemann-Hilbert Approach Yufeng Wang, Yifeng Lu, and Jinyuan Du

To Professor Heinrich Begehr on the occasion of his 80th birthday

Abstract In this article, we will deal with the asymptotics of the monic orthogonal Laurent polynomials (OLPs) on the unit circle with respect to a strictly-positive analytic weight by Riemann-Hilbert approach. We first construct a matrix RiemannHilbert problem (RHP) which is the Fokas-Its-Kitaev characterization. Then, the strong asymptotic formulas of OLPs are obtained by employing Deift-Zhou steepest descent analysis. Furthermore, the asymptotic formulas of the leading coefficient and the trailing coefficient are simultaneously obtained. Keywords Orthogonal Laurent polynomial · Riemann-Hilbert approach · Riemann-Hilbert problem · Strong asymptotics · Cauchy-type integral operator Mathematics Subject Classification (2010) 42C05, 41A35, 30G30

1 Introduction and Preliminaries As is well-known, strong asymptotics of orthogonal polynomials (OPs) can be found in the Szegö excellent monograph [1]. Recently, Riemann-Hilbert approach has been widely applied to strong asymptotic analysis of OPs with respect to various weights, see for example [2–8]. Generally, the so-called Riemann-Hilbert

Y. Wang · Y. Lu School of Mathematics and Statistics, Wuhan University, Wuhan, China e-mail: [email protected] J. Du () School of Mathematics and Statistics, Wuhan University, Wuhan, China School of Science, Linyi University, Shandong, China e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_8

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technique is just to combine Fokas-Its-Kitaev characterization with Deift-Zhou steepest-descent method. In 1992, OPs on the real line are first described as the solution of a matrix Riemann-Hilbert problem for holomorphic functions by A.S. Fokas, A.R. Its and A.V. Kitaev in their pioneering works [9]. Deift-Zhou steepestdescent method, set up by P. Deift and X. Zhou in 1993, is to change the given matrix Riemann-Hilbert problem into another model problem by an array of contour deformations and transformations [10]. An excellent overview of Riemann-Hilbert approach can be found in the review articles [11, 12] and the monographs [13, 14]. Usually, Riemann-Hilbert problem is called Riemann problem, see e. g. [15–19]. The classical theory of Riemann-Hilbert problem, also called Hilbert problem in [20], plays a prominent role in the application of Riemann-Hilbert technique [13, 14]. In this article, we always call it R-H (Riemann-Hilbert) problem. OPs are closely connected with the classical moment problem (see [21]). Since the investigation of the strong Stieltjes moment problem was initiated by W.B. Jones et al. in 1980, orthogonal Laurent polynomials (OLPs), also called orthogonal Lpolynomials, have entered into people’s horizon [22]. In order to deal with the strong Stieltjes moment problem, OLPs have been set up by Jones et al. as the rational generalization of OPs. And OLPs are a particular case of orthogonal rational functions with prescribed poles [23]. In 1984, the strong Hamburger moment problems were also discussed by Jones et al. through the study of OLPs on the real axis [24]. In contrast with OPs, OLPs have also intensively investigated because they can be applied not only to the strong Stieltjes and Hamburger moment problems but also to Padé-type approximants, continued fractions, Gauss quadrature, rational interpolation, the relativistic Toda lattice and so on [24–28]. Most of applications of OLPs are included in the review article [25]. However, unlike OPs, OLPs can also satisfy four-term or five-term recurrence relations [25–27]. In terms of Riemann– Hilbert technique, K.T.-R McLaughlin, A.H. Vartanian and X. Zhou have already discussed the asymptotic behavior of OLPs on the real line with respect to varying exponential weights [26, 27]. In addition, some algebraic properties in the theory of OLPs are investigated by the Riemann-Hilbert technique in [28]. Let T = {z : |z| = 1} be the unit circle oriented counterclockwise and D = {z : |z| < 1} be the unit disc. If a nonnegative integrable function w satisfies 9 T

w(t)|dt| > 0,

(1.1)

such a function w is usually called a weight on the unit circle. The inner product with respect to the weight w on the unit circle is defined by 9 f, g =

T

f (t)g(t)w(t)|dt|,

(1.2)

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which induces 5 5 5f 5 = 2

39

41 |f (t)| w(t)|dt| 2

T

2

.

With respect to the inner product defined by (1.2), orthonormalisation of the ordered base 6 7 1, z−1 , z, z−2 , z2 , · · · , z−n , zn , · · · leads Laurent polynomials on the unit circle, denoted : to the system of orthonormal + as ϕn : n ∈ Z+ 0 } where Z0 = {0, 1, 2, · · · } and ϕ2n (z) = c−n z−n + · · · + cn zn , cn > 0, ϕ2n+1 (z) = c−n−1 z−n−1 + c−n z−n + · · · + cn zn , c−n−1 > 0,

(1.3) (1.4)

where cn and c−n−1 are, respectively, called the leading coefficients of ϕ2n and ϕ2n+1 . Therefore, one has 9 T

ϕm (t)ϕn (t)|dt| = δm,n ,

where δm,n is Kronecker’s symbol. Furthermore, c−n in (1.3) and cn in (1.4) are respectively called the trailing coefficients of ϕ2n , ϕ2n+1 . Let κ2n = cn , κ2n+1 = c−n−1 ,

(1.5)

and hence κn is the leading coefficient of ϕn . As in [26, 27], we define the monic OLP on the unit circle πn (z) =

ϕn (z) . κn

(1.6)

Hereafter, the trailing coefficient of the monic OLP on the unit circle πn is denoted as χn .   In general, the monic OLPs on the unit circle πn n ∈ Z+ 0 possess the following basic properties: < ; 1. π2n , t  = 0 for  = −n, −n + 1, −n + 2, · · · , n − 2, n − 1; < ; 2. π2n+1 , t  = 0 for  = −n, −n + 1, · · · , n + 1, n; ; < 5 52 1 3. πn , πn = 5πn 52 = κn−2 , whence κn = > 0. πn 2

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For convenience, we define the subset of Laurent polynomials m,n =

n 

. ck z : ck ∈ C, k = m, m + 1, · · · , n k

if m ≤ n.

(1.7)

k=m

If m > n, we assume m,n = {0}. Hereafter, as in [2], we always assume that w is a strictly positive analytic weight on the unit circle T satisfying (1.1). Consequently, the weight w can be analytically extended to the biggest annular with center at the origin   6 7 D 0, ρ, ρ −1 = z : ρ < |z| < ρ −1 for a fixed ρ ∈ (0, 1). This article is organized as follows. In the next section, we will set up a matrix Riemman-Hilbert problem which characterises the monic OLPs on the unit circle. In Sect. 3, according to Deift-Zhou steepest-descent method, three reversible transformations have been constructed, and then the matrix RiemannHilbert problem constructed in Sect. 2 has been equivalently changed into a model problem. In Sect. 4, by the so-called symmetric extension, the unique solution of the model matrix Riemann-Hilbert problem has been presented as the summation of iterating Cauchy-type integrals. In the last section, strong asymptotic expansions of the monic OLPs on the unit circle are obtained from the asymptotic solution of the model matrix Riemann-Hilbert problem. At the same time, the corresponding expansions of the leading coefficient and the trailing coefficient of the monic OLPs are obtained.

2 Characterization of the Monic OLPs: Riemann-Hilbert Problem In this section, we first construct a matrix Riemman-Hilbert problem, called the Fokas-Its-Kitaev characterization, which characterises the monic OLPs of odd and even degree on the unit circle. The Fokas-Its-Kitaev characterization is the cornerstone of Riemman-Hilbert technique. We start from the matrix Riemman-Hilbert problem: find a sectionally holomorphic matrix function Y : C\T → SL2 (C) satisfying the following three conditions: (RH1) Y possesses the boundary value on the unit circle T, i.e., Y + (t) =

lim

z∈D, z→t

Y (z), Y − (t) =

lim z∈C\D, z→t

Y (z), t ∈ T

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exist, and satisfies the matrix-type boundary condition Y + (t) = Y − (t)



1 t −2n+1 w(t) 0 1

 , t ∈ T.

(2.1)

(RH2) Y has the asymptotic behavior at infinity 

z−2n 0 Y (z) 0 z2n−2



 =

 o(1) o(1) , z → ∞. o(1) o(1)

(2.2)

(RH3) Y has the asymptotic behavior near the origin  Y (z) = I +

 o(1) o(1) , z → 0, o(1) o(1)

(2.3)

where I is the identity matrix. The problem satisfying all the mentioned above conditions (RH1), (RH2) and (RH3) is called the 2 × 2 matrix Riemann-Hilbert problem, simply MRHP for Y . Let   Y11 (z) Y12 (z) , z ∈ C \ T, Y (z) = Y21 (z) Y22 (z) and the matrix Riemann-Hilbert problem (RH1–RH2–RH3) is equivalent to the following four scalar Riemann-Hilbert problems (SRHPs): ⎧ + − Y (t) = Y11 (t), t ∈ T, ⎪ ⎪ ⎨ 11 −2n z Y11 (z) = o(1), z → ∞, ⎪ ⎪ ⎩ Y11 (z) = 1 + o(1), z → 0,

(2.4)

⎧ + − − Y (t) = Y12 (t) + t −2n+1 w(t)Y11 (t), t ∈ T, ⎪ ⎪ ⎨ 12 2n−2 z Y12 (z) = o(1), z → ∞, ⎪ ⎪ ⎩ z → 0, Y12 (z) = o(1),

(2.5)

⎧ + − Y (t) = Y21 (t), t ∈ T, ⎪ ⎪ ⎨ 21 z−2n Y21 (z) = o(1), z → ∞, ⎪ ⎪ ⎩ Y21 (z) = o(1), z→0

(2.6)

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and ⎧ + − − Y (t) = Y22 (t) + t −2n+1 w(t)Y21 (t), t ∈ T, ⎪ ⎪ ⎨ 22 z2n−2 Y22 (z) = o(1), z → ∞, ⎪ ⎪ ⎩ z → 0. Y22 (z) = 1 + o(1),

(2.7)

2.1 Solutions of SRHPs (2.4) and (2.5) Obviously, Riemann-Hilbert problem (2.4) is solvable and its solution can be represented as Y11 (z) = 1 + c1 z + c2 z2 + · · · + ck zk + · · · + c2n−1 z2n−1 =: zn q2n−1 (z),

(2.8)

where q2n−1 ∈ −n,n−1 and the leading coefficient of q2n−1 is 1. Inserting (2.8) into (2.5), and setting 12 (z) = z2n−2 Y12 (z), Y

(2.9)

one immediately has ⎧ + (t) = Y − (t) + t n−1 q2n−1 (t)w(t), t ∈ T, ⎪ ⎨Y 12 12 (z) = o(1), z→∞ Y ⎪ 12 ⎩

(2.10)

and   12 (z) = o z2n−2 , z → 0. Y

(2.11)

Consequently, we have the following result and the proof is evident. 12 Lemma 2.1 If Y12 is the solution of Riemann-Hilbert problem (2.5), then Y defined by (2.9) is the solution of Riemann-Hilbert problem ((2.10)–(2.11)). Con 12 is the solution of Riemann-Hilbert problem ((2.10)–(2.11)), then versely, if Y −2n+2 12 (z) is the solution of Riemann-Hilbert problem (2.5). Y Y12 (z) = z Riemann-Hilbert problem (2.10) is solvable and its solution is 12 (z) = 1 Y 2πi

9 T

t n−1 q2n−1 (t)w(t) dt, z ∈ C \ T. t −z

(2.12)

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Since 2n−2  zk 1 z2n−1 = , t ∈ T, z ∈ D, + t −z t k+1 t 2n−1 (t − z) k=0

12 (z) given by (2.12) can be rewritten as Y 12 (z) = Y

4 9 1 t n−k−2 q2n−1 (t)w(t)dt zk 2πi T k=0 4 3 9 q2n−1 (t)w(t) 1 dt z2n−1 , z ∈ D. + 2πi T t n (t − z) 2n−2 3

(2.13)

According to the expansion (2.13), the asymptotic condition (2.11) near the origin is equivalent to 1 2πi

9 T

t n−k−2 q2n−1 (t)w(t)dt = 0, k = 0, 1, · · · , 2n − 2.

(2.14)

In terms of the inner product defined by (1.2), the conditions of solvability (2.14) can be represented as = > q2n−1 , t  = 0,  = −n + 1, −n + 2, · · · , n − 1.

(2.15)

Since q2n−1 ∈ −n,n−1 and the leading coefficient of q2n−1 is 1, the conditions of solvability (2.15) implies q2n−1 (z) = π2n−1 (z), where π2n−1 is the monic OLP defined by (1.6). To sum up the discussion above, Riemann-Hilbert problems (2.4) and (2.5) are respectively solvable, and their solutions are respectively expressed by ⎧ n z ∈ C \ T, ⎪ ⎨ Y11 (z) = z π2n−1 (z), 9 t n−1 π2n−1 (t)w(t) z−2n+2 ⎪ ⎩ Y12 (z) = dt, z ∈ C \ T, 2πi t −z T

(2.16)

where π2n−1 is just the monic OLP. In particular, by Lemma 2.1, combining (2.13) with (2.14), one has z Y12 (z) = 2πi

9 T

π2n−1 (t)w(t) dt, z ∈ D. t n (t − z)

(2.17)

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2.2 Solutions of SRHPs (2.6) and (2.7) Analogously to the discussion in Sect. 2.1, the solution of R–H problem (2.6) can be written as Y21 (z) =

2n−1 

ck zk = zn (c1 z−n+1 + · · · + c2n zn−1 ) =: zn q2n−2 (z)

(2.18)

k=1

with q2n−2 ∈ −n+1,n−1 . Putting (2.18) into (2.7), and setting 22 (z) = z2n−2 Y22 (z), z ∈ C \ T, Y

(2.19)

one easily gets ⎧ + (t) = Y − (t) + t n−1 q2n−2 (t)w(t), t ∈ T, ⎪ ⎨Y 22 22 (z) = o(1), z→∞ Y ⎪ 22 ⎩

(2.20)

and 22 (z) = 1. lim z−2n+2 Y

z→0

(2.21)

Analogously to Lemma 2.1, one also has the following result. 22 Lemma 2.2 If Y22 is the solution of Riemann-Hilbert problem (2.7), then Y defined by (2.19) is the solution of Riemann-Hilbert problem ((2.20)–(2.21)). 22 is the solution of Riemann-Hilbert problem ((2.20)–(2.21)), then Conversely, if Y 22 (z) is the solution of Riemann-Hilbert problem (2.7). Y22 (z) = z−2n+2 Y In terms of (2.20), one has 22 (z) = 1 Y 2πi

9 T

t n−1 q2n−2 (t)w(t) dt, z ∈ C \ T. t −z

Taking into account the expression 2n−3  zk 1 z2n−2 = + , t ∈ T, z ∈ D, t −z t k+1 t 2n−2 (t − z) k=0

(2.22)

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(2.22) can be represented as 22 (z) = Y

2n−3 3

4 9 1 t n−k−2 q2n−2 (t)w(t)dt zk 2πi T k=0 4 3 9 q2n−2 (t)w(t) 1 dt z2n−2 , z ∈ D. + 2πi T t n−1 (t − z)

(2.23)

Combining (2.21) with (2.23), one easily knows that (2.22) is the solution of Riemann-Hilbert problem ((2.20)–(2.21)) if and only if 9 T

t n−k−1 q2n−2 (t)w(t)

dt = 0, k = 0, 1, · · · , 2n − 3 t

(2.24)

and 1 2πi

9 T

q2n−2 (t)w(t) dt = 1. tn

(2.25)

The conditions of solvability (2.24) are equivalent to = > q2n−2 , t  = 0,  = −n + 1, −n + 2, · · · , n − 2.

(2.26)

Consequently, (2.25) leads to =

> =q > 2n−2 q2n−2 , t n−1 = 2π $⇒ , π2n−2 = 1, 2π

(2.27)

2 which in turn leads to q2n−2 (z) = 2πκ2n−2 π2n−2 (z), where κ2n−2 is the leading coefficient defined by (1.5). In general, the solutions of Riemann-Hilbert problems (2.6) and (2.7) can respectively be written as

⎧ 2 n z ∈ C \ T, ⎪ ⎨ Y21 (z) = 2πκ2n−2 z π2n−2 (z), 9 n−1 t π2n−2 (t)w(t) ⎪ 2 ⎩ Y22 (z) = −iκ2n−2 dt, z ∈ C \ T, z−2n+2 t −z T

(2.28)

where κ2n−2 is the leading coefficient of π2n−2 defined by (1.6). In addition, by Lemma 2.2, taking into account (2.23) and (2.24), one has 9 Y22 (z) =

2 −iκ2n−2

T

π2n−2 (t)w(t) dt, z ∈ D. t n−1 (t − z)

To sum up the discussion above, one has the following result.

(2.29)

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Theorem 2.1 The matrix Riemann-Hilbert problem (RH 1–RH 2–RH 3) has the unique solution expressed as Y (z) ⎛

9

⎞ t n−1 π2n−1 (t)w(t) dt ⎜ ⎟ t −z T ⎟, =⎜ 9 n−1 ⎝ ⎠ t π (t)w(t) 2n−2 2 2 2πκ2n−2 dt zn π2n−2 (z) −iκ2n−2 z−2n+2 t −z T zn π

z−2n+2 2πi

2n−1 (z)

(2.30)

z ∈ C \ T. where π2n−1 , π2n−2 are the monic OLPs on the unit circle defined by (1.6), and κ2n−2 is the leading coefficient of the OLP ϕ2n−2 defined by (1.5). In particular, ⎛

9

⎞ π2n−1 (t)w(t) dt n ⎜ ⎟ T9 t (t − z) ⎟ , z ∈ D. Y (z) =⎜ ⎝ π2n−2 (t)w(t) ⎠ 2 2 2πκ2n−2 dt zn π2n−2 (z)−iκ2n−2 n−1 (t − z) T t zn π2n−1 (z)

z 2πi

(2.31)

3 Steepest Descent Transformation of OLPs 3.1 First Transformation Y → T We will formulate an equivalent MRHP, whose coefficient matrix in the boundary condition possesses rapidly oscillating entries. To this end, let ⎧  ⎪ z−2n 0 ⎪ ⎨ , z ∈ C \ D, 0 z2n−2 U (z) = (3.1) ⎪ ⎪ ⎩ I, z ∈ D, where I is the identity matrix. Now we define the first transformation T (z) = Y (z)U (z), z ∈ C \ T.

(3.2)

Therefore, T is a sectionally holomorphic function with a jump on the unit circle T, and satisfies the following boundary condition T + (t)= Y + (t)U + (t)  −2n+1  w(t) 1t = Y − (t) 0 1

Strong Asymptotic Analysis of OLPs on the Unit Circle by Riemann-Hilbert Approach

−



= T (t) = T − (t)



t 2n 0 0 t −2n+2 t 2n tw(t) 0 t −2n+2



1 t −2n+1 w(t) 0 1

149



 , t ∈ T.

By the asymptotic condition (2.2) at infinity, one has lim T (z) = 0. Taking into z→∞

account the asymptotic condition (2.3) near the origin, one also gets lim T (z) = I. z→0

Consequently, we get the following equivalent Riemann-Hilbert problem for T : ⎧  2n  t tw(t) ⎪ + − ⎪ T (t) = T (t) , t ∈ T, ⎪ ⎪ ⎨ 0 t −2n+2 T = o(1), z → ∞, ⎪ ⎪ ⎪ ⎪ ⎩ T = I + o(1), z → 0,

(3.3)

where I is just the identity matrix.

3.2 Second Transformation T → S This transform is derived from a factorization of the coefficient matrix of boundary condition in (3.3). In fact, 

t 2n t w(t)



0 t −2n+2

   1 0 0 tw(t) 1 0 . = −2n+1 −1 w (t) 1 −tw−1 (t) 0 t t 2n−1 w−1 (t) 1 

(3.4)

? : With 0 < r < 1, we define Tr = {z : |z| = r}, T1/r = z : |z| = r −1 and − − − L = T− r + T + T1/r , where Tr and T1/r are oriented clockwise and T is oriented counterclockwise. The contour L divides the complex plane C into two parts (see Fig. 1) − + + + A − = A− 1 ∪ A2 , A = A1 ∪ A2

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Fig. 1 The contour − L = T− r + T + T1/r

with ⎧ + ⎨ A− 1 = {z : |z| < r}, A1 = {z : r < |z| < 1} ⎩ A− = {z : r −1 > |z| > 1}, A+ = :z : |z| > r −1 ?. 2 2 We define the second transformation S(z) = T (z)V (z), z ∈ C \ L,

(3.5)

where

V (z) =

⎧ ⎪ I, ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩



1

0

−z2n−1 w−1 (z)

1  0

1

z−2n+1 w−1 (z) 1

Obviously, the asymptotic conditions of S at consistent with T . And hence ⎧ + S (t) = S − (t)ϒ(t), ⎪ ⎪ ⎨ S = o(1), ⎪ ⎪ ⎩ S = I + o(1),

+ z ∈ A− 1 ∪ A2 ,

, z ∈ A+ 1,

(3.6)

, z ∈ A− 2. infinity and near the origin are t ∈ L, z → ∞, z → 0,

(3.7)

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where  ⎧ 1 0 ⎪ ⎪ ⎪ , t ∈ T− r , ⎪ ⎪ ⎪ −t 2n−1 w−1 (t) 1 ⎪ ⎪   ⎪ ⎪ ⎨ 0 tw(t) , t ∈ T, ϒ(t) = −tw−1 (t) 0 ⎪ ⎪ ⎞ ⎛ ⎪ ⎪ ⎪ ⎪ 1 0 ⎪ ⎪ ⎪ ⎠ , t ∈ T− . ⎝ ⎪ 1/r ⎩ −2n+1 −1 w (t) 1 −t

(3.8)

The matrix Riemann-Hilbert problem (3.7) for S is equivalent to that for T in Sect. 3.1. In fact, on T− r , one has ⎛

1

0

⎞

⎠ 2n−1 w −1 (t) 1 −t ⎞ ⎛ 1 0 ⎠. = S − (t) ⎝ −t 2n−1 w−1 (t) 1

S + (t) = T + (t) ⎝

Analogously, S + (t)



= T + (t)

1

0



−t 2n−1 w−1 (t) 1 ⎞ ⎛  1 0 2n tw(t) 1 0 t ⎠ ⎝ = S − (t) t −2n+1 − w(t ) 1 0 t −2n+2 − t 2n−1 1 w(t ) ⎞ ⎛ 0 tw(t) ⎠, t ∈ T = S − (t) ⎝ −1 −tw (t) 0 

and ⎛ S + (t) = T + (t) = T − (t) = S − (t)⎝

1

0

−t −2n+1 w−1 (t)

1

⎞ ⎠ , t ∈ T− . 1/r

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3.3 Third Transformation S → R Let 

1 D (z) = exp 2πi +

9 T

, log w(t) dt , z ∈ D, t −z

,  9 log w(t) 1 D − (z) = exp − dt , z ∈ C \ D, 2πi T t − z

(3.9)

(3.10)

which constitute the unique solution of the following SRHP ⎧ D(z) = 0, z ∈ C \ T, ⎪ ⎪ ⎨ + − D (t)D (t) = w(t), t ∈ T, ⎪ ⎪ ⎩ D(∞) = 1. The principal branch of the logarithm in (3.9) and (3.10) is chosen such that log 1 = 0. + − As in [1], and D (z) can be holomorphically extended to the biggest  D (z) −1 defined in Sect. 1, respectively. Hereafter, we always assume annular D 0, ρ, ρ   that D + (z) and D − (z) are holomorphic on the annular D 0, ρ, ρ −1 . Furthermore, the functions D + (z) and D − (z) are closely related with the so-called Szegö function on the unit circle ,   2π 1 eiθ + z dθ . D(z) = exp log w(eiθ ) iθ 4π 0 e −z Now, setting  ⎧ + (z) 0 −D ⎪ ⎪ ⎪ , z ∈ D, & % ⎪ ⎪ ⎨ D + (z) −1 0 W (z) =   ⎪ ⎪ 0 zD − (z) ⎪ ⎪ ⎪ , z ∈ C \ D, ⎩ 0 z[D − (z)]−1

(3.11)

one easily gets 

0

tw(t)

−tw−1 (t)

0



% &−1 , t ∈ T. = W − (t) W + (t)

(3.12)

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Fig. 2 The contour − L# = T− r + T1/r

In terms of the decomposition (3.12) of the coefficient matrix, we define the third transformation R(z) = S(z)W (z), z ∈ C \ L.

(3.13)

Taking into account (3.12), one easily gets R + (t) = R − (t), t ∈ T. In consequence, we can always assume that R given by (3.13) is holomorphic on the unit circle T hereafter. − Now, let L- = T− r + T1/r , where Tr is oriented clockwise but T1/r is oriented counterclockwise. The contour L- divides the complex plane C into two parts (see Fig. 2) 7 6 − B+ = z : r < |z| < r −1 , B− = B− 1 ∪ B2 , ? : − −1 . On T− , one has where B− r 1 = {z : |z| < r}, B2 = z : |z| > r R + (t) =



S + (t) 

=

R − (t)

−D + (t)

0



[D + (t)]−1 0

0  1 D + (t) 2n−1

 0

− D +1(t ) 0 − tw(t ) 1 ⎛ 2n−1 + ⎞ [D (t)]2 t 1 ⎠ = R − (t) ⎝ w(t) 0 1

0

−D + (t)

[D + (t)]−1

0



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i.e., ⎛ ⎜1 R + (t) = R − (t) ⎝ 0

⎞ t 2n−1 [D + (t)]2 ⎟ − w(t) ⎠ , t ∈ Tr . 1

Similarly, on T− 1/r , one has R + (t)   − (t) 0 tD = S + (t) 0 t[D − (t)]−1  −1  ⎛ − t tD (t) 1 0 0 − ⎝ = R − (t) D (t ) − −2n+1 t 0 D t (t ) − w(t ) 1 0 ⎞ ⎛ 1 0 ⎟ ⎜ = R − (t) ⎝ [D − (t)]2 ⎠ , − 2n−1 1 t w(t)

0 t

⎞ ⎠

D − (t )

which implies ⎛

1

0

⎞

⎜ ⎟ R + (t) = R − (t) ⎝ [D − (t)]2 ⎠ , 1 t 2n−1 w(t)

t ∈ T1/r .

It is obvious that the third transformation (3.13) doesn’t change the asymptotic condition near the origin, and the asymptotic condition at infinity is R(z) = O(1), z → ∞. Consequently, we obtain the model MRHP for R: ⎧ + R (t) = R − (t)G(t), t ∈ L- , ⎪ ⎪ ⎨ R(z) = O(1), z → ∞, ⎪ ⎪ ⎩ R(z) = I + o(1), z→0

(3.14)

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155

with ⎧⎛ 2n−1 + 2⎞ ⎛ 2n−1 + ⎞ D (t) t t [D (t)] ⎪ ⎪ ⎪ 1 1 ⎪ ⎝ ⎠ ⎝ − = ⎪ D (t) ⎠ , t ∈ T− w(t) r , ⎪ ⎪ ⎪ ⎨ 0 1 0 1 ⎞ ⎛ G(t) = ⎛ ⎞ 1 0 ⎪ 1 0 ⎪ ⎪ ⎪ ⎟ ⎜ ⎪ ⎠, t ∈ T1/r . ⎪ ⎝ [D − (t)]2 ⎠= ⎝ D − (t) ⎪ ⎪ 1 ⎩ 1 2n−1 + 2n−1 t D (t) t w(t)

(3.15)

Finally, one states the result, which describes the relation of solutions of MRHP (RH1–RH2–RH3) and MRHP (3.14). Lemma 3.1 The unique solution of the model MRHP (3.14) for R can be represented as R(z) = Y (z)U (z)V (z)W (z), z ∈ C \ L- ,

(3.16)

where U, V , W are respectively defined by (3.1), (3.6) and (3.11), and Y is the solution of the matrix Riemann-Hilbert problem (RH1–RH2–RH3). Proof Since all the transformations, defined respectively by (3.2), (3.5) and (3.13), are reversible, the desired expression (3.15) is obtained according to the discussion above.   Remark 3.1 In terms of the expression (3.15), G(t) = I + O(r 2n−1 ), n → ∞. This indicates that the coefficient matrix G is uniformly close to the identity matrix I as n → ∞.

4 Strong Asymptotic Solution of the Model MRHP In order to give the strong asymptotic solution of the model MRHP (3.14) for R, we need the following lemma. Lemma 4.1 If R is the solution of the model MRHP (3.14), then R(z) = R

  1 , z ∈ C \ Lz

(4.1)

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is the solution of the matrix Riemann-Hilbert problem ⎧ + ⎨ R (t) = R − (t)G- (t), t ∈ L- , ⎩

R(z) = I + o(1),

(4.2)

z → ∞,

with ⎞ ⎧⎛ D-− (t) ⎪ ⎪ ⎪ ⎜ 1 − 2n−1 + ⎟ ⎪ ⎪ ⎝ t D- (t) ⎠ , ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎨ 0 ⎞ G- (t) = ⎛ ⎪ 1 0 ⎪ ⎪⎜ ⎪ ⎟ ⎪ ⎪ ⎜ t 2n−1 D + (t) ⎟ , ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ 1 ⎩ − D-− (t)

t ∈ T1/r , (4.3) t ∈ T− r ,

where ⎧ ,  9 log w(t) dt z ⎪ + ⎪ D , z ∈ D, (z) = exp ⎪ ⎪ ⎨ 2πi T t − z t ,  9 ⎪ ⎪ log w(t) dt z ⎪ − ⎪ , z ∈ C \ D. ⎩ D- (z) = exp − 2πi T t − z t

(4.4)

Conversely, if R is the solution of the matrix Riemann-Hilbert problem (4.2), then   1 , z ∈ C \ LR(z) = R z

(4.5)

is the solution of the model MRHP (3.14). Proof If R is the solution of the model MRHP (3.14) and R is defined by (4.1), one easily has ±

R (t) =

R∓

  1 , t ∈ L- . t

Consequently, the boundary condition in (3.14) is changed to +

−

R (t) = R (t)G- (t), t ∈ L-

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157

with /  0−1 1 G- (t) = G , t

(4.6)

where the function G is given by (3.15). On the other hand, by (4.4), a simple calculation leads to D+

 ,   9 z 1 log w(t) dt = exp − = D-− (z), z ∈ C \ D z 2πi T t − z t

(4.7)

 ,   9 z 1 log w(t) dt = exp = D-+ (z), z ∈ D. z 2πi T t − z t

(4.8)

and D−

Combining (4.6)–(4.8), one can easily get the expression (4.3). Therefore, R is the solution of the model MRHP (3.14). Analogously, if R is the solution of the matrix Riemann-Hilbert problem (4.2), R defined by (4.5) is the solution of the model MRHP (3.14). This completes the proof.   Let (z) = R(z) − I, z ∈ C \ L- ,

(4.9)

and the matrix Riemann-Hilbert problem (4.2) is equivalently changed to ⎧ % & ⎨ + (t) = − (t) + R − (t) G- (t) − I , t ∈ L- . ⎩

(z) = o(1),

(4.10)

z → ∞.

Clearly, the solutions of two independent matrix Riemann-Hilbert problems (4.2) and (4.10) are connected by the relation (4.9). We need the Cauchy-type integral operator C[f ](z) =

1 2πi

9 L-

f (t) dt, z ∈ C \ L- . t −z

(4.11)

Next, we also have the following result, which describes the equivalent relation between the matrix Riemann-Hilbert problem (4.10) and the system of singular integral equations [16, 19, 20].

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Lemma 4.2 If  is the solution of the matrix Riemann-Hilbert problem (4.10), then the negative boundary-value − solves the system of singular integral equations − (t) − C − [− (G- − I )](t) = C − [G- − I ](t), t ∈ L-

(4.12)

with C − [− (G- − I )](t) =

C[− (G- − I )](z), t ∈ L- ,

lim

z∈B− ,z→t

(4.13)

where C is the Cauchy-type integral operator defined by (4.11), and B− is defined in Sect. 3.3. Conversely, if − is the solution of the system (4.12) of singular integral equations, then C[− (G- − I )](z) is the solution of the matrix Riemann-Hilbert problem (4.10). Proof We only prove the former part of the lemma, and the latter part is similarly verified. Indeed, in terms of the classical theory of Riemann-Hilbert problem, the unique solution of the matrix Riemann-Hilbert problem (4.10) can be written as "

#

1 (z) = C R (G- − I ) (z) = 2πi −

9 L-

% & − R (t) G- (t) − I dt, z ∈ C\L- , t −z (4.14)

which in turn implies that its negative boundary-value − satisfies (4.12).

 

For convenience, we define the operator C- [f ] = C − [f (G- − I )], where C − is defined by (4.13). And the system (4.12) can be reduced to (I − C- )[− ](t) = C- [I ](t), t ∈ L- , where I is the identity operator and 5 5 5C- 5 ≤ Mr 2n−1 for some M > 0, where  ·  is the usual norm of operator. In consequence, analogously to [2], the solution of the system (4.12) of singular integral equations can be represented via Neumann series − (t) =

∞  k=1

C-k [I ](t), t ∈ L- ,

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159

which leads to %

R

&−

(t) = I +

∞ 

C-k [I ](t), t ∈ L- .

(4.15)

k=1

By Lemma 4.2, inserting (4.15) into (4.14), one immediately has R(z) = I + C[G- − I ](z) + =I+

∞ 

∞ " #  C C-k [I ] · (G- − I ) (z) k=1

(4.16)

R k+1 (z), z ∈ C \ L- ,

k=0

where the convergence of the summation is locally uniform. In order to give the expression R(z) explicitly, we define two Cauchy type integral operators r C2n−1 [f ](z) = −

9

1 2πi

T− r

f − (t)

t 2n−1 D-+ (t) dt , |z| = r D-− (t) t − z

(4.17)

and 1/r

C2n−1 [f ](z) = −

1 2πi

9 T1/r

f − (t)

D-− (t) t 2n−1 D-+ (t)

dt , |z| = 1/r. t −z

(4.18)

Analogously to [2], we recursively define functions 1/r

1 r 2 1 f2n−1 (z) = C2n−1 [1](z), f2n−1 (z) = C2n−1 [f2n−1 ](z), % 2k & 1/r 2k+1 2k+2 2k+1 r f2n−1 f2n−1 (z), f2n−1 (z) = C2n−1 (z) = C2n−1 [f2n−1 ](z)

(4.19)

and 1/r

1 2 r 1 g2n−1 (z) = C2n−1 [1](z), g2n−1 (z) = C2n−1 [g2n−1 ](z), 1/r

(4.20)

2k+1 2k+2 2k+1 2k r g2n−1 (z) = C2n−1 [f2n−1 ](z), g2n−1 (z) = C2n−1 [f2n−1 ](z), 1/r

r and C2n−1 are defined, respectively, by (4.17) and (4.18). where the operators C2n−1 Following Martínez-Finkelshtein et al. in [2], we similarly have the following.

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Lemma 4.3 The unique solution of the matrix Riemann-Hilbert problem (4.2) can be expressed by ⎛ ⎞ ∞ ∞   2k+1 2k g2n−1 (z) g2n−1 (z) ⎟ ⎜1 + ⎜ ⎟ k=1 k=0 ⎟ , z ∈ C \ L- , R(z) = ⎜ (4.21) ∞ ∞ ⎟ ⎜   ⎝ ⎠ 2k+1 2k f2n−1 (z) 1 + f2n−1 (z) k=0

k=1

where the summation is uniformly convergent on any compact K ⊂ C \ L- . Proof A simple calculation leads to R 1 (z) = C[G- − I ](z) 1 = 2πi

9

1 =− 2πi

= and

L-

G- (t) − I dt t −z

9 T− r

t 2n−1 D-+ (t) dt D-− (t) t − z



00



10   9 01 D-− (t) 1 dt − 2πi T1/r t 2n−1 D-+ (t) t − z 0 0   1 (z) 0 g2n−1 1 (z) f2n−1

0

" # R 2 (z) = C C- [I ] · (G- − I ) (z) 9

−

R 1 (t){G- (t) − I } dt t −z L9  0 g 1,− (z) 00  t 2n−1 D + (t) 1 dt 2n−1 =− − 1,− − 2πi Tr f D- (t) t − z 10 0 2n−1 (z)    1,− 9 (z) 01 0 g2n−1 D-− (t) 1 dt − + 2n−1 1,− 2πi T1/r f D- (t) t − z 00 t 0 2n−1 (z)  2  g2n−1 (z) 0 = , 2 0 f2n−1 (z) =

j

j

1 2πi

where g2n−1 , f2n−1 , j = 1, 2 are defined by (4.19) and (4.20), respectively.

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161

By induction, one easily gets  R 2k−1 (z) =

0

2k−1 (z) g2n−1

2k−1 f2n−1 (z)

0

 (4.22)

and  R 2k (z) =



2k (z) g2n−1

0

0

2k (z) f2n−1

(4.23)

for k = 1, 2, 3, · · · . Inserting (4.22) and (4.23) into (4.16), we get (4.21). This completes the proof.   Finally, we get the asymptotic solution of the model Riemann-Hilbert problem (3.14). Theorem 4.1 The unique solution of the Riemann-Hilbert problem (3.14) for R can be expressed by ⎛

∞ 

⎜1 + ⎜ k=1 R(z) = ⎜ ∞ ⎜  ⎝ f 2k+1 (z)

2k (z) g2n−1

2n−1

k=0

∞ 

⎞ 2k+1 g2n−1 (z)

k=0 ∞ 

1+

2k f2n−1

⎟ ⎟ ⎟ , z ∈ C \ L- , ⎟ (z) ⎠

(4.24)

k=1

where the summation is uniformly convergent on any compact K ⊂ C \ L- and j

j

g2n−1 , f2n−1 are defined by (4.1). Proof By Lemmas 4.2 and 4.3, the unique solution of the Riemann-Hilbert problem (3.14) can be represented as (4.24).  

5 Strong Asymptotic Analysis of the Monic OLPs In this section, one comes to discuss the asymptotic behavior of OLPs in four cases. The idea is to turn the asymptotic expansion of Y into that of R, which is expressed by (4.24). Case 1 If |z| < r, taking into account (3.1), (3.6) and (3.11), one has ⎛ U (z) = I, V (z) = I, W (z) = ⎝

0

−D + (z)

[D + (z)]−1

0

⎞ ⎠.

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By Lemma 3.1, ⎛ Y (z) = R(z) ⎝ ⎛ = R(z) ⎝ ⎛

0

−D + (z)

[D + (z)]−1

0

0

D + (z)

−[D + (z)]−1

0

⎞−1 ⎠ ⎞ ⎠

(5.1)

⎞ R12 (z) + D − (z)R (z) 11 ⎜ D + (z) ⎟ ⎟ =⎜ ⎝ R22 (z) ⎠ + D (z)R21 (z) − + D (z) with ⎛ R(z) = ⎝

R11 (z) R12 (z)

⎞ ⎠.

R21 (z) R22 (z) Inserting (4.24) into (5.1), we immediately obtain .⎞ ∞ ∞  1  2k+1 + 2k g2n−1 (z) D (z) 1 + g2n−1 (z) ⎟ ⎜ − + D (z) ⎜ ⎟ k=0 k=1 ⎜ ⎟. . Y (z) =⎜ ∞ ∞ ⎟   1 ⎝ ⎠ 2k+1 2k f2n−1 (z) D + (z) f2n−1 (z) − + 1+ D (z) ⎛

k=1

(5.2)

k=0

Combining (2.31) with (5.2), one gets ∞

π2n−1 (z) = −

z−n  2k+1 g2n−1 (z), |z| < r, D + (z)

(5.3)

k=0

z 2πi

9 T

. ∞  π2n−1 (t)w(t) + 2k dt = D (z) 1 + g2n−1 (z) , |z| < r, t n (t − z)

. ∞  z−n 2k 1+ f2n−1 (z) , |z| < r, π2n−2 (z) = − 2 2πκ2n−2 D + (z) k=1 9 2 − iκ2n−2

(5.4)

k=1

T

(5.5)

∞

 π2n−2 (t)w(t) 2k+1 dt = D + (z) f2n−1 (z), |z| < r. n−1 t (t − z) k=0

(5.6)

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Case 2 If r < |z| < 1, one has ⎛ U (z) = I,

V (z) = ⎝ ⎛

1

0

−z2n−1 w−1 (z) 1

⎞ ⎠,

⎞ −D + (z) ⎠. W (z) = ⎝ + −1 0 [D (z)] 0

By Lemma 3.1, Y (z) ⎛

⎞−1⎛ ⎞−1 −D + (z) 1 0 ⎠ ⎝ ⎠ = R(z)⎝ 0 −z2n−1 w−1 (z) 1 [D + (z)]−1 ⎛ = R(z) ⎝ ⎛ = R(z) ⎝

0

0

D + (z)

−[D + (z)]−1

⎞⎛ ⎠⎝

0

1

0

z2n−1 w−1 (z)

1

z2n−1 w−1 (z)D + (z)

D + (z)

−[D + (z)]−1

0

⎞ ⎠ (5.7)

⎞ ⎠

⎞ z2n−1 R11 (z) R12 (z) + ⎜ D − (z) − D + (z) D (z)R11 (z) ⎟ ⎟ ⎜ =⎜ ⎟. ⎠ ⎝ z2n−1 R21 (z) R22 (z) + − D (z)R (z) 21 D − (z) D + (z) ⎛

By Theorem 4.1, the expression of Y (z) given by (5.7) is 0 / 0⎞ / ∞ ∞ ∞   z2n−1 1  2k+1 + 2k 2k g2n−1 (z) − + g2n−1 (z)D (z) 1+ g2n−1 (z) ⎟ ⎜ D − (z) 1+ D (z) ⎜ ⎟ k=1 k=0 k=1 ⎜ ⎟. / 0 ∞ ∞ ∞ ⎜ 2n−1  ⎟   z 1 ⎝ ⎠ 2k+1 2k+1 + 2k (z) 1+ f (z)− f D (z) f (z) 2n−1 2n−1 2n−1 D − (z) D + (z) ⎛

k=0

k=1

k=0

(5.8)

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By (2.31) and (5.8), one has / 0 ∞ ∞  zn−1 z−n  2k+1 2k 1+ π2n−1 (z) = − g2n−1 (z) − + g2n−1 (z), r < |z| < 1, D (z) D (z) k=1

z 2πi

9 T

(5.9)

k=0

/ 0 ∞  π2n−1 (t)w(t) + 2k dt = D (z) 1+ g2n−1 (z) , r < |z| < 1, t n (t − z)

(5.10)

k=1

π2n−2 (z) (r < |z| < 1) / 0 ∞ ∞   zn−1 z−n 2k+1 2k = 1+ f (z) − f2n−1 (z) , 2 2 2πκ2n−2 D − (z) k=0 2n−1 2πκ2n−1 D + (z) k=1 9 2 − iκ2n−2

(5.11)

∞

 π2n−2 (t)w(t) 2k+1 dt = D + (z) f2n−1 (z), . n−1 t (t − z)

T

(5.12)

k=0

Case 3 If 1 < |z| < 1/r, one has  U (z) =

z−2n



0

0 z2n−2  W (z) =

 , V (z) =

1

0

z−2n+1 w−1 (z) 1

 ,



zD − (z)

0

0

z[D − (z)]−1

.

Analogously, by Lemma 3.1, Y (z)= R(z)

 z−1 [D − (z)]−1 

0 z2n

0

0

z−2n+2

⎛

0



1

 0

z−1 D − (z) −z−2n+1 w−1 (z) 1 

⎞ z2n−1 0 ⎜ D − (z) ⎟ ⎟ = R(z) ⎜ − ⎝ 1 D (z) ⎠ − + D (z) z2n−1 ⎛ 2n−1 ⎞ R11 (z) R12 (z) D − (z)R12 (z) z ⎜ D − (z) − D + (z) ⎟ z2n−1 ⎟ =⎜ ⎝ z2n−1 R21 (z) R22 (z) D − (z)R22 (z) ⎠ , − + D − (z) D (z) z2n−1

(5.13)

Strong Asymptotic Analysis of OLPs on the Unit Circle by Riemann-Hilbert Approach

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which implies, by Theorem 4.1, that Y can be represented as / 0 ⎞ ∞ ∞ ∞  z2n−1 1  2k+1 D − (z)  2k+1 2k g2n−1 (z) − + g2n−1 (z) g2n−1 (z) ⎟ ⎜ D − (z) 1+ D (z) z2n−1 ⎜ ⎟ k=1 k=0 k=0 ⎜ ⎟ ⎜ ⎟. / / 0 0 ∞ ∞ ∞ ⎜ 2n−1  ⎟   1 D − (z) ⎝ z ⎠ 2k+1 2k 2k 1+ 1+ f (z)− f (z) f (z) 2n−1 2n−1 2n−1 D − (z) D + (z) z2n−1 ⎛

k=0

k=1

k=1

(5.14) Consequently, combining (2.30) with (5.14), we have 0 / ∞ ∞  zn−1 z−n  2k+1 2k π2n−1 (z)= − g2n−1 (z) − + g2n−1 (z), 1 < |z| < 1/r, 1+ D (z) D (z) k=1

z 2πi

9

(5.15)

k=0

∞

T

 t n−1 π2n−1 (t)w(t) 2k+1 dt = D − (z) g2n−1 (z), 1 < |z| < 1/r, t −z

(5.16)

k=0

π2n−2 (z) =

∞  zn−1 f 2k+1 (z) 2 2πκ2n−2 D − (z) k=0 2n−1 / 0 ∞  z−n 2k 1+ f2n−1 (z) , 1 < |z| < 1/r, − 2 2πκ2n−2 D + (z) k=1

9 2 z − iκ2n−2

T

(5.17)

/ 0 ∞  t n−1 π2n−2 (t)w(t) − 2k dt =D (z) 1+ f2n−1 (z) , 1 < |z| < 1/r. t −z k=1 (5.18)

Case 4 If |z| > 1/r, one has ⎞ ⎞ ⎛ ⎛ 0 z−2n 0 zD − (z) ⎠ , V (z) = I, W (z) =⎝ ⎠. U (z) = ⎝ 0 z2n−2 0 z[D − (z)]−1

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By Lemma 3.1, ⎛ Y (z) = R(z) ⎝

⎞⎛

z−1 [D − (z)]−1

0

0

z−1 D − (z)

⎠⎝

z2n

0

0 z−2n+2

⎞ ⎠

⎞ z2n−1 0 ⎟ ⎜ D − (z) ⎟ = R(z) ⎜ ⎝ − D (z) ⎠ 0 z2n−1 ⎛ 2n−1 ⎞ z R11 (z) D − (z)R12 (z) ⎜ D − (z) ⎟ z2n−1 ⎟ =⎜ ⎝ z2n−1 R (z) D − (z)R (z) ⎠ . 21 22 D − (z) z2n−1 ⎛

(5.19)

Inserting (4.24) into (5.19), one gets 0 / ⎞ ∞ ∞  z2n−1 D − (z)  2k+1 2k g (z) g (z) 1+ ⎜D − (z) ⎟ 2n−1 2n−1 z2n−1 ⎜ ⎟ k=1 k=0 ⎜ ⎟ Y (z)=⎜ 0⎟. / ∞ ∞ ⎜ ⎟ 2n−1 −   D (z) ⎝ z ⎠ 2k+1 2k f (z) f (z) 1+ 2n−1 2n−1 − 2n−1 D (z) z ⎛

k=0

(5.20)

k=1

In consequence, combining (2.30) with (5.20), one has . ∞  zn−1 2k π2n−1 (z) = − 1+ g2n−1 (z) , |z| > 1/r, D (z)

(5.21)

k=1

z 2πi

9 T

-∞ .  t n−1 π2n−1 (t)w(t) 2k+1 − dt = D (z) g2n−1 (z) , |z| > 1/r, t −z

(5.22)

k=0

∞

π2n−2 (z) = 9 2 z −iκ2n−2

T

 zn−1 f 2k+1 (z), |z| > 1/r, 2 2πκ2n−2 D − (z) k=0 2n−1

/ 0 ∞  t n−1 π2n−2 (t)w(t) − 2k dt = D (z) 1+ f2n−1 (z) , |z| > 1/r. t −z

In general, our main result is obtained.

k=1

(5.23)

(5.24)

Strong Asymptotic Analysis of OLPs on the Unit Circle by Riemann-Hilbert Approach

167

Theorem 5.1 Let w, ρ be defined as in Sect. 1. For a fixed r with ρ < r < 1, the following asymptotic expansion of the monic OLP of odd degree is valid: π2n−1 (z) ⎧ ∞ ⎪ z−n  2k+1 ⎪ ⎪ − g2n−1 (z), |z| < r, ⎪ ⎪ ⎪ D + (z) ⎪ ⎪ k=0 ⎪ ⎪ / 0 ⎪ ⎪ ∞ ∞ ⎨ zn−1  z−n  2k+1 2k 1+ g (z) − g2n (z), r 1/r. ⎩ 2 2πκ2n−2 D − (z) k=0 2n−1

(5.26)

Finally, the leading coefficients possesses the following expansions: ⎧ ∞  ⎪ ⎪ −2 2k+1 ⎪ κ = −2π g2n−1 (0), ⎪ ⎪ ⎨ 2n−1 k=0

∞ ⎪ ⎪ 1  2k+1 ⎪ 2 ⎪ f2n−1 (0). ⎪ ⎩ κ2n−2 = 2π

(5.27)

k=0

Similarly, the trailing coefficient of the monic OLP has the expansion χ2n−1 = 1 +

∞  k=1

2k (0). g2n−1

(5.28)

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Proof The asymptotic expansions (5.25) and (5.26) are easily deduced from the preceding discussion. Therefore, we only prove (5.27) and (5.28) in the following. First, let z → ∞ in (5.22), and we get 9

-∞ .  dt 2k+1 − t π2n−1 (t)w(t) = −2πD (∞) g2n−1 (∞) , it T n

k=0

which results in π2n−1 22 = −2π

∞ 

2k+1 g2n−1 (0).

k=0

Consequently, the first equality in (5.27) is valid. Next, in terms of (5.23), one has π2n−2 (z) 2 κ2n−2 n−1 z

. -∞  1 2k+1 = f2n−1 (z) , |z| > 1/r, 2πD − (z) k=0

which similarly implies the validity of the second equality in (5.27). Finally, (5.21) is changed to / 0 ∞  1 π2n−1 (z) 2k 1+ = − g2n−1 (z) , |z| > 1/r, zn−1 D (z) k=1

in which, letting z → ∞, the desired expansion (5.28) is obtained.

 

Acknowledgements While the corresponding author visited Free University Berlin in summer 2005 on basis of State Scholarship Fund Award of China, our group began to explore the application of Riemann-Hilbert approach, and made a debut [3]. During that time Professor H. Begehr carefully reviewed this manuscript and offered a lot of suggestions. All the authors are very grateful to Professor H. Begehr for his long-term support and help. This work was supported by NNSF for Young Scholars of China (No. 11001206) and NNSF (No. 11171260).

References 1. G. Szeg˝o, Orthogonal Polynomials, 4th edn. AMS Colloquium Publications, vol. 23 (American Mathematical Society, Providence, 1975) 2. A. Martínez-Finkelshtein, K.T-R. McLaughlin, E.B. Saff, Szeg˝o orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics. Constr. Approx. 24, 319–363 (2006) 3. Z.H. Du, J.Y. Du, Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on the unit circle. Chinese Ann. Math. 27A(5), 701–718 (2006) 4. Z.H. Du, J.Y. Du, Orthogonal trigonometric polynomials: Riemann-Hilbert analysis and relations with OPUC. Asymptot. Anal. 79, 87–132 (2012)

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5. T. Kriecherbauer, K.T-R. McLaughlin, Strong asymptotics of polynomials orthogonal with respect to Freud weights. Int. Math. Res. Not. 6, 299–333 (1999) 6. P. Deift, T. Kriecherbauer, K.T-R. McLaughlin, S. Venakides, X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52, 1491–1552 (1999) 7. R. Bo, : R.?Wong, A uniform asymptotic formula for orthogonal polynomials associated with exp −x 4 . J. Approx. Theory 98, 146–166 (1999) 8. A.B.J. Kuijlaars, T.R. Mclaughlin, W.V. Assche, M. Vanlessen, The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1]. Adv. Math. 188(2), 337–398 (2004) 9. A.S. Fokas, A.R. Its, A.V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147(2), 395–430 (1992) 10. P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problem, asymptotics for the MKdV equation. Ann. Math. 137(2), 295–368 (1993) 11. A. Martínez-Finkelshtein, Szeg˝o polynomials: a view from the Riemann-Hilbert window. Electron. Trans. Numer. Anal. 25, 369–392 (2006) 12. A.B.J. Kuijlaars, Riemann-Hilbert analysis for polynomials, in Orthogonal Polynomials and Special Functions: Leuven 2002. Lecture Notes Mathematics, vol. 1817 (Springer, Berlin, 2003), pp. 167–210 13. P. Deift, Orthogonal polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, vol. 13 (Courant Institute of Mathematical Sciences, New York, 1999) 14. A.S. Fokas, A unified approach to boundary value problems, in CBMS-NSF Region Conference Series in Applied Mathematics, vol. 78 (Society for Industrial and Applied Mathematics, Philadelphia, 2008) 15. H. Begehr, Complex Analytic Methods for Partial Differential Equation: An Introductory Text (World Scientific, Singapore, 1994) 16. J.K. Lu, Boundary Value Problems For Analytic Functions (World Scientific, Singapore, 1993) 17. Y.F. Wang, Y.J. Wang, On Riemann problems for single-periodic polyanalytic functions. Math. Nachr. 287, 1886C1915 (2014) 18. Y.F. Wang, P.J. Han, Y.J. Wang, On Riemann problem of automorphic polyanalytic functions connected with a rotation group. Complex Var. Elliptic Equ. 60(8), 1033–1057 (2015) 19. F.D. Gakhov, Boundary Value Problems (Pergamon Press, Oxford, 1966) 20. N.I. Muskhelishvili, Singular Integral Equations, 2nd edn. (Noordhoff, Groningen, 1968) 21. J.A. Shohat, J.D. Tamarkin, The Problem of Moments. American Mathematical Society Surveys, vol. II (AMS, New York, 1943) 22. W.B. Jones, W.J. Thorn, H. Waadeland, A strong Stieltjes moment problem. Trans. Am. Math. Soc. 261, 503–528 (1980) 23. A. Bultheel, P. González Vera, E. Hendriksen, O. Njåstad, Orthogonal Rational Functions. Cambridge Monographs on Applied & Computational Mathematics, vol. 5 (Cambridge University Press, Cambridge, 1999) 24. W.B. Jones, O. Njåstad, W.J. Thron, Orthogonal Laurent polynomials and the strong Hamburger moment problem. J. Math. Anal. Appl. 98, 528–554 (1984) 25. W.B. Jones, O. Njåstad, Orthogonal Laurent polynomials and strong moment theory: a survey. J. Comput. Appl. Math. 105(1–2), 51–91 (1999) 26. K.T-R. McLaughlin, A.H. Vartanian, X. Zhou, Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights. Int. Math. Res. Pap. 216, Art. ID 62815 (2006) 27. K.T-R. McLaughlin, A.H. Vartanian, X. Zhou, Asymptotics of Laurent polynomials of odd degree orthogonal with respect to varying exponential weights. Constr. Approx. 27(2), 149– 202 (2008) 28. R. Cruz-Barroso, C. Díaz Mendoza, R. Orive, Orthogonal Laurent polynomials. A new algebraic approach. J. Math. Anal. Appl. 408, 40–54 (2013)

Time Dependent Solutions for the Biot Equations Robert P. Gilbert and George C. Hsiao

Dedicated to Heinrich Begehr for his 80 Jubilee

Abstract In this paper we show that the Biot model for the ultrasound interrogation of bone rigidity, under certain restrictions, can be shown to lead to a unique solution. More precisely, we consider the classical experimental method for measuring bone parameters, that is where a bone sample in a water bath and the bone sample interrogated with an ultrasound devise. This procedure leads to an inverse problem where the ultrasound signal is measured in various positions in the water tank. In order to solve the inverse problem an accurate forward solver is necessary. It is shown that the forward problem may be formulated in terms of a boundary element method. To this end, the Biot system of equations describing the acoustic interaction with a porous material is written in a convenient, compact form. The system, and the transition conditions between the porous material, are rewritten in a Laplace transformed space. The transformed problem is reformulated as a nonlocal boundary problem. Using a variational approach it is shown that the variational problem is equivalent to a nonlocal problem and the solution is shown to be unique. We then use Lubisch’s approach to find estimates in the time domain without recourse to using the inverse Laplace transformation. Keywords Biot system · Porous media · Ultrasound interrogation · Boundary element method MSC (2010) Codes Primary: 74F10, 65M38; Secondary: 74L15, 35G46

R. P. Gilbert () Department of Material Science, Ruhr Univerität Bochum, Bochum, Germany Department of Mathematical Sciences, University of Delaware, Newark, DE, USA e-mail: [email protected] G. C. Hsiao Department of Mathematical Sciences, University of Delaware, Newark, DE, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_9

171

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R. P. Gilbert and G. C. Hsiao

1 Introduction We consider cancellous bone as a two component material consisting of a calcified bone matrix with interstitial fatty marrow. McKelvie and Palmer [23], Williams [28], and Hosokawa and Otani [12] discuss the application of Biot’s model for a poroelastic medium to cancellous bone. The Biot model treats a poroelastic medium as an elastic frame with interstitial pore fluid. The elastic frame is known as the trabeculae. Cancellous bone is anisotropic, however, as pointed out by Williams, if the acoustic waves passing through it travel in the trabecular direction an isotropic model may be acceptable. Biot’s equations may be derived using homogenization theory [7–9] which is an asymptotic method. Because of the pore size Biot’s equations should only be valid in the range below 500 kHz where we have a refraction problem. When the cancellous bone is excited by ultrasound in the frequency range above 600 kHz we are in the region of scattering rather than resonance or transmission. In the Biot model [1, 2], in additional to the displacement vector u(s)(x, t), let u(f ) (x, t) denote the displacement field of the fluid. That is, u(s) (x, t) and u(f ) (x, t) track the motions of the frame and fluid respectively. For expository reasons we explain our procedure for three dimensional problems where x = (x, y, z) ∈ R3 . For this case the following constitutive relations are assumed σ11 = 2μe11 + λe + Q*, σ22 = 2μe22 + λe + Q*,

(1.1)

σ33 = 2μe33 + λe + Q*, σij = μeij , i = j,

σ =

Qe + R*,

where e = ∇ · u(s) is the frame dilation and * = ∇ · u(f ) the fluid dilatation. Here R is a parameter measuring the pressure on the fluid required to force a certain volume of fluid into the sediment at constant volume, and Q measures the coupling of changes in the volume of the solid and fluid [1, 2]. As usual, the elastic strains are related to the displacements by eii =

∂ui , ∂xi

eij =

∂uj ∂ui + , ∂xj ∂xi

i = j, i, j = 1, 2, 3.

(1.2)

Equations (1.1) and (1.2) and an argument based upon Lagrangian dynamics are shown in [1, 2] to lead to the following equations of motion for the displacements and dilatations μu(s) + ∇[(λ + μ)e + Q*] = −∇p(f ) := ∇[Qe + R*] =

∂2 (ρ11 u(s) + ρ12 u(f ) ) ∂t 2

(1.3)

∂2 (ρ12 u(s) + ρ22 u(f ) ), ∂t 2

(1.4)

Time Dependent Solutions for the Biot Equations

173

which implies ∂ 2 u(f ) 1 = − 2 ∂t ρ22



∂ 2 u(s) ρ12 + ∇p(f ) ∂t 2

 (1.5)

Substituting (1.5) and (1.4) into (1.3) yields (s)

μu

  Q2 Q + λ+μ− ∇∇ · u(s) − ∇p(f ) R R

ρ11 + ∇p(f ) = ρ22



ρ2 ρ11 − 12 ρ22



∂ 2 u(s) ∂t 2

(1.6)

Here p(f ) is the fluid pressure; whereas, ρ11 and ρ22 are density parameters for the solid and fluid, ρ12 is a density coupling parameter. The Biot system may be therefor written as   , 4 3  2 ρ12 ∂ 2 u(s) Q2 (s) (s) ρ11 − + μ ∇∇ · u − μu + λ − ρ22 ∂t 2 R   Q ρ12 − + ∇p(f ) = 0 (1.7) R ρ22   Q ρ12 ∂ 2 ∇ · u(s) ρ22 ∂ 2 p(f ) − ρ22 + − p(f ) = 0 (1.8) R ρ22 ∂t 2 R ∂t 2 All parameters in the above equations are assumed to be constant. We now define the additional parameters and operators ρ := (ρ11 −

2 ρ12 ) > 0, ρ22

λ := (λ −

Q2 ) > 0, R

η := (

Q ρ11 − ) > 0; R ρ12

(1.9)

moreover,   ∗ u(s) := μu(s) + (λ + μ)∇∇ · u(s) .

(1.10)

Then we may rewrite the system in a more compact form as ∂ 2 u(s) − ∗ u(s) + η∇p(f ) = 0 ∂t 2

(1.11)

∂∇ · u(s) ρ22 ∂ 2 p(f ) + − p(f ) = 0. 2 ∂t R ∂t 2

(1.12)

ρ ρ22 η

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R. P. Gilbert and G. C. Hsiao

n

Fig. 1 Geometry of the problem

Ω

Γ Ω(w) := R3 \ Ω

For utilization of the transmission condition, we need to examine the fluid outside ¯ In the water (w) , we consider the the bone specimin, i.e. in (w) := R3 \ . barotropic flow of an inviscid and compressible fluid. Let v(w) := v(w) (x, t) be the velocity field and ρ (w) := ρ (w) (x, t) and p(w) := p(w) (x, t) be respectively the density and pressure of the fluid encompassing the bone sample (see Fig. 1). v(w) , p(w) and ρ (w) are small :We(w)assume that ? perturbations from the static state (w) v = 0, p = constant, ρ (w) = constant . In this case, the governing fluid equations may be linearized to yield the linear Euler equation ρ0(w)

∂v(w) + ∇p(w) = 0, ∂t 2

the linear equation of continuity ∂ρ (w) + ρ0(w) ∇ · v(w) = 0 ∂t and the linearized state equation p(w) = c2 ρ (w) in (w) × (0, T ], where c is the sound of speed defined by c2 = f (ρ0 ) and f (ρ (w) ) is a function depending on the properties of the fluid [15, 27]. It is well known that for an irrotational flow, we may introduce a velocity potential [27] φ := φ(x, t) such that (w)

v(w) = −∇φ,

and p(w) = ρ0

∂φ ∂t

(1.13)

Time Dependent Solutions for the Biot Equations

175

From this it follows that the velocity potential satisfies the wave equation ∂ 2φ − c2 φ = 0, ∂t 2

(x, t) ∈ (w) × (0, T ].

(1.14)

The time-dependent problem can be formulated as an initial-boundary-transmission problem consisting of the partial differential equations ρ

ρ22 η

∂u(s) − ∗ u(s) + η∇p(f ) = 0, ∂t 2

(x, t) ∈  × (0, T ]

∂ 2 (∇ · u(s) ) ρ22 ∂ 2 p(f ) + − p(f ) = 0, ∂t 2 R ∂t 2 ∂ 2φ − c2 φ = 0, ∂t 2

(x, t) ∈  × (0, T ]

(x, t) ∈ (w) × (0, T ]

(E1 )

(E2 )

(E3 )

together with the transmission conditions: Conservation of the Normal Component of Stress     ∂φ inc (w) ∂φ (s) (f ) + n, σ (u ) − η p I n = −ρ0 ∂t ∂t

(x, t) ∈  × (0, T ] (T C1 )

Conservation of Flux   ∂v(w) ∂φ ∂φ inc ·n = − + , ∂t ∂n ∂n

(x, t) ∈  × (0, T ]

(T C2 )

Continuity of Pore Pressure p(f ) = −βp(w) ,

(x, t) ∈  × (0, T ]

(T C3 )

and the initial conditions u(s) (x, 0) =

∂u(s) = 0, ∂t

x∈

(I1 )

p(f ) (x, 0) =

∂p(f ) = 0, ∂t

x∈

(I2 )

x ∈ (w) .

(I3 )

φ(x, 0) =

∂φ (x, 0) = 0, ∂t

Here φ inc and ∂φ inc /∂t are prescribed incident fields.

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We are going to solve the problem using Lubisch’s approach based on Laplace transform methodology. To this end we introduce the notation ∞ U (x, s) = (s)

e

∞

−st (s)

u (x, t) dt,

U

(f )

(x, s) =

0



∞

P (f ) (x, s) =

e−st u(f ) (x, t) dt

0



e−st p(f ) (x, t) dt,

∞

P (w) (x, s) =

0

e−st p(w) (x, t) dt

0

∞ (x, s) =

e−st φ(x, t) dt,

0

etc. The transformed equations now become − ∗ U(s) + ρ s 2 U(s) + η∇P (f ) = 0, ρ22 ηs 2 (∇ · U(s)) − P (f ) + −  +

ρ22 2 (f ) s P = 0, R

s2  = 0, c2

in

in ,

(Eˆ 1 )

in

(Eˆ 2 )

(w)

,

(Eˆ 3 )

    σ (U(s) ) − ηP (f ) I · n = −s ρ0(w)  + inc n

(TˆC 1 )

    ∂inc ∂ ρ12 (s) + (1 − β) + β sU · n = − ρ22 ∂n ∂n

(TˆC 2 ) (TˆC 3 )

P (f ) = −βP (w) .

2 Nonlocal Boundary Problem The next step is to reduce above transmission problem consisting of (Eˆ 1 )–(TˆC 3 ) to a nonlocal boundary problem in . We begin with (Eˆ 3 ) −  +

s2  = 0 c2

in c

(2.1)

by seeking a solution of (2.1) in the form of simple and double layer potentials, i.e.  = D(ψ) − S(ζ ),

in c ,

(2.2)

Time Dependent Solutions for the Biot Equations

where ψ = | and ζ =  := ∂. Here

∂ ∂n |

 D(ψ)(x) :=

177

are the Cauchy data of the solution  of (2.1) on

∂ Es/c (x; y)ψ(y) dy , ∂ny

x ∈ R3 \ 

(2.3)



 S(ψ)(x) :=

x ∈ R3 \ ,

Es/c (x; y)ζ(y) dy ,

(2.4)



where Es/c (x; y) :=

e−s/c (x − y) x − y

(2.5)

is the fundamental solution of the shielded Coulomb potential operator, known also as the Yukowa potential operator [22]. The Cauchy data ψ and ζ satisfy the relations [13, 16] 

ψ ζ

1

 =

2 I + K(s)

−V(s)



−W(s) ( 12 I + K(s))

ψ ζ

 x∈

,

(2.6)

Here V, K, K and W are the four basic boundary integral operators familiar from potential theory [13, 16] ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

 V(s)ζ(x) :=  K(s)ψ(x) :=

Es/c (x, y)ζ(y)dy ,

x∈



∂ Es/c (x, y)ψ(y)d y , ∂ny

x∈



 ⎪ ∂ ⎪

⎪ K (s)ψ(x) := Es/c (x, y)ζ(y)dy , x ∈  ⎪ ⎪ ⎪ ∂nx ⎪ ⎪  ⎪ ⎪ ⎪  ⎪ ⎪ ∂ ∂ ⎪ ⎪ ⎪ W(s)ψ(x) := − Es/c (x, y)ψ(y)dy , x ∈  ⎪ ⎩ ∂nx ∂ny

(2.7)



From the second equation of (2.6), we see that ∂ 1 | =: ζ = −W(s)ψ + ( I + K(s)) ζ ∂n 2 and the transmission condition (TˆC 2 ) then leads to the BIE  − s q(β) U

(s)

· n + W(s)ψ −

1 I − K(s) 2



ζ =

∂ψ inc ∂n

on

;

(2.8)

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R. P. Gilbert and G. C. Hsiao

while ψ and ζ are required to satisfy the first boundary integral equation in (2.6) as a constraint,   1 I − K(s) ψ + V(s)ζ = 0, on . 2 To simplify the representation, the coefficient in (T ˆC 2 ) has been denoted by   ρ12 q(β) := (1 − β) + β ρ22 in the BIE (2.8). We note that q(β) > 0 for β ∈ [0, 1]. With the Cauchy data ψ and ζ as new unknowns, the partial differential equation (2.1) in (w) is eliminated.: This reformulation ? leads to a nonlocal boundary problem in  for the unknowns U(s) , P (w) , ψ, ζ , consisting of the partial differential equations −∗ U(s) + ρ s 2 U(s) + η∇P (f ) = 0, ρ22 η s(∇ · U(s)) − P (f ) + s 2

(2.9)

ρ22 (f ) P = 0. R

(2.10)

and the boundary integral equations  − s q(β)U 2

(s)

· n + W(s)ψ − 

1 I − K(s) 2



ζ =

 1 I − K(s) ψ + V(s)ζ = 0, 2

∂inc ∂n

on ;

(2.11)

on ,

(2.12)

together with the transmission conditions     (w) σ (U(s)) − η P (f ) I n = −sρ0  + inc n

(2.13)

and the condition (TˆC 3 ), from which we tacitly induce to an alternative condition ∂ (f ) ∂inc P . = −βs ρ0(w) ∂n ∂n

(2.14)

  To be more precise, let us first consider the unknowns U(s) , P (f ) ∈ H1 () × H 1 (). Multiplying (2.9) by the test function U ∈ H1 () and integrating by parts, we obtain the weak formulation of (2.9) :     (s) (s) (f ) − A U , U; s − σ (U , P )n·γ U d − η P (f ) ∇ ·U dx = 0, (2.15) 



Time Dependent Solutions for the Biot Equations

179

where      σ (U(s) ) : ε(U) + s 2 ρU(s) · U dx A U(s) , U; s :=

(2.16)



is the sesquilinear form with σ (U(s) ) := λ ∇U(s) I+2 μ ε(U(s) ),

ε(U(s)) :=

 1 T ∇U(s) + ∇U(s) . 2

(2.17)

Then from the transmission condition (2.13), we obtain   (w) (w) A(U(s) , U; s) − η P (f ) , ∇ · U + ρ0 sn, γ − U = −ρ0 sinc n, γ − U 

(2.18) Similarly, multiplying (2.10) by the test function P yields ρ22 η s 2 (∇ · U(s), P ) + B(P (f ) , P ; s) = 0,

(2.19)

where the sesquilinear form B(P (f ) , P ; s) is defined by B(P

(f )

, P ; s) :=

 

∇P (f ) · ∇P +



ρ22 2 (f )  s P P dx. R

(2.20)

Now for the unknowns (ψ, ζ ) ∈ H 1/2 () × H −1/2 (), we proceed in the same manner. Multiplying (2.11) and (2.12) by the test functions ϕ¯ and ξ¯ , respectively, we obtain   1 ∂inc (s) , ϕ ¯ , ¯  + W(s)ψ, ϕ ¯  −  I − K(s) ζ, ϕ ¯  =  − s q(β)U · n, ϕ 2 ∂n (2.21)   1  I − K(s) ψ, ξ¯  + V (s)ζ, ξ¯  = 0 (2.22) 2 for ϕ¯ ∈ H 1/2() and ξ¯ ∈ H −1/2(). Finally we define the operators   As : H1 () → H1 () ,

  Bs : H 1 () → H 1 ()

(2.23)

  Bs P (f ) , P  := B P (f ) , P ; s

(2.24)

by their sesquilinear forms:   As U(s) , U := A U(s) , U; s ,

Then from (2.18), (2.20), (2.21) and (2.22), the non-local boundary problem may be formulated as a system of operator equations:

180

R. P. Gilbert and G. C. Hsiao

1 () × H 1/2 () × H −1/2 (). Then given data Let X := H1 () × H  (s) 

(d1 , d2 , d3 , d4 ) ∈ X , find U , P (f ) , ψ, ζ in X satisfying

⎛

⎞ U(s) ⎜ P (f ) ⎟ ⎜ ⎟ A ⎜ ⎟ := ⎝ ψ ⎠ ζ

⎞

(w) ⎛ (s) ⎞ As −η(∇·) sρ0 γ − n 0 ⎟ U ⎜ 2 ⎟ ⎜ (f ) ⎟ ⎜ s ρ22 η ∇· Bs 0 0 ⎟ ⎜   ⎟ ⎜ P ⎟ ⎟⎜ ⎜ 1 T − 0 W(s) − 2 I − K(s) ⎟ ⎝ ψ ⎠ ⎜ −s q(β) n γ ⎠ ⎝   1 I − K(s) ζ 0 0 V(s) 2 ⎛

= (d1 , d2 , d3 , d4 )&

(2.25)

where the data (d1 , d2 , d3 , d4 ) are given by   d1 = −sρ0(w) γ + inc n ,   (w) ∂n+ inc , d2 = −β s ρ0 d3 =

(2.26)

∂n+ inc ,

d4 = 0 Here and in the sequel γ ± represents the trace operator on  from inside (-) and

outside (+) of  and γ ± denotes the transpose of γ ± ; while ∂n± represents limits of the corresponding normal  derivatives. Our  intention is now to show that (2.25) admits a unique solution U(s) , P (f ) , ψ, ζ ∈ X. It is noteworthy to mentioning that for the time-independent case, one may simply consider the weak formulation of the corresponding nonlocal boundary problem, since the boundary integral operators involved do not depend upon the complex parameter s. However, for the time-dependent case such as (2.25), one can not analyze the weak form of (2.25) directly. As will be seen, we need to consider the weak form of (2.25) indirectly. We will pursue the idea in the next section.

3 Variational Formulation We need the definition of the following energy norms:   |||U(s)|||2|s|, := σ (U(s) ), ε(U(s))



|||P (f ) |||2|s|, := ∇P (f ) 2 +

+ ρsU(s) 2 ,

ρ22 sP (f ) 2 , R

|||||||s|,c := ∇2c + c−2 s2c ,

U(s) ∈ H1 ()

P (f ) ∈ H 1 ()  ∈ H 1 (c )

Time Dependent Solutions for the Biot Equations

181

For the complex Laplace parameter s ∈ C+ , we will denote σ := 'e s > 0,

σ := min{1, σ }

and will make use of the following equivalence relations for the norms σ |||U(s)|||1, ≤ |||U(s)||||s|, ≤

|s| |||U(s)|||1, σ

and similar relations hold for norms of P (f ) and , which may be obtained from the inequalities min {1, σ } ≤ min {1, |s|} and max {1, |s|} × min {1, σ } ≤ |s|, ∀s ∈ C+ We remark that the norms |||P (f ) |||1, and ||||||H 1 () are equivalent to P (f ) H 1 () and H 1 () respectively and so is energy norm |||U(s)|||1, equivalent to the H1 () norm of U(s) by  (s)  the second Korn equality. Now suppose that U , P (f ) , ψ, ζ in X is a solution of (2.25). Let in R3 \ .

V (s) := D(s)ψ − S(s)ζ

(3.1)

Then V ∈ H 1 (R3 \ ) is a solution of the equation − V +

s2 V = 0 in c2

R3 \ .

(3.2)

By the standard argument in potential theory (see, e.g.[13]), it can be shown that the following jump relations across  hold: 1

[γ V ] := γ + V − γ − V = ψ ∈ H 2 () [∂n V ] := ∂n+ V − ∂n− V = ζ ∈ H − 2 (), 1

It follows from (2.25) that ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

As U(s) − η(div) P (f ) + sρ0 γ − [γ V ] n = d1 (w)

s 2 ρ22 η (div U(s) ) + Bs P (f ) = d2

in 

in 

 1 ⎪ − (s) ⎪ I − K(s) U + W(s)[γ V ] − [∂n V ] = d3 −s q(β) n · γ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ ( I − K(s))[γ V ] + V(s)[∂n V ] = d4 . 2

on 

(3.3)

182

R. P. Gilbert and G. C. Hsiao

The 4th equation in (3.3), denoted by (3.3)4 , implies that −γ − V = 0, since ( 12 I − K(s))(γ + V ) + V(s)(∂n+ V ) = d4 from (2.25). This means that V given by (3.1) is a solution of (3.2) in  with homogeneous Dirichlet boundary condition on . By the uniqueness of solution to the interior Dirichlet problem of (3.2), we conclude that V ≡ 0 in . Consequently, we have [γ V ] = γ + V = ψ

and [∂n V ] = ∂n+ V = ξ

(3.4)

and the system (3.3) reduces to the following simple form ⎧

(w) (s)

(f ) ⎪ + sρ0 γ − (γ + V )n = d1 in ⎪ As U − η (div) P ⎪ ⎪ ⎪ ⎨ s 2 ρ22 η (div U(s) ) + Bs P (f ) = d2 in  ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −s q(β) (n · γ − U(s) ) − (∂n+ V ) = d3 on .

 (3.5)

To derive a variational formulation of (3.5), let us first consider the third equation (3.5)3 in (3.5). Multiplying the last term in (3.5)3 by the trace of test function Z ∈ H 1 ((w) ) , we see that    s ∇V · ∇Z + ( )2 V Z dx −∂n+ V , γ + Z = c (w) =: C(V , Z; s)

(3.6)

In the same way, we may define the operator Cs

Cs : H1 ((w) ) → H1 ((w) ) ,

(3.7)

by the sesquilinear form: (Cs V , Z)(w) := C(V , Z; s).

(3.8)

Together with the operators As and Bs , we arrive at the following variational formulation: Find (U(s), P (f ) , V ) ∈ H := H1 () × H 1 () × H 1 ((w) ) for given

Time Dependent Solutions for the Biot Equations

183

(d1 , d2 , d3 ) ∈ H such that ⎧ =    > 7  7 s¯ 6 s¯ 6 (w) ⎪ ⎪ = , As U(s) , U − η P (f ) , ∇ · U + s ρ0 (γ + V )n, γ − U d1 , U ⎪ ⎪     |s| |s| ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎨    ,  ? s¯ 1  s¯ 1 : 2 (s) (f ) d2 , P  , B η ∇ · U , P + P , P s = s 3 3 ρ ⎪   |s| ρ |s| 22 22 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = 7 > > 7 6 6= ⎪   ⎪ s¯ s¯ (w) ⎪ ⎩ , −s ρ0 n · γ − U(s) , γ + Z + a(β) Cs V , Z (w) = a(β) d3 , γ + Z   |s| |s|

(3.9) for all (U, V , Z) ∈ H, where a(β) := ρ0(w) /q(β). We notice that in the formulation (3.9), each of the equations in (3.5) has been multiplied by appropriate weight factors. It is necessary and will be transparent later.

4 Existence and Uniqueness We remark that this variational problem (3.9) is equivalent to the nonlocal problem (2.25), from which we have the following basic results. : ? Theorem 4.1 The variational problem (3.9) has a unique solution U(s), P (f ) , V ∈ H for given (d1 , d2 , d3 ) ∈ H . Moreover the following estimate holds: |||(U(s), P (f ) , V )|||H ≤ c0

|s|3 (d2 , d2 , d3 )H , σσ6

(4.1)

where c0 is a constant depending only on the physical parameters β, ρ0(w) , η, ρ22

and where H := H1 () × H 1 () × H 1 ((w) ) Proof From the system of equations in (3.9), we notice that  = >   6 s¯  (w) − η P (f ) , ∇ · U(s) + s ρ0 (γ + V )n, γ − U(s)   |s|    = 7   > s¯ s¯ + = 0. − s ρ0(w) n · γ − U(s) , γ + V + 3 s 2 η ∇ · U(s), P (f )   |s| |s| 'e

Consequently, from (3.9) we have ,      s¯ s¯ s¯  (f ) (f ) As U(s) , U(s) + a(β) C P , P + , V , V B s s (w)   |s| ρ22 |s|3 |s|   ,    s¯ s¯ ¯ ) + s¯ a(β)d , γ + V  (w) (f = 'e d d1 , U(s) + , P 2 3    |s| ρ22 |s|3 |s| (4.2)

 'e

184

R. P. Gilbert and G. C. Hsiao

Now let us examine each term on the LHS of (4.2): For the first term, we see that   7 s¯  s¯ 6 As U(s) , U(s) := σ (U(s)), ε(U(s)) + s 2 ρU(s)   |s| |s| ,   s¯ s¯ (s) (s) 2 (s) = σ (U ), ε(U ) + sU  .  |s| |s| Then, 

, s¯  σ (s) (s) As U , U |||U(s)|||2|s|, 'e = |s| |s|

(4.3)

Similarly, for the second term, we have  s¯ 1  s¯ 1 (f ) (f ) B P , P = s  |s|3 ρ22 |s|3 ρ22  1 s¯ = ∇P (f ) 2 + ρ22 |s|3

6

ρ22 2 (f ) 2 7 s P  R , s ρ22 (f ) 2 sP   |s|3 R ∇P (f ) 2 +

Thus,  'e

 , σ 1 s¯ 1  (f ) (f ) = 3 Bs P , P |||P (f ) |||2|s|,  |s|3 ρ22 |s| ρ22

(4.4)

Finally, for the third term, we obtain similarly as the first term, namely  'e

,   s¯ σ a(β) Cs , V , V (w) = a(β) |||V |||2|s|,(w) |s| |s|

(4.5)

Therefore, combining (4.3)–(4.5) and substituting into (4.2) yields σ σ 1 σ |||P (f ) |||2|s|, + |||U(s) |||2|s|, + 3 a(β) |||V |||2|s|,c |s| |s| ρ22 |s|   > , =   s¯ s¯ 1  s¯ = 'e d1 , U(s) + 3 d2 , P (f ) + a(β) d3 , γ + V    |s| |s| |s| ρ22 >     ; 1 1 ≤ | d1 , U(s) + 2 d2 , P (f ) + a(β) d3 , γ + V |    |s| ρ22

(4.6)

Time Dependent Solutions for the Biot Equations

185

However, from the definition of σ = min {1, σ }, we see that 1/|s| > σ /|s| and 1 > σ /|s|. This implies that the LHS of (4.6) satisfies the estimate, namely, σ σ 1 σ |||U(s)|||2|s|, + 3 a(β) |||V |||2|s|,c |||P (f ) |||2|s|, + |s| |s| ρ22 |s|  σ  σ 1 |||U(s)|||2|s|, + ( )2 ≥ |||P (f ) |||2|s|, + a(β) |||V |||2|s|,c |s| |s| ρ22  1 σσ2  |||P (f ) |||2|s|, + a(β) |||V |||2|s|,c |||U(s)|||2|s|, + ≥ 3 ρ22 |s| (4.7) Consequently, from (4.6), we obtain the estimates   |||U(s)|||21, + |||P (f ) |||21, + |||V |||21,(w) ≤ ≤

c c0

  >  ; 1 1  |s|3   (s) (f ) +V | | d d , U + , P + a(β) d , γ 1 2 3    σσ4 |s|2 ρ22  >    ; |s|3   (s) (f ) +V | , | d , U | + | d , P | + | d , γ 1 2 3    σσ6 (4.8)

from which we obtain finely the desired estimate |||(U(s), P (f ) , V )|||H ≤ c0

|s|3 |||(d1, d2 , d3 )|||H , σσ6

(4.9)

where c0 is a constant depending only on the physical parameters ρ0 , η, ρ22 . An alternative proof shows the following sharper estimate for the norm of |||(U(s), P (f ) , V )|||H can be established. Corollary 4.1 The estimate of (4.1) in Theorem 4.1 can be improved as |||(U(s), P (f ) , V )|||H ≤ c0

|s|2 |||(d1, d2 , d3 )|||H , σσ4

(4.10)

where c0 is a constant depending only on the physical parameters ρ0 , η, ρ22 . Proof From (4.6) in the proof of Theorem 4.1, we see that σ 1 σ σ |||U(s)|||2|s|, + 3 a(β) |||V |||2|s|,c |||P (f ) |||2|s|, + |s| |s| ρ22 |s|  >   ; 1 1  ≤ | d1 , U(s) + 2 d2 , P (f ) + a(β) d3 , γ + V |    |s| ρ22

(4.11)

186

R. P. Gilbert and G. C. Hsiao

Then  σσ2  1 1 |||U(s)|||21, + 2 |||P (f ) |||21, + a(β) |||V |||21,c |s| |s| ρ22   >  ; 1 1  d2 , P (f ) + a(β) d3 , γ + V | ≤ | d1 , U(s) + 2    |s| ρ22      > , ; 1 1 1 | d1 , U(s) | + | d2 , P (f ) | + a(β) | d3 , γ + V | , ≤    σ |s| ρ22 (4.12) from which we obtain the improved estimate immediately |s|2 |||(d1, d2 , d3 )|||H , σσ4

|||(U(s), P (f ) , V )|||H ≤ c0

(4.13)

where c0 is a constant depending only on the physical parameters ρ0 , η, ρ22 . Theorem 4.2 Let ⎧ X := H1 () × H 1 () × H 1/2() × H −1/2 (), ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

X := H1 () × H 1 () × H −1/2() × H 1/2(), : ? X0 := (d1 , d2 , d3 , d4 ) ∈ X | d4 = 0 .

Then A : X → X0 is invertible. Moreover, we have the estimate; 1

A

−1

|X 0 

X X

≤ c0

|s|3 2

1

σσ52

(4.14)

,

where c0 is a constant independent of s and σ := 'e s > 0. Proof Recall γ + V := [γ V ] = ψ ∈ H 1/2 (),

% & ∂n+ V = ∂n+ V = ζ ∈ H −1/2().

Then we have the estimates (see, e.g. [15]). ψ|2H 1/2 () = γ + V 2H 1/2 () ≤ c1 |||V |||1,|(w)

(4.15)

Similarly, an application of the Bamberger and Ha-Duong’s optimal lifting leads to the estimate ζ H −1/2 () = ∂n+ V H −1/2 () ≤



|s| σ

1/2 |||V ||||s|,c .

Time Dependent Solutions for the Biot Equations

187

Hence,  ζ 2H 1/2 () ≤ c22

 |s|3 |s| |||V |||2|s|,c ≤ c22 3 |||V |||21,c σ σ

(4.16)

From (4.11) and (4.12), we obtain the estimates 1 2

-

1 σ3 1 ψ2H 1/2 () + 2 3 ζ 2H −1/2 () c1 c2 |s|

. ≤ |||V |||21,c .

As a consequence of (4.1), it follows that   , 12 |s|2 σ3 (s) 2 (f ) 2 2 2 ≤ c0 (d1 , 0, d3 , 0)X |||U |||1, + |||P |||1, + C ψH 1/2 () + 3 ζ H −1/2() |s| σσ4

Thus we have the estimate 1

|||(U(s), P (f ) , ψ, ζ )|||X ≤ c0

|s|3 2

1

σσ52

(d1 , d2 , d3 , 0)X

from which the desired estimate (4.10) follows. This completes the proof.

5 Main Results in the Time Domain With the properties of solutions in the Laplace-transformed domains available, we may now return to the solutions in the time domain based on Lubich’s approach. We begin with a brief review of the Lubisch approach for treating boundary integral equations which has been advanced recently by the work of La liena and Sayas [19, 24]. Lubich’s approach has been adopted by many researchers for treating interesting problems (see, e.g. [14, 15, 17, 20, 21, 25, 26], to name a few). An essential feature of this approach is that estimates of properties of solutions in the time domain are obtained without the need for applying the inverse Laplace transform. Indeed the crucial result described on the Proposition below is employed to retrieve time-domain estimates from those obtained in the Laplace transformed domain. Before presenting the aforementioned results, we must introduce some notation. For Banach spaces X and Y, let B(X, Y) denote the set of bounded linear operators from X to Y. We say that an analytic function A : C+ −→ B(X, Y)

188

R. P. Gilbert and G. C. Hsiao

is an element of the class of symbols Sym (μ, B(X, Y)) , if there exists a μ ∈ R and m ≥ 0 such that A(s)X,Y ≤ CA ('e s)|s|μ for s ∈ C+ := {s ∈ C | 'e s > 0} , where CA : (0, ∞) → (0, ∞) is a non-increasing function such that CA (σ ) ≤

c , ∀σ ∈ (0, 1]. σm

In order to make the formulation of the time-domain estimates more compact, we will make use of the regularity spaces 6 7 W+k (H) := ω ∈ C k−1 (R; H) : ω ≡ 0 ∈ (−∞, 0), ω(k) ∈ L1 (R, H) , where H denotes a Banach space. The following Proposition has been established in [19, 24] Proposition 5.1 Let A = L{A} ∈ Sym (μ, B(X, Y)), with μ ≥ 0; furthermore, let k = (μ + 2), be the largest integer less than or equal to μ + 2, and let ε : k − (μ + 1) ∈ (0, 1]. If g ∈ W+k (R, X), then A 0 g ∈ C(R), Y is causal and (A 0 g)(t)Y ≤ 2

n+1

C* (t) CA (t

−1



1

)

(Pk g)(τ )X dτ

0

where Cε(t )

tε , = πε

and (Pk g)(t) =

k    k =0



q () (t).

As an immediate consequence of Proposition 5.1, we see from Theorem 4.2 that A

−1

|X

0

  1

∈ Sym 3 , B(X , X) . 2

Moreover μ = 3 12 , k = (3 12 + 2) = 5 and * = 5 − (3 12 + 1) = 12 ∈ (0, 1], and we have the following estimate : ? Theorem 5.1 Let D(t) := L−1 (d1 , d2 , d3 , 0)T belongs to W+5 (R, X˜ ). Then  T u(s) , p(f ) , φ, ∂n φ ∈ C ([0, T ], X)

Time Dependent Solutions for the Biot Equations

189

and there exists a constant c > 0 depending only on the geometry such that     T 1 1  u(s) , p(f ) , φ, ∂n φ X ≤ c t 2 σ σ 5 2 |σ = 1 t

1

(P6 D)(τ )X dτ

0

7 6 1 1 = c t 2 +1 max 1, t 5 2

1

(P6 D)(τ )X dτ.

0

Similarly, applying Proposition 5.1 to Corollary 4.1, with μ = 2, k = (μ + 2) = 4, * := k − (μ + 1) = 1,   7 6 1 4 | 1 = t max 1, t σσ4 σ= t we have the result. Theorem 5.2 Let H := H1 () × H 1 () × H 1 ((w) ) and 6 7 D(t) := L−1 (d1 , d2 , d3 )T (t) ∈ W+4 (R, H ). Then 

T u(s), p(f ) , L−1 {V } ∈ C ([0, T ], H)

and there holds the estimate  T 7 6  u(s), p(f ) , L−1 {V } (t)H ≤ c0 t 2 max 1, t 4

1

 (P4 D) (τ )H dτ,

0

where c0 > 0 is a constant.

6 Conclusion In closing, we remark that it is now a standard numerical technique based on Lubich’s approach. One may apply Galerkin’s semi-discretization in space and convolution quadrature (CQ) in time for treating interaction problem numerically, (see, for instances [14, 25]. We will pursue this direction for the fluid bone interaction problem in separate communications.

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References 1. M.A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Lower frequency range. J. Acoust. Soc Am. 28(2), 168–178 (1956) 2. M.A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. J. Acoust. Soc. Am. 28(2), 179–191 (1956) 3. J.L. Buchanan, R.P. Gilbert, Determination of the parameters of cancellous bone using high frequency acoustic measurements. Math. Comput. Model. 45, 281–308 (2007) 4. J.L. Buchanan, R.P. Gilbert, K. Khashanah, Recovery of the poroelastic parameters of cancellous bone using low frequency acoustic interrogation, in Acoustics, Mechanics, and the Related Topics of Mathematical Analysis, ed. by A. Wirgin (World Scientific, Singapore, 2002), pp. 41–47 5. J.L. Buchanan, R.P. Gilbert, K. Khashanah, Determination of the parameters of cancellous bone using low frequency acoustic measurements. J. Comput. Acoust. 12(2), 99–126 (2004) 6. S. Chaffai, F. Padilla, G. Berger, P. Languier, In vitro measurement of the frequency dependent attenuation in cancellous bone between 0.2 and 2 MHz. J. Acoust. Soc. Am. 108, 1281–1289 (2000) 7. Th. Clopeau, J.L. Ferrin, R.P. Gilbert, A. Mikeli´c, Homogenizing the acoustic properties of the seabed. Math. Comput. Model. 33, 821–841 (2001) 8. R.P. Gilbert, A. Mikelic, Homogenizing the acoustic properties of the seabed: part I. Nonlinear Anal. 40, 185–212 (2000) 9. R. Gilbert, A. Panchenko, Effective acoustic equations for a two-phase medium with microstructure. Math. Comput. Model. 39, 1431–1448 (2004) 10. R. Hodgskinson, C.F. Njeh, J.D. Currey, C.M. Langton, The ability of ultrasound velocity to predict the stiffness of cancellous bone in vitro. Bone 21, 183–190 (1997) 11. L. H˚ormander, Linear Partial Differential Operators (Springer, Berlin, 1963) 12. A. Hosokawa, T. Otani, Ultrasonic wave propagation in bovine cancellous bone. J. Acoust. Soc. Am. 101, 558–562 (1997) 13. G.C. Hsiao, W.L. Wendland, Boundary Integral Equations (Springer, Heidelberg, 2008) 14. G.C. Hsiao, T. S’anchez-Vizuet, F.-J. Sayas, Boundary and coupled boundary-finite element methods for transient wave-structure interaction. IMA J. Numer. Anal. 37, 237–265 (2016) 15. G.C. Hsiao, F.-J. Sayas, R.J. Weinacht, Time-dependent fluid-structure interaction. Math. Methods Appl. Sci. 40, 486–500 (2017). Article first published online 19 Mar 2015 in Wiley Online Library, http://dx.doi.org/10.1002/mma.3427 (http://dx.doi.org/10.102/mma. 3427, 2015) 16. G.C. Hsiao, O. Steinbach, W.L. Wendland, Boundary element methods: foundation and error analysis, in Encyclopedia of Computational Mechanics, 2nd edn., ed. by E. Stein et al. (Chichester, Wiley, 2017), pp. 1–62 17. G.C. Hsiao, T. Sánchez-Vizuet, F.-J. Sayas, R.J. Weinacht, A time-dependent fluidthermoelastic solid interaction. IMA J. Numer. Anal. 1–33 (2018). https://doi.org/10.1093/ imanum/dry016 18. D.L. Johnson, J. Koplik, R. Dashen, Theory of dynamic permeability and tortuosity in fluidsaturated porous media. J. Fluid. Mech. 176, 379–402 (1987) 19. A.R. Laliena, F.-J. Sayas, Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves. Numer. Math. 112, 637–678 (2009) 20. Ch. Lubich, On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations. Numer. Math. 67, 365–389 (1994) 21. Ch. Lubich, R. Schneider, Time discretization of parabolic boundary integral equations. Numer. Math. 63(4), 455–481 (1992) 22. P.M. Morse, H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill Book Company, New York, 1953) 23. M.L. McKelvie, S.B. Palmer, The interaction of ultrasound with cancellous bone. Phys. Med. Biol. 10, 1331–1340 (1991)

Time Dependent Solutions for the Biot Equations

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24. F.-J. Sayas, Retarded Potentials and Time Domain Boundary Integral Equations: A Road-Map. Computational Mathematics, vol. 50 (Springer, Berlin, 2016) 25. M. Schanz, Dynamic poroelasticity treated by a time domain boundary element method, in IUTAM/ACM/IABEM Symposium on Advanced Mathematical and Computational Mechanics Aspects of the Boundary Element Method, ed. by T. Burczynski (Kluwer Academic Publishers, Dordrecht, 2001), pp. 303–314 26. M. Schanz, Wave Propagation in Viscoelastic and Poroelastic Continua. Lecture Notes in Applied Mechanics, vol. 1 (Springer, Berlin, 2001) 27. G.B. Whitham, Linear and Nonlinear Waves. Pure and Applied Mathematics (Wiley, New York, 1973) 28. J.L. Williams, Prediction of some experimental results by Biot’s theory. J. Acoust. Soc. Am. 91, 1106–1112 (1992)

Schwartz-Type Boundary Value Problems for Monogenic Functions in a Biharmonic Algebra S. V. Gryshchuk and S. A. Plaksa

Dedicated to Professor Heinrich G.W. Begher on the occasion of his 80th birthday

Abstract We consider Schwartz-type boundary value problems for monogenic functions in a commutative algebra B over the field of complex numbers with the bases {e1 , e2 } satisfying the conditions (e12 + e22 )2 = 0, e12 + e22 = 0. The algebra B is associated with the biharmonic equation, and considered problems have relations to the plane elasticity. We develop methods of its solving which are based on expressions of solutions by hypercomplex integrals analogous to the classic Schwartz and Cauchy integrals. Keywords Biharmonic equation · Biharmonic algebra · Biharmonic plane · Monogenic function · Schwartz-type boundary value problem Mathematics Subject Classification (2010) Primary 30G35; Secondary 31A30

1 Monogenic Functions in a Biharmonic Algebra Definition 1.1 An associative commutative two-dimensional algebra B with the unit 1 over the field of complex numbers C is called biharmonic (see [1, 2]) if in B there exists a basis {e1 , e2 } satisfying the conditions (e12 + e22 )2 = 0,

e12 + e22 = 0 .

Such a basis {e1 , e2 } is also called biharmonic.

S. V. Gryshchuk () · S. A. Plaksa Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_10

193

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In the paper [2] I.P. Mel’nichenko proved that there exists the unique biharmonic algebra B, and he constructed all biharmonic bases in B. Note that the algebra B is isomorphic to four-dimensional over the field of real numbers R algebras considered by A. Douglis [3] and L. Sobrero [4]. In what follows, we consider a biharmonic basis {e1 , e2 } with the following multiplication table (see [1]): e1 = 1,

e22 = e1 + 2ie2 ,

(1.1)

where i is the imaginary complex unit. We consider also a basis {1, ρ} (see [2]), where a nilpotent element ρ = 2e1 + 2ie2

(1.2)

satisfies the equality ρ 2 = 0 . ' We use the Euclidean norm a := |z1 |2 + |z2 |2 in the algebra B, where a = z1 e1 + z2 e2 and z1 , z2 ∈ C. Consider a biharmonic plane μe1 ,e2 := {ζ = x e1 + y e2 : x, y ∈ R} which is a linear span of the elements e1 , e2 of the biharmonic basis (1.1) over the field R. With a domain D of the Cartesian plane xOy we associate the congruent domain Dζ := {ζ = xe1 + ye2 ∈ μe1 ,e2 : (x, y) ∈ D} in the biharmonic plane μe1 ,e2 and the congruent domain Dz := {z = x + iy : (x, y) ∈ D} in the complex plane C. Its boundaries are denoted by ∂D, ∂Dζ and ∂Dz , respectively. Let Dζ (or Dz , D) be the closure of domain Dζ (or Dz , D, respectively). In what follows, ζ = x e1 + y e2 , z = x + iy, where (x, y) ∈ D, and ζ0 = x0 e1 + y0 e2 , z0 = x0 + iy0 , where (x0 , y0 ) ∈ ∂D. Any function  : Dζ −→ B has an expansion of the type (ζ ) = U1 (x, y) e1 + U2 (x, y) ie1 + U3 (x, y) e2 + U4 (x, y) ie2 ,

(1.3)

where Ul : D −→ R, l = 1, 2, 3, 4, are real-valued component-functions. We shall use the following notation: Ul [] := Ul , l = 1, 2, 3, 4. Definition 1.2 A function  : Dζ −→ B is monogenic in a domain Dζ if it has the classical derivative  (ζ ) at every point ζ ∈ Dζ :  (ζ ) :=

lim

h→0, h∈μe1 ,e2

  (ζ + h) − (ζ ) h−1 .

It is proved in [1] that a function  : Dζ −→ B is monogenic in Dζ if and only if its each real-valued component-function in (1.3) is real differentiable in D and the following analog of the Cauchy–Riemann condition is fulfilled: ∂(ζ ) ∂(ζ ) = e2 . ∂y ∂x

(1.4)

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Rewriting the condition (1.4) in the extended form, we obtain the system of four equations (cf., e.g., [1, 5]) with respect to component-functions Uk , k = 1, 4, in (1.3): ∂U3 (x, y) ∂U1 (x, y) = , ∂y ∂x

(1.5)

∂U2 (x, y) ∂U4 (x, y) = , ∂y ∂x ∂U3 (x, y) ∂U1 (x, y) ∂U4 (x, y) = −2 , ∂y ∂x ∂x ∂U4 (x, y) ∂U2 (x, y) ∂U3 (x, y) = +2 . ∂y ∂x ∂x All component-functions Ul , l = 1, 2, 3, 4, in the expansion (1.3) of any monogenic function  : Dζ −→ B are biharmonic functions (cf., e.g., [5, 6]), i.e., satisfy the biharmonic equation in D: 2 U (x, y) ≡

∂ 4 U (x, y) ∂ 4 U (x, y) ∂ 4 U (x, y) +2 + = 0. 4 ∂x ∂x 2 ∂y 2 ∂y 4

At the same time, every biharmonic in a simply-connected domain D function U (x, y) is the first component U1 ≡ U in the expression (1.3) of a certain function  : Dζ −→ B monogenic in Dζ and, moreover, all such functions  are found in [5, 6] in an explicit form. Every monogenic function  : Dζ −→ B is expressed via two corresponding analytic functions F : Dz −→ C, F0 : Dz −→ C of the complex variable z in the form (cf., e.g., [5, 6]):  (ζ ) = F (z)e1 −

 iy F (z) − F0 (z) ρ 2

∀ ζ ∈ Dζ .

(1.6)

The equality (1.6) establishes one-to-one correspondence between monogenic functions  in the domain Dζ and pairs of complex-valued analytic functions F, F0 in the domain Dz . Using the equality (1.2), we rewrite the expansion (1.6) for all ζ ∈ Dζ in the basis {e1 , e2 }:     (ζ ) = F (z) − iyF (z) + 2F0 (z) e1 + i 2F0 (z) − iyF (z) e2 .

(1.7)

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2 Schwartz-Type BVP’s for Monogenic Functions Consider a boundary value problem on finding a function  : Dζ −→ B which is monogenic in a domain Dζ when limiting values of two component-functions in (1.3) are given on the boundary ∂Dζ , i.e., the following boundary conditions are satisfied: Uk (x0 , y0 ) = uk (ζ0 ) ,

Um (x0 , y0 ) = um (ζ0 )

∀ ζ0 ∈ ∂Dζ

for 1 ≤ k < m ≤ 4, where Ul (x0 , y0 ) =

lim

ζ →ζ0 ,ζ ∈Dζ

Ul [ (ζ )] ,

l ∈ {k, m},

and uk , um are given continuous functions. We demand additionally the existence of finite limits lim

ζ →∞, ζ ∈Dζ

Ul [(ζ )] ,

l ∈ {k, m},

in the case where the domain Dζ is unbounded as well as the assumption that every given function ul , l ∈ {k, m}, has a finite limit ul (∞) :=

lim

ζ →∞, ζ ∈∂Dζ

ul (ζ )

(2.1)

if ∂Dζ is unbounded. We shall call such a problem by the (k-m)-problem. V.F. Kovalev [7] considered (k-m)-problems with additional assumptions that the sought-for function  : Dζ −→ B is continuous in Dζ and has the limit lim

ζ →∞, ζ ∈Dζ

(ζ ) =: (∞) ∈ B

in the case where the domain Dζ is unbounded. He named such problems as biharmonic Schwartz problems owing to their analogy with the classic Schwartz problem on finding an analytic function of a complex variable when values of its real part are given on the boundary of domain. We shall call problems of such a type as (k-m)-problems in the sense of Kovalev. Note, that in previous papers [5, 6, 8–13] we interpret the (k-m)-problem as the (k-m)-problem in the sense of Kovalev. It was established in [7] that all (k-m)-problems are reduced to the main three problems: with k = 1 and m ∈ {2, 3, 4}, respectively. It is shown (see [7–9]) that the main biharmonic problem is reduced to the (1–3)problem. In [8], we investigated the (1–3)-problem for cases where Dζ is either a

Schwartz-Type Boundary Value Problems for Monogenic Functions in a. . .

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half-plane or a unit disk in the biharmonic plane. Its solutions were found in explicit forms with using of some integrals analogous to the classic Schwartz integral. In [9, 10], using a hypercomplex analog of the Cauchy type integral, we reduced the (1–3)-problem to a system of integral equations and established sufficient conditions under which this system has the Fredholm property. It was made for the case where the boundary of domain belongs to a class being wider than the class of Lyapunov curves that was usually required in the plane elasticity theory (cf., e.g., [14–18]). The similar is done for the (1–4)-problem in [12]. In [12, 13], there is considered a relation between (1–4)-problem and boundary value problems of the plane elasticity theory. Namely, there is considered a problem on finding an elastic equilibrium for isotropic body occupying D with given limiting ∂v values of partial derivatives ∂u ∂x , ∂y for displacements u = u(x, y) , v = v(x, y) on the boundary ∂D. In particular, it is shown in [13] that such a problem is reduced to (1–4)-problem.

3 (1–3)-Problem and a Biharmonic Problem A biharmonic problem (cf., e.g., [14, p. 13]) is a boundary value problem on finding a biharmonic function V : D −→ R with the following boundary conditions: ∂V (x, y) = u1 (x0 , y0 ) , (x,y)→(x0,y0 ), (x,y)∈D ∂x lim

(3.1) ∂V (x, y) = u3 (x0 , y0 ) lim (x,y)→(x0,y0 ), (x,y)∈D ∂y

∀ (x0 , y0 ) ∈ ∂D .

It is well-known a great importance of the biharmonic problem in the plane elasticity theory (see, e.g., [14, 19]). Let 1 be monogenic in Dζ function having the sought-for function V (x, y) of the problem (3.1) as the first component: 1 (ζ ) = V (x, y) e1 + V2 (x, y) ie1 + V3 (x, y) e2 + V4 (x, y) ie2 . Differentiating the previous equality with respect to x and using a condition of the type (1.5) for the monogenic function 1 , we obtain  1 (ζ ) =

∂V (x, y) ∂V2 (x, y) ∂V (x, y) ∂V4 (x, y) e1 + ie1 + e2 + ie2 ∂x ∂x ∂y ∂x

and, as consequence, we conclude that the biharmonic problem with boundary conditions (3.1) is reduced to the (1–3)-problem for monogenic functions with the same boundary data.

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In what follows, let us agree to use the same denomination u for functions u : ∂D −→ R, u : ∂Dz −→ R, u : ∂Dζ −→ R taking the same values at corresponding points of boundaries ∂D, ∂Dz , ∂Dζ , respectively, i.e., u(x0 , y0 ) = u(z0 ) = u(ζ0 ) for all (x0 , y0 ) ∈ ∂D . A necessary condition of solvability of the (1–3)-problem as well as the biharmonic problem (3.1) is the following (cf., e.g., [9]):  u1 (x, y) dx + u3 (x, y) dy = 0.

(3.2)

∂D

Below, we state assumptions, under which the condition (3.2) is also sufficient for the solvability of the (1–3)-problem.

4 Boundary Value Problems Associated with a (1–4)-Problem Now, we assume that D is a bounded simply connected domain in the Cartesian plane xOy. For a function u : D −→ R we denote a limiting value at a point (x0 , y0 ) ∈ ∂D by   u(x, y)

(x0 ,y0 )

:=

lim

(x,y)∈D,(x,y)→(x0,y0 )

u(x, y) ,

if there exists such a finite limit. Consider a boundary value problem: to find in D partial derivatives V1 := ∂u ∂x , for displacements u = u(x, y), v = v(x, y) of an elastic isotropic body V2 := ∂v ∂y occupying D, when their limiting values are given on the boundary ∂D:   Vk (x, y)

(x0 ,y0 )

= hk (x0 , y0 )

∀ (x0 , y0 ) ∈ ∂D,

k = 1, 2,

(4.1)

where hk : ∂D −→ R, k = 1, 2, are given functions. We shall call this problem as the (ux , vy )-problem. This problem has been posed in [13]. For a biharmonic function W : D −→ R we denote Ck [W ](x, y) := −Wk (x, y) + κ0 W0 (x, y)

∀ (x, y) ∈ D,

where W1 (x, y) :=

∂ 2 W (x, y) , ∂x 2

W2 (x, y) :=

∂ 2 W (x, y) , ∂y 2

W0 (x, y) := W1 (x, y) + W2 (x, y) ,

k = 1, 2,

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λ+2μ κ0 := 2(λ+μ) , μ and λ are Lamé constants (cf., e.g., [19, p. 2]). The following equalities are valid in D (cf., e.g., [19, pp. 8–9],[14, p. 5]):

2μ Vk (x, y) = Ck [W ](x, y)

∀ (x, y) ∈ D,

k = 1, 2.

Then solving the (ux , vy )-problem is reduced to finding the functions Ck [W ], k = 1, 2, in D with an unknown biharmonic function W : D −→ R, when their limiting values satisfy the system   = 2μ hk (x0 , y0 ) ∀ (x0 , y0 ) ∈ ∂D, k = 1, 2. (4.2) Ck [W ](x, y) (x0 ,y0 )

Consider some auxiliary statements. Lemma 4.1 ([13]) Let W be a biharmonic function in a domain D and ∗ be a monogenic in Dζ function such that U1 [∗ ] = W . Then the following equalities are true: ∂ 2 W (x, y) = U1 [(ζ )] , ∂x 2

∂ 2 W (x, y) = U1 [(ζ )] − 2U4 [(ζ )] , ∂y 2

(4.3)

for every (x, y) ∈ D, where  := 

∗ . Lemma 4.2 ([13]) The (ux , vy )-problem is equivalent to a boundary value problem 2

2

on finding in D the second derivatives ∂ W∂x(x,y) , ∂ W∂y(x,y) of a biharmonic function 2 2 W , which have limiting values at all (x0 , y0 ) ∈ ∂D and satisfy the boundary data:  ∂ 2 W (x, y)  = λ h1 (x0 , y0 ) + (λ + 2μ) h2 (x0 , y0 ),  ∂x 2 (x0 ,y0 )  ∂ 2 W (x, y)  = (λ + 2μ) h1 (x0 , y0 ) + λ h2 (x0 , y0 ).  ∂y 2 (x0 ,y0 ) Then the general solution of (ux , vy )-problem is expressed by the formula: Vk (x, y) =

1 Ck [W ](x, y) 2μ

∀(x, y) ∈ D,

k = 1, 2.

(4.4)

The following theorem establishes relations between solutions of (ux , vy )problem and corresponding (1–4)-problem. Theorem 4.3 Let W be a biharmonic function satisfying the boundary conditions (4.2). Then W rebuilds the general solution of (ux , vy )-problem with boundary data (4.1) by the formula (4.4). The general solution  of (1–4)-problem with boundary data u1 = λ h1 + (λ + 2μ) h2 ,

u4 = −μ h1 + μ h2 ,

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2

generates the second order derivatives ∂∂xW2 , ∂∂yW2 in D by the formulas (4.3). The general solution of (ux , vy )-problem is expressed for every (x, y) ∈ D by the equalities 2μ

μ ∂u(x, y) λ + 2μ = U1 [(ζ )] − U4 [(ζ )] , ∂x λ+μ λ+μ

2μ

μ ∂v(x, y) λ = U1 [(ζ )] + U4 [(ζ )] . ∂y λ+μ λ+μ

A theorem analogous to Theorem 4.3 is proved in [13, Theorem 4] in assumption that the (1–4)-problem is understood in the sense of Kovalev. But it is still valid with the same proof for the (1–4)-problem formulated in this paper. It happens due to Lemmas 4.1, 4.2 and the fact that the left-hand sides of (4.3) have limiting values on ∂D if and only if U1 [], U4 [] have limiting values on ∂Dζ . The elastic equilibrium in terms of displacements and stresses can be found by use of the generalized Hooke’s law and solutions V1 , V2 of the (ux , vy )-problem (see [13, sect. 5]).

5 Solving Process of (1–4)-Problem via Analytic Functions of a Complex Variable A method for solving the (1–4)-problem by means of its reduction to classic Schwartz boundary value problems for analytic functions of a complex variable is delivered in [11]. Let us formulate some results of such a kind. In what follows, we assume that the domain Dz is simply connected (bounded or unbounded), and in this case we shall say that the domains D and Dζ are also simply connected. For a function F : Dz −→ C we denote its limiting value at a point z0 ∈ ∂Dz by F + (z0 ) if it exists. The classic Schwartz problem is a problem on finding an analytic function F : Dz −→ C of a complex variable when values of its real part are given on the boundary of domain, i.e., (Re F )+ (t) = u(t)

∀ t ∈ ∂Dz ,

(5.1)

where u : ∂Dz −→ R is a given continuous function. Theorem 5.1 Let ul : ∂Dζ −→ R, l ∈ {1, 4}, be continuous functions and F be a solution of the classic Schwartz problem with boundary condition: (Re F )+ (t) = u1 (t) − u4 (t)

∀ t ∈ ∂Dz .

(5.2)

Schwartz-Type Boundary Value Problems for Monogenic Functions in a. . .

201

and, furthermore, the function   ∗ (z) := Re −iyF (z) F

∀ z ∈ Dz

have continuous limiting values on ∂Dz . Then a solution of the (1–4)-problem is expressed by the formula (1.6) or, the same, by the formula (1.7), where the function F0 is a solution of the classic Schwartz problem with boundary condition: (Re F0 )+ (t) =

 +  1 ∗ (t) u4 (t) − F 2

∀ t ∈ ∂Dz .

(5.3)

Proof It follows from the expression (1.7) that the (1–4)-problem is reduced to finding a pair of analytic in Dz functions F , F0 satisfying the following boundary conditions: -   ∗ + 2F0 + (t) = u1 (t) ∀ t ∈ ∂Dz , Re F + F (5.4)    ∗ + 2F0 + (t) Re F = u4 (t) ∀ t ∈ ∂Dz . ∗ has continuous limiting values on ∂Dz , the In the case where the function F conditions (5.4) are equivalent to the boundary conditions (5.2), (5.3) of classic Schwartz problems.   Theorem 5.2 The general solution of the homogeneous (1–4)-problem for an arbitrary simply connected domain Dζ is expressed by the formula (ζ ) = a1 ie1 + a2 e2 ,

(5.5)

where a1 , a2 are any real constants Proof By Theorem 5.1, a solving process of the homogeneous (1–4)-problem consists of consecutive finding of solutions of two homogeneous classic Schwartz problems, viz.: a) to find an analytic in Dz function F satisfying the boundary condition (Re F )+ (t) = 0 for all t ∈ ∂Dz . As a result, we have F (z) = ai, where a is an arbitrary real constant; b) to find similarly an analytic in Dz function F0 satisfying the boundary condition (Re F0 )+ (t) = 0 for all t ∈ ∂Dz . Consequently, getting a general solution of the homogeneous (1–4)-problem in the form (1.7), we can rewrite it in the form (5.5).   Remark 5.3 A statement similar to Theorem 5.2 is proved for homogeneous (1–4)problem in the sense of Kovalev in [11], where the formula of solutions is the same as (5.5).

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Remark 5.4 Considering the functions (z) := −iyF (z) ∀ z ∈ Dz , F ) := −iy (ζ ) ρ (ζ

∀ ζ ∈ Dζ

and taking into account the equalities (1.6), (1.2), we obtain the relations   iy   ∂(ζ )



(z)ρ = ρ = −iy F (z)e1 − F (z) − F0 (z) ρ ρ = F (ζ ) = −iy ∂x 2 (z)e2 (z)e1 + 2i F = 2F

∀ ζ ∈ Dζ .

% & ∗ (z) = 1 U1 (ζ ) F 2

∀ z ∈ Dz ,

Thus,

% & ∗ )+ exist and continuous on ∂Dz if and only if U1  is and limiting values (F continuously extended on ∂D . Remark 5.5 Theorem 5.2 shows an example of the (1–4)-problem (with u1 = u4 ≡ 0 and with no extra assumptions on a domain Dζ ) when the condition on existence  + ∗ is satisfied. Evidently, we have another similar of continuous limiting values F trivial case, where u1 and u4 are real constants. In the next section we consider   else ∗ + a nontrivial case of the (1–4)-problem when the continuous limiting values F exist.

6 (1–4)-Problem for a Half-Plane Consider the (1–4)-problem in the case where the domain Dζ is the half-plane + := {ζ = xe1 + ye2 : y > 0}. Consider the biharmonic Schwartz integral for the half-plane + : 1 S+ [u](ζ ) := πi

+∞ −∞

u(t)(1 + tζ ) (t − ζ )−1 dt (t 2 + 1)

∀ ζ ∈ + .

Here and in what follows, all integrals along the real axis are understood in the sense of their Cauchy principal values, i.e. +∞ g(t, ·) dt := −∞

N lim

N→+∞ −N

g(t, ·) dt ,

Schwartz-Type Boundary Value Problems for Monogenic Functions in a. . .

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The function S+ [u](ζ ) is the principal extension (see [20, p. 165]) into the halfplane + of the complex Schwartz integral 1 S[u](z) := πi

+∞ −∞

u(t)(1 + tz) dt , (t 2 + 1)(t − z)

which determines a holomorphic function in the half-plane {z = x + iy : y > 0} of the complex plane C with the given boundary values u(t) of real part on the real line R. Furthermore, the equality y ρ S+ [u](ζ ) = S[u](z)e1 − 2π

∞ −∞

u(t) dt (t − z)2

∀ ζ ∈ +

(6.1)

holds. The following relations were proved within the proof of Theorem 1 in [8]: ∞ y −∞

u(t) d t ≤ 4 ωR (u, 2y) + 2 y (t − z)2

∞

ωR (u, η) dη → 0, η2

z→ξ,

∀ξ ∈ R,

2y

(6.2) where ωR (u, ε) =

sup

t1 ,t2 ∈R:|t1 −t2 |≤ε

|u(t1 ) − u(t2 )|

is the modulus of continuity of the function u . In addition, ∞ y −∞

u(t) d t → 0, (t − z)2

z → ∞.

(6.3)

It follows from the equality (6.1) and the relations (6.2), (6.3) that " # U1 S+ [u](ζ ) → u(ξ ) ,

z → ξ,

∀ ξ ∈ R ∪ {∞}

and 

∗ F

+

(ξ ) = 0

∀ ξ ∈ R ∪ {∞}

∗ defined in Theorem 5.1. for the function F

(6.4)

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Theorem 6.1 Let every function ul : R −→ R, l ∈ {1, 4}, have a finite limit of the type (2.1). Then the general solution of the (1–4)-problem for the half-plane + is expressed by the formula (ζ ) = S+ [u1 ](ζ ) e1 + S+ [u4 ](ζ ) ie2 + a1 ie1 + a2 e2 ,

(6.5)

where a1 , a2 are any real constants. Proof It follows from the relation (6.4) that the function 1,4 (ζ ) = S+ [u1 ](ζ ) e1 + S+ [u4 ](ζ ) ie2

(6.6)

is a solution of the (1–4)-problem for the half-plane + . The general solution of the (1–4)-problem in the form (6.5) is obtained by summarizing the particular solution (6.6) of the inhomogeneous (1–4)-problem and the general solution (5.5) of the homogeneous (1–4)-problem.   Remark 6.2 In Theorem 3 [11] we obtain the general solution of (1–4)-problem in the sense of Kovalev in the form (6.5) but under complementary assumptions that for every given function ul : R −→ R, l ∈ {1, 4}, its modulus of continuity and the local centered (with respect to the infinitely remote point) modulus of continuity satisfy Dini conditions.

7 Solving Process of (1–4)-Problem for Bounded Simply Connected Domain with Use of the Complex Green Function Now, for solving the (1–4)-problem we shall use solutions of the classic Schwartz problem for analytic functions of a complex variable in the form of an appropriate Schwartz operator involving the complex Green function. Here we assume that Dz is a bounded simply connected domain with a smooth boundary ∂Dz . Let g(z, z0 ) be the Green function of Dz for the Laplace operator (cf, e.g., [21, p. 22]). It is well-known that the general solution of the Schwartz boundary value problem for analytic functions with boundary datum (5.1) is expressed in the form (cf, e.g., [21, p. 52]) F (z) = (Su)(z) + ia0 ,

(7.1)

with an arbitrary real number a0 and the Schwartz operator (Su)(z) having the form (Su)(z) := −

1 2π

 u(t) ∂Dz

∂M(t, z) dst ∂nt

∀z ∈ Dz ,

(7.2)

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where the complex Green function M (cf, e.g., [21, p. 32]) is of the form M(w, z) = g(w, z) + ih(w, z), h is a conjugate harmonic function to the Green function g with respect to w ∈ Dz : w = z, nt is the outward normal unit vector at the point t ∈ ∂Dz , st is an arc coordinate of the point t . ∗ defined in Theorem 5.1 takes the form Now, the function F     ∗ (z) = Re −iy S(u1 − u4 ) (z) ∀ z ∈ Dz , (7.3) F where u1 and u4 are given functions of the (1–4) problem. Therefore, using expressions of solutions of the classic Schwartz problems with the boundary conditions (5.2), (5.3) in the form (7.1) via appropriate Schwartz operators of the type (7.2), by Theorem 5.1 we obtain the following statement.  + ∗ on Theorem 7.1 Let the function (7.3) have the continuous limiting values F ∂Dz . Then the general solution of (1–4)-problem is expressed in the form     (ζ ) = F (z) − iyF (z) + 2F0 (z) e1 + i 2F0 (z) − iyF (z) e2 + +a1 ie1 + a2 e2

∀ z ∈ Dz ,

where   F (z) = S(u1 − u4 ) (z),

F0 (z) =

 +  1  ∗ S u4 − F (z) 2

and a1 , a2 are any real constants. In the next section we develop a method for solving the inhomogeneous (1–4) ∗ have problem without an essential in Theorem 5.1 assumption that the function F continuous limiting values on the boundary ∂Dz .

8 Solving BVP’s by Means Hypercomplex Cauchy-Type Integrals Let the boundary ∂Dζ of the bounded domain Dζ be a closed smooth Jordan curve. Below, we show a method for reducing (1–3)-problem and (1–4)-problem to systems of the Fredholm integral equations. Such a method was developed in [9, 12]. Obtained results are appreciably similar for the mentioned problems, however, in contrast to (1–3)-problem, which is solvable in a general case if and only if a certain natural condition is satisfied, the (1–4)-problem is solvable unconditionally. We use the modulus of continuity of a continuous function ϕ given on ∂Dζ : ω(ϕ, ε) :=

sup

τ1 ,τ2 ∈∂Dζ : τ1 −τ2 ≤ε

ϕ(τ1 ) − ϕ(τ2 ) .

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We assume that ω(ϕ, ε) satisfies the Dini condition: 1

ω(ϕ, η) d η < ∞. η

(8.1)

0

Consider the biharmonic Cauchy type integral: B[ϕ](ζ ) :=

1 2πi



ϕ(τ )(τ − ζ )−1 dτ

∀ζ ∈ μe1 ,e2 \ ∂Dζ .

(8.2)

∂Dζ

It is proved in Theorem 4.2 [9] that the integral (8.2) has limiting values B + [ϕ](ζ0 ) :=

lim

ζ →ζ0 , ζ ∈Dζ

(ζ ),

B − [ϕ](ζ0 ) :=

lim

ζ →ζ0 , ζ ∈μe1 ,e2 \Dζ

(ζ )

in every point ζ0 ∈ ∂Dζ that are represented by the Sokhotski–Plemelj formulas: B + [ϕ](ζ0 ) =



1 1 ϕ(ζ0 ) + 2 2πi

ϕ(τ )(τ − ζ0 )−1 dτ ,

(8.3)

∂Dζ

1 1 B [ϕ](ζ0) = − ϕ(ζ0 ) + 2 2πi −



ϕ(τ )(τ − ζ0 )−1 dτ ,

∂Dζ

where a singular integral is understood in the sense of its Cauchy principal value:  ϕ(τ )(τ − ζ0 ) ∂Dζ

−1

 dτ := lim

ε→0 {τ ∈∂Dζ :τ −ζ0 >ε}

ϕ(τ )(τ − ζ0 )−1 dτ.

We assume that boundary functions uk , k ∈ {1, 3} or k ∈ {1, 4}, of the (1–3) problem or the (1–4) problem, respectively, satisfy Dini conditions of the type (8.1). We seek solutions in a class of functions represented in the form (ζ ) = B[ϕ](ζ ) ∀ ζ ∈ Dζ , where ϕ(ζ ) = ϕ1 (ζ ) e1 + ϕ3 (ζ ) e2

∀ ζ ∈ ∂Dζ

(8.4)

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for the (1–3) problem or ϕ(ζ ) = ϕ1 (ζ ) e1 + ϕ4 (ζ ) ie2

∀ ζ ∈ ∂Dζ

(8.5)

for the (1–4) problem, and every function ϕk : ∂Dζ −→ R, k ∈ {1, 3, 4}, satisfies a Dini condition of the type (8.1). We use a conformal mapping z = τ (t) of the upper half-plane {t ∈ C : Im t > 0} onto the domain Dz . Denote τ1 (t) := Re τ (t), τ2 (t) := Im τ (t). Inasmuch as the mentioned conformal mapping is continued to a homeomorphism between the closures of corresponding domains, the function τ (s) := τ1 (s)e1 + τ2 (s)e2

∀s ∈ R

generates a homeomorphic mapping of the extended real axis R := R ∪ {∞} onto the curve ∂Dζ . Introducing the function g(s) := ϕ ( τ (s))

∀s ∈ R,

we rewrite the equality (8.3) in the form (cf. [9]) 1 1 B [ϕ](ζ0) = g(t) + 2 2πi +

∞ −∞

1 g(s)k(t, s) ds + 2πi

∞ g(s) −∞

1 + st ds , (s − t)(s 2 + 1)

where k(t, s) = k1 (t, s)e1 + iρ k2 (t, s) , k1 (t, s) := k2 (t, s) :=

1 + st τ (s) − , τ (s) − τ (t) (s − t)(s 2 + 1)

  τ (s) τ2 (s) − τ2 (t) τ2 (s) ,  2 −  2 τ (s) − τ (t) 2 τ (s) − τ (t)

and a correspondence between the points ζ0 ∈ ∂Dζ \ { τ (∞)} and t ∈ R is given by the equality ζ0 = τ (t). Evidently, g(s) = g1 (s)e1 + g3 (s)e2 for the (1–3) problem and g(s) = g1 (s)e1 + g4 (s)ie2 for the (1–4) problem, where gl (s) := ϕl ( τ (s)) for all s ∈ R, l ∈ {1, 3, 4}. % & Now, in the case of (1–3)-problem, we single out components Ul B + [ϕ](ζ0) , l ∈ {1, 3}, and after the substitution them into the boundary conditions of the

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(1–3)-problem, we shall obtain the following system of integral equations for finding the functions g1 and g3 : %

&

1 1 U1 B [ϕ](ζ0 ) ≡ g1 (t) + 2 2π +

−

1 π

∞ −∞

1 1 U3 B + [ϕ](ζ0 ) ≡ g3 (t) + 2 2π −

1 π

  g1 (s) Im k1 (t, s) + 2Re k2 (t, s) ds−

−∞

g3 (s)Im k2 (t, s) ds = u1 (t),

&

%

∞

∞

∞

  g3 (s) Im k1 (t, s) − 2Re k2 (t, s) ds−

−∞

g1 (s)Im k2 (t, s) ds = u3 (t)

∀t ∈ R,

−∞

(8.6)   τ (t) , l ∈ {1, 3}. where ul (t) := ul % Similarly, in the case of (1–4)-problem, we single out components Ul B + [ϕ] & (ζ0 ) , l ∈ {1, 4}, and after the substitution them into the boundary conditions of the (1–4)-problem, we shall obtain the following system of integral equations for finding the functions g1 and g4 : %

&

1 1 U1 B [ϕ](ζ0 ) ≡ g1 (t) + 2 2π +

1 − π %

∞ −∞

1 1 U4 B + [ϕ](ζ0 ) ≡ g4 (t) + 2 2π 1 π

  g1 (s) Im k1 (t, s) + 2Re k2 (t, s) ds−

−∞

g4 (s)Re k2 (t, s) ds = u1 (t),

&

+

∞

∞

∞

  g4 (s) Im k1 (t, s) − 2Re k2 (t, s) ds+

−∞

g1 (s)Re k2 (t, s) ds = u4 (t)

∀t ∈ R,

−∞

(8.7)   where ul (t) := ul τ (t) , l ∈ {1, 4}. Let C(R ) denote the Banach space of functions g∗ : R −→ C that are continuous on the extended real axis R with the norm g∗ C(R ) := sup |g∗ (t)|. t ∈R

In Theorem 6.13 [9] there are conditions which are sufficient for compactness of integral operators on the left-hand sides of equations of the systems (8.6), (8.7) in the space C(R ).

Schwartz-Type Boundary Value Problems for Monogenic Functions in a. . .

209

To formulate such conditions, consider the conformal mapping σ (T ) ofthe unit disk {T ∈ C : |T | < 1} onto the domain Dz such that τ (t) = σ tt −i +i for all t ∈ {t ∈ C : Im t > 0}. Thus, it follows from Theorem 6.13 [9] that if the conformal mapping σ (T ) have the nonvanishing continuous contour derivative σ (T ) on the unit circle  := {T ∈ C : |T | = 1}, and its modulus of continuity ω (σ , ε) :=

sup

T1 ,T2 ∈, |T1 −T2 |≤ε

|σ (T1 ) − σ (T2 )|

satisfies a condition of the type (8.1), then the integral operators in the systems (8.6), (8.7) are compact in the space C( R ). Let D(R) denote the class of functions g∗ ∈ C(R ) whose the modulus of continuity ωR (g∗ , ε) and the local centered (with respect to the infinitely remote point) modulus of continuity ωR,∞ (g∗ , ε) =

sup

τ ∈R:|τ |≥1/ε

|g∗ (τ ) − g∗ (∞)|

satisfy the Dini conditions 1

ωR (g∗ , η) d η < ∞, η

0

1

ωR,∞ (g∗ , η) d η < ∞. η

0

Since the sought-for function ϕ in (8.2) has to satisfy the condition (8.1), it is necessary to require that the corresponding functions g1 , g3 in (8.4) or g1 , g4 in (8.5) should belong to the class D(R). In the next theorems we state a condition on the conformal mapping σ (T ), under which all solutions of the system (8.6), (8.7) satisfy the mentioned requirement. Theorem 8.1 Assume that the functions ul : ∂Dζ −→ R, l ∈ {1, 3}, satisfy conditions of the type (8.1). Also, assume that the conformal mapping σ (T ) has the nonvanishing continuous contour derivative σ (T ) on the circle , and its modulus of continuity ω (σ , ε) satisfies the condition 2

3 ω (σ , η) ln dη < ∞. η η

(8.8)

0

Then all functions g1 , g3 ∈ C( R ) satisfying the system of Fredholm integral equations (8.6) belong to the class D(R), and the corresponding function ϕ in (8.4) satisfies the Dini condition (8.1).

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Assume additionally: 1) all solutions (g1 , g3 ) ∈ C( R ) × C( R ) of the homogeneous system of equations (8.6) (with uk ≡ 0 for k ∈ {1, 3}) are differentiable on R; 2) for every mentioned%solution (g1 , g3 ) of the homogeneous system of equations & (8.6), the integral B ϕ is finite in Dζ and μe1 ,e2 \ Dζ , and the functions % % & & % % & & U1 B ϕ (ζ ) − U4 B ϕ (ζ ) % % & & % % & & U2 B ϕ (ζ ) + U3 B ϕ (ζ )

∀ ζ ∈ Dζ ,

∀ ζ ∈ μe1 ,e2 \ Dζ

are bounded, where ϕ is the contour derivative of the corresponding function ϕ in (8.4), i.e., ϕ (ζ ) ≡ ϕ( τ (s)) := g1 (s)e1 + g3 (s)e2 for all s ∈ R. Then the following assertions are true: (i) the number of linearly independent solutions of the homogeneous system of equations (8.6) is equal to 1; (ii) the non-homogeneous system of equations (8.6) is solvable if and only if the condition (3.2) is satisfied. Theorem 8.2 Assume that the functions ul : ∂Dζ −→ R, l ∈ {1, 4}, satisfy conditions of the type (8.1). Also, assume that the conformal mapping σ (T ) has the nonvanishing continuous contour derivative σ (T ) on the circle , and its modulus of continuity satisfy the condition (8.8). Then the following assertions are true: (i) the system of Fredholm integral equations (8.7) has the unique solution in C( R ); (ii) all functions g1 , g4 ∈ C( R ) satisfying the system (8.7) belong to the class D(R), and the corresponding function ϕ in (8.5) satisfies the Dini condition (8.1). Remark 8.3 Generalizing Theorem 6.13 [9], Theorem 8.1 is proved similarly. Theorem 8.2 is proved in [12] if a (1–4)-problem is understood in the sense of Kovalev but it is still valid for a (1–4)-problem considered in this paper. Acknowledgements This research is partially supported by the State Program of Ukraine (Project No. 0117U004077) and Grant of Ministry of Education and Science of Ukraine (Project No. 0116U001528).

References 1. V.F. Kovalev, I.P. Mel’nichenko, Biharmonic functions on the biharmonic plane. Rep. Acad. Sci. USSR Ser. A. 8, 25–27 (1981, in Russian) 2. I.P. Melnichenko, Biharmonic bases in algebras of the second rank. Ukr. Mat. Zh. 38(2), 224– 226 (1986, in Russian). English transl. (Springer) in Ukr. Math. J. 38(2), 252–254 (1986)

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3. A. Douglis, A function-theoretic approach to elliptic systems of equations in two variables. Commun. Pure Appl. Math. 6(2), 259–289 (1953) 4. L. Sobrero, Nuovo metodo per lo studio dei problemi di elasticità, con applicazione al problema della piastra forata. Ricerche Ingegneria 13(2), 255–264 (1934, in Italian) 5. S.V. Grishchuk, S.A. Plaksa, Monogenic functions in a biharmonic algebra. Ukr. Mat. Zh. 61(12), 1587–1596 (2009, in Russian). English transl. (Springer) in Ukr. Math. J. 61(12), 1865– 1876 (2009) 6. S.V. Gryshchuk, S.A. Plaksa, Basic properties of monogenic functions in a biharmonic plane, in Complex Analysis and Dynamical Systems V, Contemporary Mathematics, vol. 591 (American Mathematical Society, Providence, 2013), pp. 127–134 7. V.F. Kovalev, Biharmonic Schwarz Problem. Preprint No. 86.16, Institute of Mathematics, Acad. Sci. USSR (Inst. of Math. Publ. House, Kiev, 1986, in Russian) 8. S.V. Gryshchuk, S.A. Plaksa, Schwartz-type integrals in a biharmonic plane. Int. J. Pure Appl. Math. 83(1), 193–211 (2013) 9. S.V. Gryshchuk, S.A. Plaksa, Monogenic functions in the biharmonic boundary value problem. Math. Methods Appl. Sci. 39(11), 2939–2952 (2016) 10. S.V. Gryshchuk, One-dimensionality of the kernel of the system of Fredholm integral equations for a homogeneous biharmonic problem. Zb. Pr. Inst. Mat. NAN Ukr. 14(1), 128–139 (2017, in Ukrainian). English summary 11. S.V. Gryshchuk, S.A. Plaksa, A Schwartz-type boundary value problem in a biharmonic plane. Lobachevskii J. Math. 38(3), 435–442 (2017) 12. S.V. Gryshchuk, S.A. Plaksa, Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations. Open Math. 15(1), 374–381 (2017) 13. S.V. Gryshchuk, B-valued monogenic functions and their applications to boundary value problems in displacements of 2-D elasticity, in Analytic Methods of Analysis and Differential Equations: AMADE 2015, Belarusian State University, Minsk, Belarus, ed. by S.V. Rogosin, M.V. Dubatovskaya (Cambridge Scientific Publishers, Cambridge, 2016), pp. 37–47. ISBN (paperback): 978-1-908106-56-8 14. S.G. Mikhlin, The plane problem of the theory of elasticity, in Trans. Inst. of Seismology, Acad. Sci. USSR. No. 65 (Acad. Sci. USSR Publ. House, Moscow, 1935, in Russian) 15. N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Fundamental Equations, Plane Theory of Elasticity, Torsion and Bending. English transl. from the 4th Russian edition by R.M. Radok (Noordhoff International Publishing, Leiden, 1977) 16. A.I. Lurie, Theory of Elasticity. Engl. transl. by A. Belyaev (Springer, Berlin, 2005) 17. N.S. Kahniashvili, Research of the plain problems of the theory of elasticity by the method of the theory of potentials, in Trudy Tbil. Univer. 50 (Tbil. Univer., Tbilisi, 1953, in Russian) 18. Yu.A. Bogan, On Fredholm integral equations in two-dimensional anisotropic theory of elasticity. Sib. Zh. Vychisl. Mat. 4(1), 21–30 (2001, in Russian) 19. L. Lu, Complex Variable Methods in Plane Elasticity/Series in Pure Mathematics, vol. 22 (World Scientific, Singapore, 1995) 20. E. Hille, R.S. Phillips, Functional Analysis and Semi-Groups. Colloquium Publications, vol. 31 (American Mathematical Society, Providence, 2000) 21. H. Begehr, Complex Analytic Methods for Partial Differential Equations. An Introductory Text (World Scientific, Singapore, 1994)

The String Equation for Some Rational Functions Björn Gustafsson

Dedicated to Heinrich Begehr, on the occasion of his 80th birthday

Abstract For conformal maps defined in the unit disk one can define a certain Poisson bracket that involves the harmonic moments of the image domain. When this bracket is applied to the conformal map itself together with its conformally reflected map the result is identically one. This is called the string equation, and it is closely connected to the governing equation, the Polubarinova-Galin equation, for the evolution of a Hele-Shaw blob of a viscous fluid (or, by another name, Laplacian growth). For non-univalent analytic functions the Poisson bracket may become ambiguous, hence the string equation need not make sense. In the present paper we show that for a certain class of (non-univalent) rational functions related to quadrature Riemann surfaces, the string equation does make sense, and holds. Keywords Polubarinova-Galin equation · String equation · Poisson bracket · Harmonic moments · Branch points · Hele-Shaw flow · Laplacian growth · Quadrature Riemann surface Mathematics Subject Classification (2010) Primary 30C55; Secondary 31A25, 34M35, 37K05, 76D27

1 Introduction This paper originates in some recent developments related to evolutions of HeleShaw blobs of viscous fluids, also referred to as Laplacian growth. The history of this subject is beautifully summarized in [36], and general expositions are given in

B. Gustafsson () Department of Mathematics, KTH, Stockholm, Sweden e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_11

213

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B. Gustafsson

[8, 9, 35]. The subject started with some experiments and a short paper by Henry Selby Hele-Shaw in 1898. The mathematical model in form of a two dimensional ideal flow was derived by H. Lamb [17], starting from the actual three dimensional fluid description involving a highly viscous fluid squeezed between two plates. For the corresponding free and moving boundary problem, an equation for the conformal map from the unit disk was obtained P.Ya. Polubarinova-Kochina and L.A. Galin around 1945. Slightly later, in 1948, U.P. Vinogradov and P.P. Kufarev were able to prove local existence of solutions of the appropriate initial value problem, under the necessary analyticity conditions. This was a considerable achievement at that time, and not until the 1990s further progress in that respect were made by M. Reissig, L. Wolfersdorf [23], J. Escher, G. Simmonett [2] and others. See also [18, 25, 31]. Slightly later, around 2000, a group of Russian mathematical physicists, led by M. Mineev-Weinstein, P. Wiegmann, A. Zabrodin, considered the Hele-Shaw problem from the point of view of integrable systems, and the PolubarinovaGalin equation then reappears under the name “string equation”. See for example [14, 15, 21, 22, 37]. The integrable system approach appears as a consequence of the discovery 1972 by S. Richardson [24] that the Hele-Shaw problem has a complete set of conserved quantities, namely the harmonic moments. The name string equation can be traced back to theories of integrable hierarchies, such as the 2D Toda hierarchy, for which there appears a pair L, L¯ of “Lax operators” satisfying what in this context is called the string equation, namely ¯ = h¯ , [L, L] where h¯ > 0 is Planck’s constant. As h¯ → 0 this equation is taken over, in the contexts described in [21, 37] (for example), by our “dispersionless” string equation (see (1.1) below) for a conformal map. In this way the subject of Laplacian growth connects to topological gravity and matrix models of 2D gravity. See papers mentioned above, and in addition [13, 19, 20], to mention just a few. Related to this are also connections to ensembles of random normal matrices, quantum Hall regimes and Coulomb gas ensembles, see [11, 30, 38, 39], for example. The string equation for conformal maps is deceptively simple and beautiful. It reads {f, f ∗ } = 1,

(1.1)

in terms of a special Poisson bracket referring to harmonic moments and with f any normalized conformal map from some reference domain, in our case the unit disk, to the fluid domain for the Hele-Shaw flow. The main question for this paper now is: if such a beautiful equation as (1.1) holds for all univalent functions, shouldn’t it also hold for non-univalent functions? The answer is that the Poisson bracket does not (always) make sense in the nonunivalent case, but that one can extend its meaning, actually in several different ways, and after such a step the string equation indeed holds. Thus the problem is not

The String Equation for Some Rational Functions

215

that the string equation is difficult to prove, the problem is that the meaning of the string equation is ambiguous in the non-univalent case. In this paper we show that the string equation makes sense and holds for a class of rational functions related to quadrature Riemann surfaces. In a related paper [4] (see also [3]) it is shown that the string equation makes good sense and holds, but in a different way, for polynomials.

2 The String Equation for Univalent Conformal Maps We consider analytic functions f (ζ ) defined in a neighborhood of the closed unit disk and normalized by f (0) = 0, f (0) > 0. In addition, we always assume that f has no zeros on the unit circle. It will be convenient to write the Taylor expansion around the origin on the form f (ζ ) =

∞ 

aj ζ j +1

(a0 > 0).

j =0

If f is univalent it maps D = {ζ ∈ C : |ζ | < 1} onto a domain  = f (D). The harmonic moments for this domain are  1 Mk = zk dxdy, k = 0, 1, 2, . . . . π  The integral here can be pulled back to the unit disk and pushed to the boundary there. This gives 1 Mk = 2πi



1 f (ζ ) |f (ζ )| d ζ¯ dζ = 2πi D k



f (ζ )k f ∗ (ζ )f (ζ )dζ,

2

(2.1)

∂D

where f ∗ (ζ ) = f (1/ζ¯ )

(2.2)

denotes the holomorphic reflection of f in the unit circle. In the form in (2.1) the moments make sense also when f is not univalent. Computing the last integral in (2.1) by residues gives Richardson’s formula [24] for the moments:  (j0 + 1)aj0 · · · ajk a¯ j0 +...+jk +k , (2.3) Mk = (j1 ,...,jk )≥(0,...,0)

This is a highly nonlinear relationship between the coefficients of f and the moments, and even if f is a polynomial of low degree it is virtually impossible to invert it, to obtain ak = ak (M0 , M1 , . . . ), as would be desirable in many situations.

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Still there is, quite remarkably, an explicit expressions for the Jacobi determinant of the change (a0 , a1 , . . . ) → (M0 , M1 , . . . ) when f restricted to the class of polynomials of a fixed degree. This formula, which was proved by to O. Kuznetsova and V. Tkachev [16] and V. Tkachev [32] after an initial conjecture of C. Ullemar [33], is briefly discussed in Theorem 5.1 below. There are examples of different simply connected domains having the same harmonic moments, see for example [26, 27, 40]. Restricting to domains having analytic boundary the harmonic moments are however sensitive for at least small variations of the domain. This can easily be proved by potential theoretic methods. Indeed, arguing on an intuitive level, an infinitesimal perturbation of the boundary can be represented by a signed measure sitting on the boundary (this measure representing the speed of infinitesimal motion). The logarithmic potential of that measure is a continuous function in the complex plane, and if the harmonic moments were insensitive for the perturbation then the exterior part of this potential would vanish. At the same time the interior potential is a harmonic function, and the only way all these conditions can be satisfied is that the potential vanishes identically, hence also that the measure on the boundary vanishes. On a more rigorous level, in the polynomial case the above mentioned Jacobi determinant is indeed nonzero. Compare also discussions in [25]. The conformal map, with its normalization, is uniquely determined by the image domain  and, as indicated above, the domain is locally encoded in the sequence the moments M0 , M1 , M2 , . . . . Thus the harmonic moments can be viewed as local coordinates in the space of univalent functions, and we may write f (ζ ) = f (ζ ; M0 , M1 , M2 , . . . ). In particular, the derivatives ∂f /∂Mk make sense. Now we are in position to define the Poisson bracket. Definition 2.1 For any two functions f (ζ ) = f (ζ ; M0 , M1 , M2 , . . . ), g(ζ ) = g(ζ ; M0 , M1 , M2 , . . . ) which are analytic in a neighborhood of the unit circle and are parametrized by the moments we define {f, g} = ζ

∂f ∂g ∂g ∂f −ζ . ∂ζ ∂M0 ∂ζ ∂M0

(2.4)

This is again a function analytic in a neighborhood of the unit circle and parametrized by the moments. The Schwarz function [1, 29] of an analytic curve  is the unique holomorphic function defined in a neighborhood of  and satisfying S(z) = z¯ ,

z ∈ .

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217

When  = f (∂D), f analytic in a neighborhood of ∂D, the defining property of S(z) becomes S ◦ f = f ∗,

(2.5)

holding then identically in a neighborhood of the unit circle. Notice that f ∗ and S depend on the moments M0 , M1 , M2 . . . , like f . The string equation asserts that {f, f ∗ } = 1

(2.6)

in a neighborhood of the unit circle, provided f is univalent in a neighborhood of the closed unit disk. This result was first formulated and proved in [37] for the case of conformal maps onto an exterior domain (containing the point of infinity). For conformal maps to bounded domains a proof based on somewhat different ideas and involving explicitly the Schwarz function was given in [9]. For convenience we briefly recall this proof below. Writing (2.5) more explicitly as f ∗ (ζ ; M0 , M1 , . . . ) = S(f (ζ ; M0 , M1 , . . . ); M0 , M1 , . . . ) and using the chain rule when computing {f, f ∗ } = ζ

∂f ∗ ∂M0

gives, after simplification,

∂S ∂f ·( ◦ f ). ∂ζ ∂M0

(2.7)

Next one notices that the harmonic moments are exactly the coefficients in the expansion of a certain Cauchy integral at infinity: 1 2πi



∞

∂

 Mk wdw ¯ = z−w zk+1

(|z| >> 1).

k=0

Combining this with the fact that the jump of this Cauchy integral across ∂ is z¯ it follows that S(z) equals the difference between the analytic continuations of the exterior (z ∈ e ) and interior (z ∈ ) functions defined by the Cauchy integral. Therefore S(z; M0 , M1 , . . . ) =

∞  Mk + function holomorphic in , zk+1 k=0

and so, since M0 , M1 , . . . are independent variables, ∂S 1 (z; M0 , M1 , . . . ) = + function holomorphic in . ∂M0 z

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Inserting this into (2.7) one finds that {f, f ∗ } is holomorphic in D. Since the Poisson bracket is invariant under holomorphic reflection in the unit circle it follows that {f, f ∗ } is holomorphic in the exterior of D (including the point of infinity) as well, hence it must be constant. And this constant is found to be one, proving (2.6). In the forthcoming sections we wish to allow non-univalent analytic functions in the string equation. Then the basic ideas in the above proof still work, but what may happen is that f and S are not determined by the moments M0 , M1 , . . . alone. Since ∂f/∂M0 is a partial derivative one has to specify all other independent variables in order to give a meaning to it. So there may be more variables, say f (ζ ) = f (ζ ; M0 , M1 , . . . ; B1 , B2 , . . . ). This does not change the proof very much, but the meaning of the string equation depends on the choice of these extra variables. Natural choices turn out to be locations of branch points, i.e., one takes Bj = f (ωj ), where the ωj ∈ D denote the zeros of f inside D. One good thing with choosing the branch points as additional variables is that keeping these fixed, as is implicit then in the notation ∂/∂M0 , means that f in this case can be viewed as a conformal map into a fixed Riemann surface, which will be a branched covering over the complex plane. There are also other possibilities for giving a meaning to the string equation, like restricting f to the class of polynomials of a fixed degree, as in Theorem 5.1 below. But then one must allow the branch points to move, so this gives a different meaning to ∂/∂M0 .

3 Intuition and Physical Interpretation in the Non-univalent Case As indicated above we shall consider also non-univalent analytic functions as conformal maps, then into Riemann surfaces above C. In general these Riemann surfaces will be branched covering surfaces, and the non-univalence is then absorbed in the covering projection. It is easy to understand that such a Riemann surface, or the corresponding conformal map, will in general not be determined by the moments M0 , M1 , M2 , . . . alone. As a simple example, consider an oriented curve  in the complex plane encircling the origin twice (say). In terms of the winding number, or index, ν (z) =

1 2πi

9 

dζ ζ −z

(z ∈ C \ ),

(3.1)

this means that ν (0) = 2. Points far away from the origin have index zero, and some other points may have index one (for example). Having only the curve 

The String Equation for Some Rational Functions

219

available it is natural to define the harmonic moments for the multiply covered (with multiplicities ν ) set inside  as Mk =

1 π

 C

zk ν (z)dxdy =

1 2πi

 zk z¯ dz,

k = 0, 1, 2, . . . .



It is tempting to think of this integer weighted set as a Riemann surface over (part of) the complex plane. However, without further information this is not possible. Indeed, since some points have index ≥ 2 such a covering surface will have to have branch points, and these have to be specified in order to make the set into a Riemann surface. And only after that it is possible to speak about a conformal map f . Thus f is in general not determined by the moments alone. In the simplest non-univalent cases f will be (locally) determined by the harmonic moments together with the location of the branch points. There are actually more problems in the non-univalent case. Even if we specify all branch points, the test class of functions 1, z, z2 , . . . used in defining the moments may be too small since each of these functions take the same value on all sheets above any given point in the complex plane. In order for the Riemann surface and the conformal map to be determined one would need all analytic functions on the Riemann surface itself as test functions. There are several ways out of these problems, and we shall consider, in this paper, two such ways: • Restrict f to the class of polynomials of a fixed degree. This turns out to work well, even without specifying the branch points. • Restrict f to rational functions which map D onto quadrature Riemann surfaces admitting a quadrature identity of a special form. Then the branch points have to be specified explicitly, but it turns out that the presence of a quadrature identity resolves the problem of the test functions zk being unable to distinguish between sheets. Quadrature Riemann surfaces were introduced in [28] and, as will become clear in the forthcoming sections, they naturally enter the picture. The physical interpretation of the string equation is most easily explained with reference to general variations of analytic functions in the unit disk. Consider an arbitrary smooth variation f (ζ ) = f (ζ, t), depending on a real parameter t. We always keep the normalization f (0, t) = 0, f (0, t) > 0, and f is assumed to be analytic in a full neighborhood of the closed unit disk, with f = 0 on ∂D. Then one may define a corresponding Poisson bracket written with a subscript t: {f, g}t = ζ

∂g ∂f ∂f ∂g −ζ . ∂ζ ∂t ∂ζ ∂t

(3.2)

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B. Gustafsson

This Poisson bracket is itself an analytic function in a neighborhood of ∂D. It is determined by its values on ∂D, where we have {f, f ∗ }t = 2 Re[f˙ ζf ]. The classical Hele-Shaw flow moving boundary problem, also called Laplacian growth, is a particular evolution, characterized (in the univalent case) by the harmonic moments being conserved, except for the first one which increases linearly with time, say as M0 = 2t + constant. This means that f˙ = 2∂f/∂M0 , which makes {f, f ∗ }t = 2{f, f ∗ } and identifies the string equation (2.6) with the PolubarinovaGalin equation Re [f˙(ζ, t) ζf (ζ, t)] = 1,

ζ ∈ ∂D,

(3.3)

for the Hele-Shaw problem. Dividing (3.3) by |f | gives 1 ζf ]= Re [f˙ · |ζf | |ζf |

on ∂D.

Here the left member can be interpreted as the inner product between f˙ and the unit normal vector on ∂ = f (∂D), and the right member as the gradient of a suitably normalized Green’s function of  = f (D) with pole at the origin. In fact, taking that Green’s function to be log |ζ | when pulled back to D and differentiating this with respect to z = f (ζ ), to get the gradient, we have   ∂ 1 |∇G | = 2 log |ζ | = . ∂z |ζf (ζ )| Thus (3.3) says that ∂ moves in the normal direction with velocity |∇G |, and for the string equation we then have 2

∂G ∂f  = normal ∂M0 ∂n

on ∂,

(3.4)

the subscript “normal” signifying normal component when considered as a vector on ∂. The above interpretations remain valid in the non-univalent case, with G interpreted as the Green’s function of  regarded as a Riemann surface. However, as already remarked, the moments Mk do not determine  as a Riemann surface in this case, also specification of the branch points is needed. Thus the string equation represents a whole family of domain evolutions in the non-univalent case. The most natural of these is the one for which the branch points remain fixed, because this case represents a pure expansion of an initially given Riemann surface which does not change internally during the evolution.

The String Equation for Some Rational Functions

221

A different way to express the string equation is to separate f into real and imaginary parts. On writing f (ζ, t) = z = x + iy, Eq. (3.3) becomes, with ζ = eiθ , ∂(x, y) = 1, ∂(t, θ ) or simply dt ∧ dθ = dx ∧ dy, holding then on the boundary f (eiθ , t) = x + iy. One can use the Poisson bracket to also set up equations for the derivatives of f with respect to the other moments, namely ∂f/∂Mk , k ≥ 1. This has been done (in the univalent case) in [14, 15, 22, 37], and in our setting in [9]. Let for this purpose W (z) = W (z; M0 , M1 , . . . ) be an appropriately normalized primitive function of the Schwarz function S(z) = S(z; M0 , M1 , . . . ) and define a k:th order Hamiltonian function by Hk (ζ ; M0 , M1 , . . . ) = −

∂W (z; M0 , M1 , . . . ) , ∂Mk

where z = f (ζ ; M0 , M1 , . . . ). Thus z is to be kept fixed under the differentiation, while the Hamiltonian is to be considered as a function of ζ . Note that the multi-valued term M0 log z in W (z) disappears under the differentiation. Then one shows that ∂f = {f, Hk }. ∂Mk We finally point out the important role of the general Poisson bracket (3.2) when differentiating the formula (2.1) for the moments Mk with respect to t for a given evolution. To give a more general statement in this respect we replace the function f (ζ )k appearing in (2.1) by a function g(ζ, t) which is analytic in ζ and depends on t in the same way as h(f (ζ, t)) does, where h is analytic, for example h(z) = zk . This means that g = g(ζ, t) has to satisfy f˙(ζ, t) g(ζ, ˙ t) = , g (ζ, t) f (ζ, t)

(3.5)

saying that g “flows with” f and locally can be regarded as a time independent function in the image domain of f . We then have (cf. Lemma 4.1 in [5]) Lemma 3.1 Assume that g(ζ, t) is analytic in ζ in a neighborhood of the closed unit disk and depends smoothly on t in such a way that (3.5) holds. Then 1 d 2πi dt



1 g(ζ, t)|f (ζ, t)|2 d ζ¯ dζ = 2π D

the last integrand being evaluated at ζ = eiθ .



2π 0

g(ζ, t){f, f ∗ }t dθ,

(3.6)

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B. Gustafsson

As a special case, with g(ζ, t) = h(f (ζ, t)), we have Corollary 3.2 If h(z) is analytic in a fixed domain containing the closure of f (D, t) then 1 d 2πi dt



1 h(f (ζ, t))|f (ζ, t)| d ζ¯ dζ = 2π D



2π

2

h(f (ζ, t)){f, f ∗ }t dθ.

0

Proof The proof of (3.6) is straight-forward: differentiating under the integral sign and using partial integration we have d dt



   ∗  d ∗ ¯ gf ˙ f + g f˙∗ f + gf ∗ f˙ dζ g|f | d ζ dζ = gf f dζ = dt ∂D D ∂D   ∗  = gf ˙ f + g f˙∗ f − g f ∗ f˙ − g(f ∗ ) f˙ dζ

2

∂D



  (gf ˙ − f˙g )f ∗ + g(f˙∗ f − (f ∗ ) f˙) dζ =

= ∂D



g · {f, f ∗ }t ∂D

dζ , ζ  

which is the desired result.

4 An Example 4.1 General Case For constants a, b, c ∈ C with 0 < |a| < 1 < |b|, c = 0, consider the rational function f (ζ ) = c ·

ζ (ζ − a) . ζ −b

(4.1)

Here the derivative f (ζ ) = c ·

ζ 2 − 2bζ + ab (ζ − ω1 )(ζ − ω2 ) =c· (ζ − b)2 (ζ − b)2

vanishes for @ ω1,2 = b(1 ±

1−

a ), b

(4.2)

The String Equation for Some Rational Functions

223

where ω1 ω2 = ab, 12 (ω1 + ω2 ) = b. The constant c is to be adapted according to the normalization f (0) =

ac > 0. b

This fixes the argument of c, so the parameters a, b, c represent 5 real degrees of freedom for f . We will be interested in choices of a, b, c for which one of the roots ω1,2 is in the unit disk, say |ω1 | < 1. Then |ω2 | > 1. The function f is in that case not locally univalent, but can be considered as a conformal map onto a Riemann surface over C having a branch point over @  2 c ω12 a f (ω1 ) = cb 1 − 1 − , = b b where we, as a matter of notation, let ω1 correspond to the minus sign in (4.2) (this is natural in the case 0 < a < 1 < b). The holomorphically reflected function is f ∗ (ζ ) = c¯ ·

1 − aζ ¯ . ¯ ) ζ (1 − bζ

Let h(z) be any analytic (test) function defined in a neighborhood of the closure of f (D), for example h(z) = zk , k ≥ 0. Then, denoting by νf the index of f (∂D), see (3.1), we have    1 1 1 hνf dxdy = h(f (ζ ))|f (ζ )|2 d ζ¯ dζ = h(f (ζ ))f ∗ (ζ )f (ζ )dζ π C 2π i D 2π i ∂D = Res h(f (ζ ))f ∗ (ζ )f (ζ )dζ + Res h(f (ζ ))f ∗ (ζ )f (ζ )dζ ζ =1/b¯

ζ =0

 = |c|2

 ¯ ¯ 2) (a¯ − b)(1 − 2|b|2 + a b|b| a ¯ h(f (0)) + h(f (1/b)) . ¯ − |b|2 )2 b b(1

In summary, 1 π

 C

¯ hνf dxdy = Ah(f (0)) + Bh(f (1/b)),

(4.3)

where a A = |c|2 , b

B = |c|2

¯ ¯ 2) (a¯ − b)(1 − 2|b|2 + a b|b| . ¯ − |b|2 )2 b(1

(4.4)

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B. Gustafsson

For the harmonic moments (with respect to the weight νf ) this gives M0 =A + B, ¯ k, Mk =Bf (1/b)

k = 1, 2, . . . .

Here only M0 , M1 , M2 are needed since the Mk lie in geometric progression from ¯ can be determined provided k = 1 on. From these three moments, A, B and f (1/b) M1 = 0, and after that a, b, c can be found, at least generically. Thus the moments M0 , M1 , M2 actually suffice to locally determine f , and we can write f (ζ ) = f (ζ ; M0, M1 , M2 ), provided f is known a priori to be of the form (4.1) with M1 = 0. However, as will be seen below, when we specialize to the case M1 = 0 things change. The quadrature Riemann surface picture enters when one starts from the second member in the computation above and considers g(ζ ) = h(f (ζ )) as an independent test function on the Riemann surface, thus allowing g(ζ1 ) = g(ζ2 ) even when f (ζ1 ) = f (ζ2 ). The quadrature identity becomes 1 2πi

 D

¯ g(ζ )|f (ζ )|2 d ζ¯ dζ = Ag(0) + Bg(1/b),

(4.5)

for g analytic and integrable (with respect to the weight |f |2 ) in the unit disk.

4.2 First Subcase There are two cases in this example which are of particular interest. These represent ¯ This does not change (4.5) instances of M1 = 0. The first case is when a = 1/b. very much, it is only that the two weights become equal: 1 2πi

 D

¯ g(ζ )|f (ζ )|2d ζ¯ dζ = Ag(0) + Ag(1/b),

where A = B = |c|2 /|b|2 and the normalization for c becomes c > 0. However, (4.3) changes more drastically because the two quadrature nodes now lie over the ¯ = 0 = f (0), so (4.3) effectively becomes same point in the z-plane. Indeed f (1/b) a one point identity: 1 π

 C

hνf dxdy = 2Ah(0)

The String Equation for Some Rational Functions

225

and, for the moments, M0 = 2A,

M1 = M2 = · · · = 0.

(4.6)

Clearly knowledge of these are not enough to determine f . This function originally had 5 real degrees of freedom. Two of them were used in condition ¯ but there still remain three, and M0 = 2A is only one real equation. a = 1/b, So something more would be needed, for example knowledge of the location of the branch point B1 = f (ω1 ). We have A   @ M0 1 , ω1 = b 1 − 1 − 2 , c = |b| |b| 2 by which @

M0 B1 = b|b| 2

A

 1−

1 1− 2 |b|

2 .

(4.7)

This equation can be solved for b in terms of B1 and M0 . Indeed, by some elementary calculations one finds that   B1 2|B0 |2 1/4 2|B0 |2 −1/4 b= ( , ) +( ) 2|B1 | M0 M0 ¯ b and c one then has f explicitly on the form and after substitution of a = 1/b, f (ζ ) = f (ζ ; M0 ; B1 ). By (4.6) the moment sequence is the same as that for the disk D(0, way to understand that is to observe that

√ 2A). One

¯ c 1 − bζ f (ζ ) = − · ζ · ζ −b b¯ √ A) covered is a function which maps D onto the disk D(0, √ √ twice. In other words, νf = 2χD(0,√A) . Note that the disk D(0, A) = D(0, M0 /2) depends only on M0 , not on B1 , so varying just B1 keeps f (∂D) fixed as a set.

4.3 Second Subcase ¯ This means that the quadrature node The second interesting case is when ω1 = 1/b. 1/b¯ is at the same time a branch point. What also happens is that the quadrature node looses its weight: one gets B = 0. The quadrature node is still there, but it is only “virtual”. (In principle it can be restored by allowing meromorphic test functions

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B. Gustafsson

with a pole at the point, as the weight |f |2 certainly allows, but we shall not take such steps.) Thus we have again a one node quadrature identity, but this time in a more true sense, namely that it is such a quadrature identity on the Riemann surface itself:  1 g(ζ )|f (ζ )|2d ζ¯ dζ = Ag(0), (4.8) 2πi D where A=2

|c|2 |c|2 − 4 = |c|2|ω1 |2 (2 − |ω1 |2 ). 2 |b| |b|

Of course we also have 

1 π

C

hνf dxdy = Ah(0).

and M0 = A,

M1 = M2 = · · · = 0.

(4.9)

Here again the moments do not suffice to identify f . Indeed, we have now a one parameter family of functions f satisfying (4.9) with the same value of A. Explicitly ¯ which we keep as the free parameter, this becomes, in terms of ω1 = 1/b, f (ζ ) = c ·

ζ (2|ω1 |2 − |ω1 |4 − ω¯ 1 ζ ) , 1 − ω¯ 1 ζ

with c related to ω1 and M0 by √ M0 ' . c= |ω1 | 2 − |ω1 |2 The branch point is √ ω1 |ω1 | M0 . B1 = f (ω1 ) = ' 2 − |ω1 |2 Since |ω1 | < 1 we have |B1 | < give ω1 in terms of B1 and M0 :

√ M0 . The above relationship can be inverted to

B A C 2 | |B1 |4 B1 C |B 2|B1 |2 1 D− ω1 = + + . |B1 | 2M0 M0 4M02

The String Equation for Some Rational Functions

227

Thus one can explicitly write f on the form f (ζ ) = f (ζ ; M0; B1 ) also in the present case. It is interesting to also compute f (ζ ). One gets f (ζ ) = c ·

(ζ − ω1 )(ζ − (ζ −

2 ω¯ 1 1 2 ω¯ 1 )

+ ω1 )

,

which is, up to a constant factor, the contractive zero divisor in Bergman space corresponding to the zero ω1 ∈ D, alternatively, the reproducing kernel for those L2 -integrable analytic functions in D which vanish at ω1 . See [10, 12] for these concepts in general. Part of the meaning in the present case is simply that (4.8) holds.

5 The String Equation for Polynomials If one stays strictly within the class of polynomials of a fixed degree, then it turns out that the string equation holds without any reference to branch points (which then are allowed to move freely). This is somewhat remarkable, and we shall here summarize the main statement, Theorem 5.1 below. Proof details are given in [4] (see also [3]). So we consider polynomials, of a fixed degree n + 1: f (ζ ) =

n 

aj ζ j +1 ,

a0 > 0.

(5.1)

j =0

It is obvious from Definition 2.1 that whenever the Poisson bracket (2.4) makes sense (i.e., whenever ∂f/∂M0 makes sense), it will vanish if f has zeros at two points which are reflections of each other with respect to the unit circle. Thus the string equation cannot hold in such cases. For polynomial maps Theorem 5.1 says that this is the only exception: the string equation makes sense and holds whenever f and f ∗ have no common zeros. Two polynomials having common zeros is something which can be tested by the classical resultant, which vanishes exactly in this case. Now f ∗ is not really a polynomial, only a rational function, but one may work with the polynomial ζ n f ∗ (ζ ) instead. Alternatively, one may use the meromorphic resultant, which applies to meromorphic functions on a compact Riemann surface, in particular rational functions. Very briefly expressed, the meromorphic resultant R(g, h) between two meromorphic functions g and h is defined as the multiplicative action of one of the functions on the divisor of the other. The second member of (5.2) below gives an example of the multiplicative action of h on the divisor of g. See [6] for further details.

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B. Gustafsson

What will be needed is only the meromorphic resultant  in the case of two rational functions of the form g(ζ ) = nj=0 bj ζ j and h(ζ ) = nk=0 ck ζ −k , and in this case it is closely related to the ordinary polynomial resultant Rpol (see [34]) for the two polynomials g(ζ ) and ζ n h(ζ ). Indeed, denoting by ω1 , . . . , ωn the zeros of g, the divisor of g is the formal sum 1 · (ω1 ) + · · · + 1 · (ωn ) − n · (∞), noting that g has a pole of order n at infinity. This gives the meromorphic resultant, and its relation to the polynomial resultant, as R(g, h) =

h(ω1 ) · · · · · h(ωn ) 1 = n n Rpol (g(ζ ), ζ n h(ζ )). n h(∞) b 0 c0

(5.2)

The theorem is an interplay between the Poisson bracket, the resultant and the Jacobi determinant between the moments and the coefficients of f in (5.1). It is mainly due to O. Kuznetsova and V. Tkachev [16] and V. Tkachev [32], only the statement about the string equation is (possibly) new. Theorem 5.1 With f a polynomial as in (5.1), the identity 2 ∂(M¯ n , . . . M¯ 1 , M0 , M1 , . . . , Mn ) = 2a0n +3n+1 R(f , f ∗ ) ∂(a¯ n , . . . , a¯ 1 , a0 , a1 , . . . , an )

holds generally. It follows that the derivative ∂f/∂M0 makes sense whenever R(f , f ∗ ) = 0, and then also the string equation {f, f ∗ } = 1 holds.

6 The String Equation on Quadrature Riemann Surfaces Here we shall extend the above result for polynomials to certain kinds of rational functions, and thereby illustrate the general role played by the branch points as being independent variables for analytic functions, besides the harmonic moments. This will actually not be a pure generalization of the polynomial case since we really do have to bother about the branch points, which eventually will be kept fixed in the main statement, Theorem 6.2. One way to handle the problem, mentioned in the beginning of Sect. 3, that the harmonic moments represent too few test functions because functions in the z-plane cannot not distinguish points on different sheets on the Riemann surface above it, is to turn to the class of quadrature Riemann surfaces. Such surfaces have been introduced and discussed in a special case in [28] (see also [7]), and we shall only need them in that special case. In principle, a quadrature Riemann surface is a Riemann surface provided with a Riemannian metric such that a finite quadrature identity holds for the corresponding area integral of integrable analytic functions on the surface. The special case which we shall consider is that the Riemann surface is a bounded simply connected branched covering surface of the complex plane. The Riemannian

The String Equation for Some Rational Functions

229

metric is then obtained by pull back from the complex plane via the covering map. In addition, we shall only consider one point quadrature identities, but then of arbitrary high order. All this amounts to a generalization of polynomial images of the unit disk. Being simply connected means that the quadrature Riemann surface is the conformal image of an analytic function f in D, and the one point quadrature identity then is of the form, when pulled back to D, 1 2πi

 D

g(ζ )|f (ζ )|2 d ζ¯ dζ =

n 

cj g (j ) (0).

(6.1)

j =0

This is to hold for analytic test functions g which are integrable with respect to the weight |f |2 . The cj are fixed complex constants (c0 necessarily real and positive), and we assume that cn = 0 to give the integer n a definite meaning. Such an identity (6.1) holds whenever f is a polynomial, and it is well known that in case f is univalent, being a polynomial is actually necessary for an identity (6.1) to hold. However, for non-univalent functions f it is different. As we have already seen in Sect. 4.3, rational functions which are not polynomials can also give an identity (6.1) under certain conditions. Indeed, if (6.1) holds then f has to be a rational function. This has been proved in [28], and under our assumptions it is easy to give a direct argument: applying (6.1) with g(ζ ) =

1 z−ζ

for |z| > 1 makes the right member become an explicit rational function (see the right member of (6.2) below), while the left member equals the Cauchy transform of the density |f |2 χD . This Cauchy transform is a continuous function in all of C with the z¯ -derivative equal to f (z)f (z) in D, thus being there of the form f (z)f (z) + h(z), with h(z) holomorphic in D. The continuity then gives the matching condition f (z)f (z) + h(z) =

n  k!ck , zk+1

z ∈ ∂D.

(6.2)

k=0

Here f (z) can be replaced by f ∗ (z), and it follows that this function extends to be meromorphic in D. Hence f ∗ (and f ) are meromorphic on the entire Riemann sphere, and so rational. We point out that quadrature Riemann surfaces as above are dense in the class of all bounded simply connected branched covering surfaces over C. This is obvious since each polynomial f produces such a surface. Therefore the restriction to quadrature Riemann surfaces is no severe restriction. The reason that (6.1) is a useful identity in our context is that it reduces the information of f (D) as a multi-sheeted surface to information concentrated at one single point on it, and near that point it

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does not matter that the test functions 1, z, z2 , . . . cannot distinguish different sheets from each other. Having an identity (6.1) we can easily compare the constants (c0 , c1 , . . . , cn ) with the moments (M0 , M1 , . . . , Mn ). It is just to choose g(ζ ) = f (ζ )k to obtain Mk , and this gives a linear relationship mediated by a non-singular triangular matrix. Thus we have a one-to one correspondence (M0 , M1 , . . . , Mn ) ↔ (c0 , c1 , . . . , cn ).

(6.3)

The relations between the cj and f are obtained by reading off, from (6.2) with f¯ replaced by f ∗ , the Laurent expansion of f ∗ f at the origin: f ∗ (ζ )f (ζ ) =

n  k!ck + holomorphic in D. ζ k+1

(6.4)

k=0

This is to be combined with (6.3). We see that the information about moments are now encoded in local information of f at the origin and infinity. If f has zeros ω1 , . . . , ωm in D then (6.4) means that f ∗ is allowed to have poles at these points, in addition to the necessary pole of order n + 1 at the origin, which is implicit in (6.4) since f (0) = 0. We may now start counting parameters. Taking into account the normalization at the origin, f has from start (1 + 2m + 2n) + 2m real parameters (numerator plus denominator when writing f as a quotient). These shall be matched with the 1 + 2n parameters in the Mk or ck . Next, each pole of f ∗ in D \ {0} has to be a zero of f , which give 2m equations for the parameters. Now there remain 2m free parameters, and we claim that these can be taken to be the locations of the branch points, namely Bj = f (ωj )

j = 1, . . . , m.

(6.5)

Thus we expect that f can be parametrized by the Mk and the Bj : f (ζ ) = f (ζ ; M0 , . . . , Mn ; B1 , . . . , Bm ).

(6.6)

In particular, ∂f/∂M0 then makes sense, with the understanding that B1 , . . . , Bm , as well as M1 , . . . , Mn , are kept fixed under the derivation. Clearly the parameters Mk and Bj depend smoothly on f . This dependence can be made explicit by obvious residue formulas: 1 Mk = 2πi 1 Bj = 2πi



f (ζ )k f ∗ (ζ )f (ζ )dζ = Res f (ζ )k f ∗ (ζ )f (ζ )dζ,

(6.7)

f (ζ )f

(ζ ) f (ζ )f

(ζ ) dζ = Res dζ. ζ =ωj f (ζ ) f (ζ )

(6.8)

∂D

9

|ζ −ωj |=ε

ζ =0

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(ζ ) Thus the branch points are exactly the residues of f (ζf)f dζ in D.

(ζ ) We summarize part of the above discussion in Proposition 6.1 below, and then formulate the main result of the paper, Theorem 6.2.

Proposition 6.1 Let f be analytic in a neighborhood of the closed unit disk, normalized by f (0) = 0, f (0) > 0 and satisfying f = 0 on ∂D. Let ω1 , . . . , ωm denote the zeros of f in D, these zeros assumed to be simple. Then a quadrature identity of the kind (6.1) holds, for some choice of coefficients c0 , c1 , . . . , cn with cn = 0, if and only if f is a rational function such that f has a pole of order n + 1 at infinity and possibly finite poles at the reflected points 1/ω¯ k of the zeros of f in D. This means that f is of the form f (ζ ) =

a0 ζ + a1 ζ 2 + · · · + am+n ζ m+n+1 . (1 − ω¯ 1 ζ ) . . . (1 − ω¯ m ζ )

(6.9)

Theorem 6.2 For functions f as in Proposition 6.1, the coefficients a0 , . . . , am+n can be viewed as free parameters (local coordinates for the class of functions f considered), subject only to a0 > 0, with the roots ω1 , . . . , ωm being determined by the conditions f (ωk ) = 0. The map (a0 , a1 , . . . , am+n ) → (M0 , M1 , . . . , Mn ; B1 , . . . , Bm ) obtained from (6.7), (6.8) (or (6.5)) then can be viewed as a change of local coordinates. In terms of the latter coordinates the partial derivative ∂f/∂M0 , and hence the Poisson bracket (2.4), makes sense. Finally, the string equation {f, f ∗ } = 1 holds. Proof The parameters M0 , M1 , . . . , Mn ; B1 , . . . , Bm represent as many data as there are independent coefficients in (6.9), namely the a0 , a1 , . . . , am+n , and we take for granted that they are indeed independent coordinates. From that point on there is a straight-forward argument proving the theorem based on existence result for the Hele-Shaw flow moving boundary problem (or Laplacian growth). Indeed, it is known that there exists, given f (·, 0), an evolution t → f (·, t) such that M1 , . . . , Mn ; B1 , . . . , Bm remain fixed under the evolution, and {f, f ∗ }t = 1,

(6.10)

holds. The branch points being kept fixed means that the evolution is to take place on a fixed Riemann surface (which however has to be extended during the evolution), and then at least a weak solution forward in time (t ≥ 0) can be guaranteed by potential theoretic methods (partial balayage or obstacle problems). Under the present assumptions, involving only rational functions, local solutions in both time directions can also be obtained by direct approaches which reduce (6.10) to finite dimensional dynamical systems. Both these methods are developed in detail in [5]. Below we give some further details related to the direct approach with rational functions.

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From (6.10) alone follows that the moments M1 , . . . , Mn are preserved. This can be seen from Corollary 3.2, which gives  1 d d f (ζ, t)k |f (ζ, t)|2 d ζ¯ dζ = (6.11) Mk = dt 2πi dt D =

1 2π



2π

f (ζ, t)k {f, f ∗ }t dθ =

0

1 2π



2π

f k dθ = 0

0

for k ≥ 1. Equation (6.10) is equivalent to 2Re [f˙(ζ, t) ζf (ζ, t)] = 1 holding for ζ ∈ ∂D, essentially the Polubarinova-Galin equation (3.3), and on dividing by |f (ζ, t)|2 this becomes Re

1 f˙(ζ, t) = , ζf (ζ, t) 2|f (ζ, t)|2

ζ ∈ ∂D.

(6.12)

In the case that f has no zeros in D, (6.12) gives that f˙(ζ, t) = ζf (ζ, t)P (ζ, t)

(ζ ∈ D),

(6.13)

where P (ζ, t) is the Poisson-Schwarz integral 1 P (ζ, t) = 2π



2π 0

1 eiθ + ζ dθ. 2|f (eiθ , t)|2 eiθ − ζ

(6.14)

In the case that f has zeros in D one may add to P (ζ, t) rational functions which are purely imaginary on ∂D and whose poles in D are killed by the zeros of the factor ζf (ζ, t) in front. The result is f˙(ζ, t) = ζf (ζ, t) (P (ζ, t) + R(ζ, t)) , where R(ζ, t) is any function of the form   m m  b¯j (t)ζ bj (t)  bj (t) R(ζ, t) = i Im + − . ωj (t) ζ − ωj (t) 1 − ω¯ j (t)ζ j =1

(6.15)

(6.16)

j =1

The first term here is just to ensure normalization, namely Im R(0, t) = 0. Thus (6.15), together with (6.14) and (6.16), is equivalent to (6.10). The coefficients bj ∈ C in (6.16) are arbitrary, and they are actually proportional to the speed of the branch points under variation of t: using (6.15) and (6.16) we see that d Bj = f (ωj , t)ω˙ j + f˙(ωj , t) = f˙(ωj , t) = ωj bj f

(ωj , t). dt Here f

(ωj ) = 0 since we assumed that the zeros are simple.

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Thus the term R(ζ, t) represents motions of the branch points. However, we are interested in the case that the branch points do not move, since the interpretation of ∂/∂M0 in the string equation amounts to all other variables M1 , . . . , Mn ; B1 , . . . , Bm being kept fixed. Thus we have only Eq. (6.13) to deal with, and as mentioned at least local existence of solutions of this can be guaranteed by potential theoretic or by complex analytic methods using (6.9) as an “Ansatz”, see [5]. Remark 6.3 The last part of the above proof identifies f˙, given by (6.13), with the partial derivative ∂f/∂M0 . The right member of (6.13) can on the other hand be viewed as defining a directional derivative ∇(0)f , independent of any coordinates, so that (6.13) reads f˙ = ∇(0)f . This “Hele-Shaw derivative” ∇(a), for points a ∈  in general, is systematically discussed in previously mentioned papers, such as [21]. Acknowledgement The author is grateful to an anonymous referee for constructive comments on the paper.

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Newtonian and Single Layer Potentials for the Stokes System with L∞ Coefficients and the Exterior Dirichlet Problem Mirela Kohr, Sergey E. Mikhailov, and Wolfgang L. Wendland

Dedicated to Professor H. Begehr on the occasion of his 80th birthday

Abstract A mixed variational formulation of some problems in L2 -based Sobolev spaces is used to define the Newtonian and layer potentials for the Stokes system with L∞ coefficients on Lipschitz domains in R3 . Then the solution of the exterior Dirichlet problem for the Stokes system with L∞ coefficients is presented in terms of these potentials and the inverse of the corresponding single layer operator. Keywords Stokes system with L∞ coefficients · Newtonian and layer potentials · Variational approach · Inf-sup condition · Sobolev spaces Mathematics Subject Classification (2010) Primary 35J25, 35Q35, 42B20, 46E35; Secondary 76D, 76M

M. Kohr Faculty of Mathematics and Computer Science, Babe¸s-Bolyai University, Cluj-Napoca, Romania e-mail: [email protected] S. E. Mikhailov Department of Mathematics, Brunel University London, Uxbridge, UK e-mail: [email protected] W. L. Wendland () Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart, Stuttgart, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_12

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1 Introduction Let u be an unknown vector field, π be an unknown scalar field, and f be a given vector field defined on an exterior Lipschitz domain − ⊂ R3 . Let also E(u) be the symmetric part of the gradient of u, ∇u. Then the equations Lμ (u, π) := div (2μE(u)) − ∇π = f, div u = 0 in −

(1.1)

determine the Stokes system with a known viscosity coefficient μ ∈ L∞ (− ). This linear PDE system describes the flows of viscous incompressible fluids, when the inertia of such a fluid can be neglected. The coefficient μ is related to the physical properties of the fluid (for further details we refer the reader to the books [45] and [23] and the references therein). The methods of layer potential theory have a main role in the analysis of boundary value problems for elliptic partial differential equations (see, e.g., [13, 17, 30, 32, 39, 42, 48]). Fabes, Kenig and Verchota [21] obtained mapping properties of layer potentials for the constant coefficient Stokes system in Lp spaces. Mitrea and Wright [42] have used various methods of layer potentials in the analysis of the main boundary problems for the Stokes system with constant coefficients in arbitrary Lipschitz domains in Rn . The authors in [34] have obtained mapping properties of the constant coefficient Stokes layer potential operators in standard and weighted Sobolev spaces by exploiting results of singular integral operators. Gatica and Wendland [24] used the coupling of mixed finite element and boundary integral methods for solving a class of linear and nonlinear elliptic boundary value problems. The authors in [33] used the Stokes and Brinkman integral layer potentials and a fixed point theorem to show an existence result for a nonlinear Neumann-transmission problem for the Stokes and Brinkman systems with data in Lp , Sobolev, and Besov spaces (see also [35, 36]). All above results are devoted to elliptic boundary value problems with constant coefficients. Potential theory plays also a main role in the study of elliptic boundary value problems with variable coefficients. Dindo˘s and Mitrea [19] have obtained wellposedness results in Sobolev spaces for Poisson problems for the Stokes and Navier-Stokes systems with Dirichlet condition on C 1 and Lipschitz domains in compact Riemannian manifolds by using mapping properties of Stokes layer potentials in Sobolev and Besov spaces. Chkadua, Mikhailov and Natroshvili [14] obtained direct segregated systems of boundary-domain integral equations for a mixed boundary value problem of Dirichlet-Neumann type for a scalar secondorder divergent elliptic partial differential equation with a variable coefficient in an exterior domain of R3 (see also [13]). Hofmann, Mitrea and Morris [29] considered layer potentials in Lp spaces for elliptic operators of the form L = −div(A∇u) acting in the upper half-space Rn+ , n ≥ 3, or in more general Lipschitz graph domains, with an L∞ coefficient matrix A, which is t-independent, and solutions of the equation Lu = 0 satisfy interior De Giorgi-Nash-Moser estimates. They obtained a Calderón-Zygmund type theory associated to the layer potentials, and

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well-posedness results of boundary problems for the operator L in Lp and endpoint spaces. Our variational approach is inspired by that developed by Sayas and Selgas in [46] for the constant coefficient Stokes layer potentials on Lipschitz boundaries, and is based on the technique of Nédélec [44]. Girault and Sequeira [26] obtained a well-posed result in weighted Sobolev spaces for the Dirichlet problem for the standard Stokes system in exterior Lipschitz domains of Rn , n = 2, 3. B˘acu¸ta˘ , Hassell and Hsiao [7] developed a variational approach for the standard Brinkman single layer potential and used it in the analysis of the time dependent exterior Stokes problem with Dirichlet boundary condition in Rn , n = 2, 3. Barton [8] constructed layer potentials for strongly elliptic differential operators in general settings by using the Lax-Milgram theorem, and generalized various properties of layer potentials for harmonic and second order elliptic equations. Brewster et al. in [9] have used a variational approach and a deep analysis to obtain wellposedness results for boundary problems of Dirichlet, Neumann and mixed type for higher order divergence-form elliptic equations with L∞ coefficients in locally (*, δ)-domains and in Besov and Bessel potential spaces. Choi and Lee [15] have studied the Dirichlet problem for the Stokes system with nonsmooth coefficients, and proved the unique solvability of the problem in Sobolev spaces on a bounded Lipschitz domain  ⊂ Rn (n ≥ 3) with a small Lipschitz constant when the coefficients have vanishing mean oscillations with respect to all variables. Choi and Yang [16] obtained the existence and pointwise bound of the fundamental solution for the Stokes system with measurable coefficients in Rn , n ≥ 3, whenever the weak solutions of the system are locally Hölder continuous. Alliot and Amrouche [3] have used a variational approach to obtain weak solutions for the exterior Stokes problem in weighted Sobolev spaces. Also, Amrouche and Nguyen [5] proved existence and uniqueness results in weighted Sobolev spaces for the Poisson problem with Dirichlet boundary condition for the Navier-Stokes system in exterior Lipschitz domains in R3 . The purpose of this work is to show the well-posedness result of the Poisson problem of Dirichlet type for the Stokes system with L∞ coefficients in L2 -based Sobolev spaces on an exterior Lipschitz domain in R3 . We use a variational approach that reduces this boundary value problem to a mixed variational formulation. A similar variational approach is used to define the Newtonian and layer potentials for the Stokes system with L∞ coefficients on Lipschitz surfaces in R3 , by using the weak solutions of some transmission problems in L2 -based Sobolev spaces. Finally, the mapping properties of these layer potentials are used to construct explicitly the solution of the exterior Dirichlet problem for the Stokes system with L∞ coefficients. The analysis developed in this paper confines to the case n = 3, due to its practical interest, but the extension to the case n ≥ 3 can be done with similar arguments.

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2 Functional Setting and Useful Results Let + :=  ⊂ R3 be a bounded Lipschitz domain, i.e., an open connected set whose boundary ∂ is locally the graph of a Lipschitz function. Assume that ∂ ˚± is connected. Let − := R3 \ + denote the exterior Lipschitz domain. Let E denote the operators of extension by zero outside ± .

2.1 Standard L2 -Based Sobolev Spaces and Related Results Let F and F −1 denote the Fourier transform and its inverse defined on the space of tempered distributions S ∗ (R3 ) (i.e., the topological dual of the space S(R3 ) of all rapidly decreasing infinitely differentiable functions on R3 ). The Lebesgue space of (equivalence classes of) measurable, square integrable functions on R3 is denoted by L2 (R3 ), and by L∞ (R3 ) we denote the space of (equivalence classes of) essentially bounded measurable functions on R3 . Let H 1 (R3 ) and H 1 (R3 )3 denote the L2 based Sobolev (Bessel potential) spaces 5 5 : ? 1 H 1 (R3 ) := f ∈ S ∗ (R3 ) : f H 1 (R3 ) = 5F −1 [(1+|ξ |2) 2 F f ]5L2 (R3 ) < ∞ , (2.1) H 1 (R3 )3 := {f = (f1 , f2 , f3 ) : fj ∈ H 1 (R3 ), j = 1, 2, 3}.

(2.2)

The topological dual of a linear space X is denoted by X∗ . Now let  be + , − or R3 . We denote by D( ) := C0∞ ( ) the space of infinitely differentiable functions with compact support in  , equipped with the inductive limit topology. Let D∗ ( ) denote the corresponding space of distributions on  , i.e., the dual space of D( ). Let us consider the space H 1 ( ) := {f ∈ D∗ ( ) : ∃ F ∈ H 1 (R3 ) such that F | = f } ,

(2.3)

1 ( ) is the closure of D( ) where ·| is the restriction operator to  . The space H 1 3 in H (R ). This space can be also characterized as 7 6 1 ( ) := f ∈ H 1 (R3 ) : supp f ⊆  . (2.4) H 1 ( )3 are the spaces of vector-valued Similar to definition (2.2), H 1 ( )3 and H 1 ( ), functions whose components belong to the scalar spaces H 1 ( ) and H 1

( ) can be identified with the respectively (see, e.g., [38]). The Sobolev space H ˚1 ( ) of D( ) in the norm of H 1 ( ) (see, e.g., [43, (3.11), (3.13)], [38, closure H Theorem 3.33]). The space D( ) is dense in H 1 ( ), and the following spaces can be isomorphically identified (cf., e.g., [38, Theorem 3.14]) −1 ( ), H −1 ( ) = (H 1( ))∗ . (H 1 ( ))∗ = H

(2.5)

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For s ∈ [0, 1], the Sobolev space H s (∂) on the boundary ∂ can be defined by using the space H s (R2 ), a partition of unity and the pull-backs of the local parametrization of ∂, and H −s (∂) = (H s (∂))∗ . All the above spaces are Hilbert spaces. For further properties of Sobolev spaces on bounded Lipschitz domains and Lipschitz boundaries, we refer to [1, 31, 38, 42, 47]. A useful result for the next arguments is given below (see, e.g., [17], [31, Proposition 3.3]). Lemma 2.1 Assume that  := + ⊂ R3 is a bounded Lipschitz domain with connected boundary ∂ and denote by − := R3 \  the corresponding exterior domain. Then there exist linear and bounded trace operators γ± : H 1 (± ) → 1 H 2 (∂) such that γ± f = f |∂ for any f ∈ C ∞ (± ). These operators are surjective and have (non-unique) bounded linear right inverse operators γ±−1 : 1

H 2 (∂) → H 1 (± ). The jump of a function u ∈ H 1 (R3 \ ∂) across ∂ is denoted by [γ (u)] := 1 (R3 ), [γ (u)] = 0. The trace operator γ : H 1 (R3 ) → γ+ (u) − γ− (u). For u ∈ Hloc 1

H 2 (∂) can be also considered and is linear and bounded.1 If X is either an open subset or a surface in R3 , then we use the notation ·, ·X for the duality pairing of two dual Sobolev spaces defined on X.

2.2 Some Weighted Sobolev Spaces and Related Results For a point x = (x1 , x2 , x3 ) ∈ R3 , its distance to the origin is denoted by |x| = 1 (x12 + x22 + x32 ) 2 . Let ρ denote the weight function 1

ρ(x) = (1 + |x|2 ) 2 .

(2.6)

For λ ∈ R, we consider the weighted space L2 (ρ λ ; R3 ) given by f ∈ L2 (ρ λ ; R3 ) ⇐⇒ ρ λ f ∈ L2 (R3 ),

(2.7)

which is a Hilbert space when it is endowed with the inner product and the associated norm,  (f, g)L2 (ρ λ ;R3 ) := fgρ 2λ dx, f 2L2 (ρ λ ;R3 ) := (f, f )L2 (ρ λ ;R3 ) . (2.8) R3

trace operators defined on Sobolev spaces of vector fields on ± or R3 are also denoted by γ± and γ , respectively.

1 The

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We also consider the weighted Sobolev space 7 6 H1 (R3 ) : = f ∈ D (R3 ) : ρ −1 f ∈ L2 (R3 ), ∇f ∈ L2 (R3 )3 7 6 = f ∈ L2 (ρ −1 ; R3 ) : ∇f ∈ L2 (R3 )3 ,

(2.9)

which is a Hilbert space with respect to the inner product (f, g)H1 (R3 ) := (f, g)L2 (ρ −1 ;R3 ) + (∇f, ∇g)L2 (R3 )3

(2.10)

and the associated norm 5 52 5 5 f 2H1 (R3 ) := 5ρ −1 f 5 2

L (R3 )

+ ∇f 2L2 (R3 )3

(2.11)

(cf. [28]; see also [5]). The spaces L2 (ρ λ ; − ) and H1 (− ) can be similarly defined, and D(− ) is dense in H1 (− ) (see, e.g., [28, Theorem I.1], [27, Ch.1, Theorem 2.1]). The seminorm |f |H1 (− ) := ∇f L2 (− )3

(2.12)

is equivalent to the norm of H1 (− ) defined as in (2.11), with − in place of R3 (see, e.g., [18, Chapter XI, Part B, §1, Theorem 1]). The weighted spaces L2 (ρ −1 ; + ) and H1 (+ ) coincide with the standard spaces L2 (+ ) and H 1 (+ ), respectively (with equivalent norms). Note that the result in Lemma 2.1 extends also to the weighted Sobolev space H1 (− ). Therefore, there exists a linear bounded exterior trace operator 1

γ− : H1 (− ) → H 2 (∂),

(2.13)

which is also surjective (see [46, p. 69]). Moreover, the embedding of the space H 1 (− ) into H1 (− ) and Lemma 2.1 show the existence of a (non-unique) linear 1 and bounded right inverse γ−−1 : H 2 (∂) → H1 (− ) of operator (2.13) (see [34, Lemma 2.2], [40, Theorem 2.3, Lemma 2.6]). ˚1 (− ) ⊂ H1 (− ) denote the closure of D(− ) in H1 (− ). This space Let H can be characterized as : ? ˚1 (− ) = v ∈ H1 (− ) : γ− v = 0 on ∂ H

(2.14)

1 (− ) ⊂ H1 (R3 ) denote the closure of (cf., e.g., [38, Theorem 3.33]). Also let H 1 3 D(− ) in H (R ). This space can be also characterized as 1 (− ) = {u ∈ H1 (R3 ) : supp u ⊆ − }, H

(2.15)

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˚1 (− ) via the extension by and can be isomorphically identified with the space H ˚1 (− ) (cf., e.g., [38, Theorem 3.29 (ii)]). ˚− , i.e., H ˚− H 1 (− ) = E zero operator E In addition, consider the spaces (see, e.g., [5, p. 44], [37, Theorem 2.4])  ∗  ∗  1  (− ) ∗ , H −1 (− ) := H1 (− ) . H−1 (R3 ) := H1 (R3 ) , H−1 (− ) := H

3 The Conormal Derivative Operators for the Stokes System with L∞ Coefficients In the sequel we assume that the viscosity coefficient μ of the Stokes system (1.1) belongs to L∞ (R3 ) and there exists a constant cμ > 0, such that cμ−1 ≤ μ ≤ cμ a.e. in R3 .

(3.1)

Let E(u) := 12 (∇u + (∇u)& ) be the strain rate tensor. If (u, π) ∈ C 1 (± )3 × we can define the classical interior and exterior conormal derivatives (i.e., the boundary traction fields) for the Stokes system (1.1) with continuously differentiable viscosity coefficient μ by the well-known formula C 0 (± ),

tc± μ (u, π) := γ± (−πI + 2μE(u)) ν,

(3.2)

where ν is the outward unit normal to + , defined a.e. on ∂, and the symbol ± refers to the limit and conormal derivative from ± . Then for any function ϕ ∈ D(R3 )3 we obtain the first Green identity ; < ± tc± μ (u, π), ϕ

∂

; < =2μE(u), E(ϕ)± − π, div ϕ± + Lμ (u, π), ϕ ± .

This formula suggests the following weak definition of the generalized conormal derivative for the Stokes system with L∞ coefficients in the setting of L2 -weighted Sobolev spaces (cf., e.g., [17, Lemma 3.2], [34, Lemma 2.9], [35, Lemma 2.2], [40, Definition 3.1, Theorem 3.2], [42, Theorem 10.4.1]). Definition 3.1 Let μ ∈ L∞ (R3 ) satisfy assumption (3.1). Let 6 −1 (± )3 : H1 (± , Lμ ) := (u± , π± , ˜f± ) ∈ H1 (± )3 × L2 (± ) × H 7 Lμ (u± , π± ) = ˜f± |± and div u± = 0 in ± .

(3.3)

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1 −2 Then define the conormal derivative operator t± (∂)3 , μ : H (± , Lμ ) → H 1

− 21 ˜ H1 (± , Lμ ) / (u± , π± , ˜f± ) −→ t± (∂)3 , μ (u± , π± ; f± ) ∈ H = > ˜ ± t± := 2μE(u± ), E(γ±−1 )± μ (u± , π± ; f± ), 

(3.4)

∂

− π± , div(γ±−1 )±

+ ˜f± , γ±−1 ± , ∀  ∈ H 2 (∂)3 , 1

(3.5)

where γ±−1 : H 2 (∂)3 → H1 (± )3 is a (non-unique) bounded right inverse of the 1

1

trace operator γ± : H1 (± )3 → H 2 (∂)3 . ± We use the simplified notation t± μ (u± , π± ) for tμ (u± , π± ; 0). The following assertion can be proved similar to [41, Theorem 5.3], [34, Lemma 2.9].

Lemma 3.2 Let μ ∈ L∞ (R3 ) satisfy assumption (3.1). Then for all w± ∈ H1 (± )3 and (u± , π± , ˜f± ) ∈ H1 (± , Lμ ) the following identity holds ; < ˜ ± t± μ (u± , π± ; f± ), γ± w±

∂

= 2μE(u± ), E(w± )± − π± , div w± ± + ˜f± , w± ± .

(3.6)

1

Let γ denote the trace operator from H1 (R3 )3 to H 2 (∂)3 (cf., e.g., [40, Theorem 2.3, Lemma 2.6], [7, (2.2)]). For (u± , π± , ˜f± ) ∈ H1 (± , Lμ ), let ˚+ u+ + E ˚− u− , π := E ˚+ π+ + E ˚− π− , f := ˜f+ + ˜f− u := E

(3.7)

− ˜ ˜ [tμ (u, π; f)] := t+ μ (u+ , π+ ; f+ )−tμ (u− , π− ; f− ).

(3.8)

Moreover, if f = 0, we define − [tμ (u, π)] := [tμ (u, π; 0)] = t+ μ (u+ , π+ )−tμ (u− , π− ).

(3.9)

Then Lemma 3.2 leads to the following result. Lemma 3.3 Let μ ∈ L∞ (R3 ) satisfy assumption (3.1). Also let (u± , π± , f˜± ) ∈ H1 (± , Lμ ) and let (u, π, f) be defined as in (3.7). Then for all w ∈ H1 (R3 )3 , the following formula holds ; < [tμ (u, π; f)], γ w

∂

=2μE(u+ ), E(w)+ + 2μE(u− ), E(w)− − π, div wR3 + f, wR3 .

We also need the following particular case of Lemma 3.3 when f = 0.

(3.10)

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Lemma 3.4 Let μ ∈ L∞ (R3 ) satisfy assumption (3.1). Also let (u± , π± , 0) ∈ H1 (± , Lμ ). Let u and π defined as in formula (3.7). Then for all w ∈ H1 (R3 )3 , ;

[tμ (u, π)], γ w

< ∂

=2μE(u+ ), E(w)+ + 2μE(u− ), E(w)− − π, div wR3 .

(3.11)

4 Newtonian and Single Layer Potentials for the Stokes System with L∞ Coefficients Recall that the function μ ∈ L∞ (R3 ) satisfies conditions (3.1). Next, we define the Newtonian and single layer potentials for the L∞ coefficient Stokes system (1.1).

4.1 Variational Solution of the Variable-Coefficient Stokes System in R3 First we show the following useful well-posedness result. Lemma 4.1 Let aμ (·, ·) : H1 (R3 )3 × H1 (R3 )3 → R and b(·, ·) : H1 (R3 )3 × L2 (R3 ) → R be the bilinear forms given by aμ (u, v) := 2μE(u), E(v)R3 , ∀ u, v ∈ H1 (R3 )3 , b(v, q) := −div v, qR3 , ∀ v ∈ H1 (R3 )3 , ∀ q ∈ L2 (R3 ).

(4.1) (4.2)

Also let  : H1 (R3 )3 → R be a linear and bounded map. Then the mixed variational formulation 

aμ (u, v) + b(v, p) = (v), ∀ v ∈ H1 (R3 )3 , b(u, q) = 0, ∀ q ∈ L2 (R3 )

(4.3)

is well-posed. Hence, (4.3) has a unique solution (u, p) ∈ H1 (R3 )3 × L2 (R3 ) and there exists a constant C = C(cμ ) > 0 such that uH1 (R3 )3 + pL2 (R3 ) ≤ CH−1 (R3 )3 .

(4.4)

Proof By using conditions (3.1) and definition (2.11) of the norm of the weighted Sobolev space H1 (R3 ) we obtain that |aμ (u, v)| ≤ 2cμ E(u)L2 (R3 )3×3 E(v)L2 (R3 )3×3 ≤ 2cμ uH1 (R3 )3 vH1 (R3 )3 , ∀ u, v ∈ H1 (R3 )3 .

(4.5)

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Moreover, by using the Korn type inequality for functions in H1 (R3 )3 , 1

grad vL2 (R3 )3×3 ≤ 2 2 E(v)L2 (R3 )3×3

(4.6)

(cf., e.g., [46, (2.2)]) and since the seminorm |g|H1 (R3 ) := ∇gL2 (R3 )3

(4.7)

is a norm in H1 (R3 )3 equivalent to the norm defined by (2.11) (see, e.g., [18, Chapter XI, Part B, §1, Theorem 1]), there exists a constant c1 > 0 such that aμ (u, u) ≥ 2cμ−1 E(u)2L2 (R3 )3×3 ≥ cμ−1 ∇u2L2 (R3 )3×3 ≥ cμ−1 c1 u2H1 (R3 )3 , ∀ u ∈ H1 (R3 )3 .

(4.8)

Inequalities (4.5) and (4.8) show that aμ (·, ·) : H1 (R3 )3 × H1 (R3 )3 → R is a bounded and coercive bilinear form. Moreover, since the divergence operator div : H1 (R3 )3 → L2 (R3 )

(4.9)

is bounded, then the bilinear form b(·, ·) : H1 (R3 )3 × L2 (R3 ) → R is bounded as well. In addition, the operator in (4.9) is surjective (cf. [2, Proposition 2.1], [46, Proposition 2.4]) and also 6 7 1 Hdiv (R3 )3 := w ∈ H1 (R3 )3 : div w = 0 7 6 = w ∈ H1 (R3 )3 : b(w, q) = 0, ∀ q ∈ L2 (R3 ) . In addition, the operator in (4.9) is surjective (cf. [2, Proposition 2.1], [46, Proposition 2.4]), and hence the operator 1 −div : H1 (R3 )3 /Hdiv (R3 )3 → L2 (R3 )

is an isomorphism. Then by Lemma 2(ii) the bounded bilinear form b(·, ·) : H1 (R3 )3 × L2 (R3 ) → R satisfies the inf-sup condition (7). Hence, there exists β0 ∈ (0, ∞) such that inf

q∈L2 (R3 )\{0}

b(w, q) ≥ β0 . w H1 (R3 )3 qL2 (R3 ) w∈H1 (R3 )3 \{0} sup

(4.10)

1 (R3 )3 , we By applying Theorem 4, with X = H1 (R3 )3 , M = L2 (R3 ), V = Hdiv conclude that the mixed variational formulation (4.3) has a unique solution (u, p) ∈ H1 (R3 )3 × L2 (R3 ) and there exists a constant C = C(cμ ) > 0 such that (u, p) satisfies inequality (4.4).  

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Next we use the result of Lemma 4.1 in order to show the well-posedness of the L∞ coefficient Stokes system in the space H1 (R3 )3 × L2 (R3 ) (see also [2, Theorem 3] for the constant-coefficient case). Theorem 4.2 Let μ ∈ L∞ (R3 ) satisfy conditions (3.1). Then the L∞ coefficient Stokes system 

div (2μE(u)) − ∇π = ,  ∈ H−1 (R3 )3 , div u = 0, in R3 ,

(4.11)

has a unique solution (u, p) ∈ H1 (R3 )3 × L2 (R3 ), and there exists a constant C0 = C0 (cμ ) > 0 such that uH1 (R3 )3 + pL2 (R3 ) ≤ C0 H−1 (R3 )3 .

(4.12)

Proof Note that the Stokes system (4.11) is equivalent to the variational problem (4.3) as follows from the density of D(R3 )3 in the space H1 (R3 )3 (cf., e.g., [28], [46, Proposition 2.1]). Then the well-posedness result of the Stokes system with L∞ coefficients (4.11) follows from Lemma 4.1.  

4.2 Newtonian Potential for the Stokes System with L∞ Coefficients The well-posedness of problem (4.11) allows us to define the Newtonian potential for the Stokes system with L∞ coefficients as follows. Definition 4.3 For any  ∈ H−1 (R3 )3 , we define the Newtonian velocity and pressure potentials for the Stokes system with L∞ coefficients as N μ;R3  := u, Qμ;R3  := π,

(4.13)

where (u, π) ∈ H1 (R3 )3 × L2 (R3 ) is the unique solution of problem (4.11) with the given datum . Moreover, the well-posedness of problem (4.11) yields the continuity of the above operators as stated in the following assertion (cf. also [34, Lemma A.3] for μ = 1). Lemma 4.4 The Newtonian velocity and pressure potential operators N μ;R3 : H−1 (R3 )3 → H1 (R3 )3 , Qμ;R3 : H−1 (R3 )3 → L2 (R3 ) are linear and continuous.

(4.14)

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4.3 Single Layer Potential for the Stokes System with L∞ Coefficients 1

For a given ϕ ∈ H − 2 (∂)3, we now consider the following transmission problem for the Stokes system with L∞ coefficients   ⎧ ⎨ div 2μE(uϕ ) − ∇πϕ = 0 in R3 \ ∂, in R3 \ ∂, div uϕ = 0 & ⎩% tμ (uϕ , πϕ ) = ϕ on ∂,

(4.15)

  and show that this problem has a unique solution uϕ , πϕ ∈ H1 (R3 )3 ×L2 (R3 ) (cf. also [46, Proposition 5.1] for μ = 1). Note that the membership of uϕ in H1 (R3 )3 implies the transmission condition %

& γ (uϕ ) = 0 on ∂ ,

(4.16)

& % and the first equation in (4.15) implies also that the jump tμ (uϕ , πϕ ) is well defined. Theorem 4.5 Let ϕ ∈ H − 2 (∂)3 be given. Then the transmission problem (4.15) has the following equivalent mixed variational formulation: Find (uϕ , πϕ ) ∈ H1 (R3 )3 × L2 (R3 ) such that 1



2μE(uϕ ), E(v)R3 −πϕ , div vR3 = ϕ, γ v∂ , ∀ v ∈ H1 (R3 )3 , div uϕ , qR3 = 0, ∀ q ∈ L2 (R3 ).

(4.17)

Moreover, problem (4.17) is well-posed. Hence (4.17) has a unique solution (uϕ , πϕ ) ∈ H1 (R3 )3 × L2 (R3 ), and there exists a constant C = C(cμ ) such that uϕ H1 (R3 )3 + πϕ L2 (R3 ) ≤ Cϕ

H

−1 2 (∂)3

.

(4.18)

Proof The equivalence between the transmission problem (4.15) and the variational problem (4.17) follows from the density of the space D(R3 )3 in H1 (R3 )3 and formula (3.11), while the well-posedness of the variational problem (4.17) is an immediate consequence of Lemma 4.1 with the linear and continuous form  : H1 (R3 )3 → R given by (v) := ϕ, γ v∂ = γ ∗ ϕ, vR3 , ∀ v ∈ H1 (R3 )3 , 1

and hence  = γ ∗ ϕ, where γ ∗ : H − 2 (∂)3 → H−1 (R3 )3 is the adjoint of the trace 1 operator γ : H1 (R3 )3 → H 2 (∂)3 .   Theorem 4.5 leads to the following definition (cf. [46, p. 75] for μ = 1).

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Definition 4.6 For any ϕ ∈ H − 2 (∂)3 , we define the single layer velocity and pressure potentials for the Stokes system with L∞ coefficients (1.1) as 1

Vμ;∂ ϕ := uϕ , Qsμ;∂ ϕ := πϕ ,

(4.19)

and the potential operators V μ;∂ : H − 2 (∂)3 → H 2 (∂)3 and K∗μ;∂ : 1

1

1

1

H − 2 (∂)3 → H − 2 (∂)3 as V μ;∂ ϕ := γ uϕ , K∗μ;∂ ϕ :=

 1+ tμ (uϕ , πϕ ) + t− μ (uϕ , πϕ ) , 2

(4.20)

where (uϕ , πϕ ) is the unique solution of problem (4.15) in H1 (R3 )3 × L2 (R3 ). The next result shows the continuity of single layer velocity and pressure potential operators for the variable coefficient Stokes system (cf. [46, Proposition 5.2], [34, Lemma A.4, (A.10), (A.12)] and [42, Theorem 10.5.3] in the case μ = 1). Lemma 4.7 The following operators are linear and continuous 1

1

Vμ;∂ : H − 2 (∂)3 → H1 (R3 )3 , Qsμ;∂ : H − 2 (∂)3 → L2 (R3 ), 1

1

1

(4.21)

1

V μ;∂ : H − 2 (∂)3 → H 2 (∂)3, K∗μ;∂ : H − 2 (∂)3 → H − 2 (∂)3 . (4.22) Proof The continuity of operators (4.21) and (4.22) follows from the wellposedness of the transmission problem (4.15) and Definition 4.6.   The next result yields the jump relations of the single layer potential and its conormal derivative across ∂ (see also [46, Proposition 5.3] for μ = 1). 1

Lemma 4.8 Let ϕ ∈ H − 2 (∂)3 . Then almost everywhere on ∂, %

& γ Vμ;∂ ϕ = 0,  "  #  1 s ∗ tμ Vμ;∂ ϕ, Qsμ;∂ ϕ = ϕ, t± μ Vμ;∂ ϕ, Qμ;∂ ϕ = ± ϕ+Kμ;∂ ϕ. 2

(4.23) (4.24)

Proof Formulas (4.23) and (4.24) follow from Definition 4.6 and the transmission condition in (4.16), as well as the transmission condition in the third line of (4.15).   Let Rν = {cν : c ∈ R}. Let Ker {T : X → Y } := {x ∈ X : T (x) = 0} denote the null space of the map T : X → Y . We next obtain the main properties of the single layer potential operator (cf., e.g., [42, Theorem 10.5.3], and [7, Proposition 3.3(c)] and [46, Proposition 5.4] for μ = 1 and α ∈ [0, ∞)).

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Lemma 4.9 The following properties hold Vμ;∂ ν = 0 in R3 , Qsμ;∂ ν = −χ+ 7 6 1 1 Ker V μ;∂ : H − 2 (∂)3 → H 2 (∂)3 = Rν, 1

1

V μ;∂ ϕ ∈ Hν2 (∂)3 , ∀ ϕ ∈ H − 2 (∂)3 ,

(4.25) (4.26) (4.27)

where χ+ = 1 in + , χ+ = 0 in − , and 1 : ? 1 Hν2 (∂)3 := φ ∈ H 2 (∂)3 : ν, φ∂ = 0 .

(4.28)

Proof First, we consider the transmission problem (4.15) with the datum ϕ = ν ∈ 1 H − 2 (∂)3 . Then the solution of this problem is given by   (uν , πν ) = 0, −χ+ ∈ H1 (R3 )3 × L2 (R3 ).

(4.29)

Indeed, the pair (uν , πν ) satisfies the equations and the transmission condition in (4.15), as well as the transmission condition (4.16), and, in view of formula (3.11) and the divergence theorem, [tμ (uν , πν )], γ v∂ = −πν , div vR3 = ν, γ v∂ , ∀ v ∈ D(R3 )3 .

(4.30)

Then by formula (2.3), Lemma 2.1, the dense embedding of the space D(R3 )3 in H 1 (R3 )3 , and the above equality, we obtain that [tμ (uν , πν )], ∂ = ν, ∂ 1 for any  ∈ H 2 (∂)3 . Hence, [tμ (uν , πν )] = ν, as asserted. Then Definition 4.6 : implies relations (4.25). Moreover, V μ;∂ ν = 0, i.e., Rν ⊆ Ker V μ;∂ : ? 1 1 H − 2 (∂)3 → H 2 (∂)36 . 7 1 1 Now let ϕ 0 ∈ Ker V μ;∂ : H − 2 (∂)3 → H 2 (∂)3 . Let (uϕ 0 , πϕ 0 ) =   Vμ;∂ ϕ 0 , Qsμ;∂ ϕ 0 ∈ H1 (R3 )3 ×L2 (R3 ) be the unique solution of problem (4.15) with datum ϕ 0 . Since γ uϕ 0 = 0 on ∂, formula (3.11) yields that   0 = [tμ (uϕ 0 , πϕ 0 )], γ uϕ 0 ∂ = aμ uϕ 0 , uϕ 0 ,

(4.31)

and hence uϕ 0 = 0, πϕ0 = cχ+ in R3 , where c ∈ R. In view of formula (3.11), [tμ (uϕ 0 , πϕ 0 )], γ w∂ = −πϕ0 , div wR3 = −cν, γ w∂ , ∀ w ∈ D(R3 )3 , and, thus, ϕ 0 = [tμ (uϕ , πϕ 0 )] = −cν. Hence, formula (4.26) follows. 1

Now let ϕ ∈ H − 2 (∂)3 . By using the first formula in (4.20), we obtain for ; < ; < ; < 1 any ϕ ∈ H − 2 (∂)3 that V μ;∂ ϕ, ν ∂ = γ uϕ , ν ∂ = div uϕ , 1  = 0, where uϕ = Vμ;∂ ϕ. Thus, we get relation (4.27).  

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Next we use the notation [[·]] for the equivalence classes of the space 1 1 H − 2 (∂, 1 T M)/Rν. Thus, any [[ϕ]] ∈ H − 2 (∂)3 /Rν can be written as 1 [[ϕ]] = ϕ+Rν, where ϕ ∈ H − 2 (∂)3 . Exploiting properties (4.26) and (4.27), we now show the following invertibility result (cf. [42, Theorem 10.5.3], [7, Proposition 3.3(d)], [46, Proposition 5.5] for μ = 1 and α ≥ 0 constant). Lemma 4.10 The following operator is an isomorphism 1

V μ;∂ : H − 2 (∂)3 /Rν → Hν2 (∂)3. 1

(4.32)

Proof We use arguments similar to those for Proposition 5.5 in [46]. First, Lemma 4.7 and the membership relation (4.27) imply that the linear operator 1 in (4.32) is continuous. We show that this operator is also H − 2 (∂)3/Rν-elliptic, i.e., that there exists a constant c = c(∂) > 0 such that ;

< V μ;∂ [[ϕ]] , [[ϕ]] ∂ ≥ c [[ϕ]]2

H

1

−1 2 (∂)3 /Rν

, ∀ [[ϕ]] ∈ H − 2 (∂)3 /Rν. (4.33)

1

1

Let [[ϕ]] ∈ H − 2 (∂)3/Rν. Thus, [[ϕ]] = ϕ +Rν, where ϕ ∈ H − 2 (∂)3 . In view of formula (3.11), Definition 4.6, relations (4.26), (4.27), and inequality (4.8), ;

< ; < V μ;∂ ([[ϕ]]), [[ϕ]] ∂ = V μ;∂ (ϕ), ϕ ∂ = γ uϕ , [tμ (uϕ , πϕ )]∂ = aμ (uϕ , uϕ ) ≥ cμ−1 uϕ 2H 1 (R3 )3 ,

(4.34)

where uϕ = Vμ;∂ ϕ and πϕ = Qsμ;∂ ϕ. Now we use the property that the trace operator 1

1 γ : Hdiv (R3 )3 → Hν2 (∂)3

(4.35)

1

1 (R3 )3 (cf., e.g., is surjective having a bounded right inverse γ −1 : Hν2 (∂)3 → Hdiv 1

[46, Proposition 4.4]). Hence, for any ∈ Hν2 (∂)3 , we have that w = γ −1 ∈ 1 (R3 )3 . Then there exists c ≡ c (∂) ∈ (0, ∞) such that Hdiv |[[ϕ]] , ∂ | = |ϕ, ∂ | = |[tμ (uϕ , πϕ )], γ w∂ | = |aμ (uϕ , w)|

(4.36)

≤ 2cμ uϕ H1 (R3 )3 γ −1 H1 (R3 )3 ≤ 2cμ c uϕ H1 (R3 )3  

1

H 2 (∂)3

,

where the first equality in (4.36) follows from the relation [[ϕ]] = ϕ + Rν and the 1

membership of in Hν2 (∂)3 , the second equality follows from Definition 4.6, and

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the third equality is a consequence of formula (3.11). Since the space Hν2 (∂)3 is 1 the dual of the space H − 2 (∂)3/Rν, formula (4.36) yields that  [[ϕ]] 

H

−1 2

(∂)3 /Rν

≤ 2cμ c uϕ H1 (R3 )3 .

(4.37)

Then by (4.34) and (4.37) we obtain inequality (4.33), and the Lax-Milgram lemma yields that operator (4.32) is an isomorphism.   Remark 4.11 The fundamental solution of the constant-coefficient Stokes system in R3 is well known and leads to the construction of Newtonian and boundary layer potentials via the integral approach (see, e.g., [17, 32, 42, 48]). In view of Theorems 4.2 and 4.5, the Newtonian and single layer potentials provided by the variational approach (in the case μ = 1) coincide with classical ones expressed in terms of the fundamental solution, since they satisfy the same boundary value problems (4.11) and (4.15), respectively (see also [46, Proposition 5.1] for μ = 1). The assumption μ = 1 is a particular case of a more general case of L∞ coefficients analyzed in this paper. We also note that an alternative approach, reducing various boundary value problems for variable-coefficient elliptic partial differential equations to boundary-domain integral equations, by employing the explicit parametrix-based integral potentials, was explored in, e.g., [12–14].

5 Exterior Dirichlet Problem for the Stokes System with L∞ Coefficients In this section we analyze the exterior Dirichlet problem for the Stokes system with L∞ coefficients ⎧ ⎨ div (2μE(u)) − ∇π = f in − , (5.1) div u = 0 in − , ⎩ on ∂, γ− u = φ 1

with given data (f, φ) ∈ H−1 (− )3 × H 2 (∂)3 .

5.1 Variational Approach First, we use a variational approach and show that problem (5.1) has a unique solution (u, π) ∈ H1 (− )3 × L2 (− ) (see also [26, Theorem 3.4] and [3, Theorem 3.16] for the constant-coefficient Stokes system).

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Theorem 5.1 Assume that μ ∈ L∞ (− ) satisfies conditions (3.1). Then for all 1 given data (f, φ) ∈ H−1 (− )3 × H 2 (∂)3 the exterior Dirichlet problem for ∞ the L coefficient Stokes system (5.1) is well posed. Hence problem (5.1) has a unique solution (u, π) ∈ H1 (− )3 × L2 (− ) and there exists a constant C ≡ C(∂; cμ ) > 0 such that uH1 (− )3 + πL2 (− )

 ≤ C fH−1 (− )3 + φ

 1

H 2 (∂)3

.

(5.2)

1 (− )3 implies Proof First, we note that the density of the space D(− )3 in H that the exterior Dirichlet problem (5.1) has the following equivalent variational formulation: Find (u, π) ∈ H1 (− )3 × L2 (− ) such that ⎧ 1 (− )3 , ⎨ 2μE(u), E(˜v)− − π, div v˜ − = −f, v˜ − , ∀ v˜ ∈ H 2 div u, q− = 0, ∀ q ∈ L (− ), ⎩ γ− (u) = φ on ∂.

(5.3)

Next, we consider u0 ∈ H1 (− )3 such that 

div u0 = 0 in − , γ− u0 = φ on ∂.

(5.4)

Particularly, we can choose u0 as the solution of the Dirichlet problem for a constant-coefficient Brinkman system 

(# − αI)u0 − ∇π0 = 0, div u0 = 0 in − , on ∂, γ− u0 = φ

(5.5)

where α > 0 is an arbitrary constant. The solution is given by the double layer potential  u0 = Wα;∂ 1

1 I + Kα;∂ 2

−1

φ,

(5.6)

1

where Kα;∂ : H 2 (∂)3 → H 2 (∂)3 is the corresponding Brinkman double-layer boundary potential operator. Note that  (Wα h)j (x) := Sijα  (x, y)ν (y)hi (y)dσy . (5.7) ∂

The explicit form of the kernel Sijα  (x, y) can be found in [48, (2.14)–(2.18)] and [32, Section 3.2.1]. 1 1 In addition, the operator 12 I + Kα;∂ : H 2 (∂)3 → H 2 (∂)3 is an isomorphism, and u0 belongs to the space H 1 (− )3 and satisfies (5.5), and hence (5.4).

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Moreover, the embedding H 1 (− )3 ⊂ H1 (− )3 shows that u0 belongs also to the space H1 (− )3 (see also [26, Lemma 3.2, Remark 3.3]). ˚1 (− )3 , the variational Then with the new variable ˚ u := u − u0 ∈ H problem (5.3) reduces to the following mixed variational formulation (c.f. Problem (Q) in p. 324 of [26] for the constant-coefficient Stokes system): Find (˚ u, π) ∈ ˚1 (− )3 × L2 (− ) such that H -

˚1 (− )3 , aμ;− (˚ u, v) + b− (v, π) = Fμ;u0 (v), ∀ v ∈ H 2 b− (˚ u, q) = 0, ∀ q ∈ L (− ),

(5.8)

˚1 (− )3 × H ˚1 (− )3 → R and b : H ˚1 (− )3 × L2 (− ) → R where aμ;− : H − are the bilinear forms given by ˚1 (− )3 , aμ;− (w, v) := 2μE(w), E(v)− , ∀ v, w ∈ H ˚1 (− )3 , q ∈ L2 (− ), b− (v, q) := −div v, q− , ∀ v ∈ H

(5.9) (5.10)

˚1 (− )3 → R is the linear form given by and Fμ;u0 : H   ˚1 (− )3 . ˚− v− + 2μE(u0 ), E(v)− , ∀ v ∈ H Fμ;u0 (v) := − f, E

(5.11)

˚1 (− )3 and H 1 (− )3 can be identified Here we took into account that the spaces H 1 3 1 3 ˚ ˚ through the isomorphism E− : H (− ) → H (− ) . Note that 7 6 ˚1 (− )3 : = v ∈ H ˚1 (− )3 : div v = 0 in − H div 6 7 ˚1 (− )3 : b (v, q) = 0, ∀ q ∈ L2 (− ) . = v∈H −

(5.12)

Now, formula (2.11), inequality (3.1) and the Hölder inequality yield that |aμ;− (v1 , v2 )| ≤ 2cμ E(v1 )L2 (− )3×3 E(v2 )L2 (− )3×3 ˚1 (− )3 . ≤ 2cμ v1 H1 (− )3 vH1 (− )3 , ∀ v1 , v2 ∈ H

(5.13)

Moreover, the formula 2E(v)2L2 (

−)

3×3

= grad v2L2 (

−)

3×3

+div v2L2 ( ) , ∀ v ∈ D(− )3 −

(5.14)

(cf., e.g., the proof of Corollary 2.2 in [46]), and the density of the space D(− )3 ˚1 (− )3 show that the same formula holds also for any function in H ˚1 (− )3 . in H Therefore, we obtain the following Korn type inequality ˚1 (− )3 . grad vL2 (− )3×3 ≤ 2 2 E(v)L2 (− )3×3 , ∀ v ∈ H 1

(5.15)

Newtonian and Single Layer Potentials for the Stokes System with L∞ . . .

255

Then by using inequality (5.15), the equivalence of seminorm (2.12) to the norm (2.11) in the space H1 (− )3 , and assumption (3.1) we deduce that there exists a constant C = C(− ) > 0 such that u2H1 (− )3 ≤ Cgrad u2L2 (− )3×3 ≤ 2CE(u)2L2 (− )3×3 ≤ 2Ccμ μE(u)2L2 (

−)

3×3

˚1 (− )3 , = 2Ccμ aμ;− (u, u), ∀ u ∈ H

and accordingly that aμ;− (u, u) ≥

1 ˚1 (− )3 . u2H1 ( )3 , ∀ u ∈ H − 2Ccμ

(5.16)

In view of inequalities (5.13) and (5.16) it follows that the bilinear form aμ;− (·, ·) : ˚1 (− )3 × H ˚1 (− )3 → R is bounded and coercive. Moreover, arguments similar H ˚1 (− )3 × to those for inequality (5.13) imply that the bilinear form b− (·, ·) : H 2 1 3 3 ˚ (R ) → R given by (5.10) and (5.11), L (− ) → R and the linear form Fμ;u0 : H are also bounded. Since the operator ˚1 (− )3 → L2 (− ) div : H

(5.17)

is surjective (cf., e.g., [26, Theorem 3.2]), then by Lemma 2, the bounded bilinear ˚1 (− )3 × L2 (− ) → R satisfies the inf-sup condition form b− (·, ·) : H inf

sup

q∈L2 (− )\{0} v∈H ˚1 (− )3 \{0}

b− (v, q) vH ˚1 (

−)

3

qL2 (− )

≥ βD

(5.18)

with some constant βD > 0 (cf. [26, Theorem 3.3]). Then Theorem 4 (with X = ˚1 (− )3 and M = L2 (− )) implies that the variational problem (5.8) has a unique H ˚1 (− )3 × L2 (− ). Moreover, the pair (u, π) = (˚ solution (˚ u, π) ∈ H u + u0 , π) ∈ 1 3 2 H (− ) × L (− ), where u0 ∈ H1 (− )3 satisfies relations (5.4), is the unique solution of the mixed variational formulation (5.3) and depends continuously on the 1 given data (f, φ) ∈ H−1 (− )3 ×H 2 (∂)3 . The equivalence between the variational problem (5.3) and the exterior Dirichlet problem (5.1) shows that problem (5.1) is also well-posed, as asserted.  

5.2 Potential Approach Theorem 5.1 asserts the well-posedness of the exterior Dirichlet problem for the Stokes system with L∞ coefficients. However, if the given data (f, φ) belong to the 1

space H−1 (− )3 × Hν2 (∂)3 , then the solution can be expressed in terms of the

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Newtonian and single layer potential and of the inverse of the single layer operator as follows (cf. [26, Theorem 3.4] for μ > 0 constant, [22, Theorem 10.1] and [37, Theorem 5.1] for the Laplace operator). 1

Theorem 5.2 If f ∈ H−1 (− )3 and φ ∈ Hν2 (∂)3 then the exterior Dirichlet problem (5.1) has a unique solution (u, π) ∈ H1 (− )3 × L2 (− ), given by     ˜ u = N μ;R3 (˜f)|− + Vμ;∂ V −1 ( f) , φ − γ N 3 − μ;R μ;∂     ˜ in − , π = Qμ;R3 (˜f)|− + Qsμ;∂ V −1 μ;∂ φ − γ− N μ;R3 (f)

(5.19) (5.20)

where ˜f is an extension of f to an element of H1 (R3 )3 . Proof The result follows from Definition 4.3 and Lemmas 4.7, 4.8, and 4.10.

 

Appendix: Mixed Variational Formulations and Their Well-Posedness Property Here we make a brief review of well-posedness results due to Babu˘ska [6] and Brezzi [10] for mixed variational formulations related to bounded bilinear forms in reflexive Banach spaces. We follow [20, Section 2.4], [11], and [25, §4]. Let X and M be reflexive Banach spaces, and let X∗ and M∗ be their dual spaces. Let a(·, ·) : X ×X → R, b(·, ·) : X ×M → R be bounded bilinear forms. Then we consider the following abstract mixed variational formulation. For f ∈ X∗ , g ∈ M∗ given, find a pair (u, p) ∈ X × M such that 

a(u, v) + b(v, p) = f (v), ∀ v ∈ X, b(u, q) = g(q), ∀ q ∈ M.

(1)

Let A : X → X∗ be the bounded linear operator defined by Av, w = a(v, w), ∀ v, w ∈ X,

(2)

where ·, · := X∗ ·, ·X is the duality pairing of the dual spaces X∗ and X. We also use the notation ·, · for the duality pairing M∗ ·, ·M . Let B : X → M∗ and B ∗ : M → X∗ be the bounded linear and transpose operators given by Bv, q = b(v, q), v, B ∗ q = Bv, q, ∀ v ∈ X, ∀ q ∈ M.

(3)

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In addition, we consider the spaces V := Ker B = {v ∈ X : b(v, q) = 0, ∀ q ∈ M} , : ? V ⊥ := T ∈ X∗ : T , v = 0, ∀ v ∈ V .

(4) (5)

Then the following well-posedness result holds (cf., e.g., [20, Theorem 2.34]). Theorem 1 Let X and M be reflexive Banach spaces, f ∈ X∗ and g ∈ M∗ , and a(·, ·) : X × X → R and b(·, ·) : X × M → R be bounded bilinear forms. Let V be the subspace of X defined by (4). Then the variational problem (1) is well-posed if and only if a(·, ·) satisfies the conditions ⎧ a(u, v) ⎨ ∃ λ > 0 such that inf sup ≥ λ, u∈V \{0} v∈V \{0} uX vX ⎩ {v ∈ V : a(u, v) = 0, ∀ u ∈ V } = {0},

(6)

and b(·, ·) satisfies the inf-sup (Ladyzhenskaya-Babu˘ska-Brezzi) condition, ∃ β > 0 such that

b(v, q) ≥ β. q∈M\{0} v∈X\{0} vX qM inf

sup

(7)

Moreover, there exists a constant C depending on β, λ and the norm of a(·, ·), such that the unique solution (u, p) ∈ X × M of (1) satisfies the inequality uX + pM ≤ C (f X∗ + gM∗ ) .

(8)

In addition, we have (see [20, Theorem A.56, Remark 2.7], [4, Theorem 2.7]). Lemma 2 Let X, M be reflexive Banach spaces. Let b(·, ·) : X × M → R be a bounded bilinear form. Let B : X → M∗ and B ∗ : M → X∗ be the operators defined by (3), and let V = Ker B. Then the following results are equivalent: (i) There exists a constant β > 0 such that b(·, ·) satisfies condition (7). (ii) B : X/V → M∗ is an isomorphism and BwM∗ ≥ βwX/V for any w ∈ X/V . (iii) B ∗ : M → V ⊥ is an isomorphism and B ∗ qX∗ ≥ βqM for any q ∈ M. Remark 3 Let X be a reflexive Banach space and V be a closed subspace of X. If a bounded bilinear form a(·, ·) : V × V → R is coercive on V , i.e., there exists a constant ca > 0 such that a(w, w) ≥ ca w2X , ∀ w ∈ V , then the conditions (6) are satisfied as well (see, e.g., [20, Lemma 2.8]).

(9)

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The next result known as the Babu˘ska-Brezzi theorem is the version of Theorem 1 for Hilbert spaces (see [6], [10, Theorems 0.1, 1.1, Corollary 1.2]). Theorem 4 Let X and M be two real Hilbert spaces. Let a(·, ·) : X × X → R and b(·, ·) : X × M → R be bounded bilinear forms. Let f ∈ X∗ and g ∈ M∗ . Let V be the subspace of X defined by (4). Assume that a(·, ·) : V × V → R is coercive and that b(·, ·) : X × M → R satisfies the inf-sup condition (7). Then the variational problem (1) is well-posed. Acknowledgements The research has been supported by the grant EP/M013545/1: “Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs” from the EPSRC, UK. Part of this work was done in April/May 2018, when M. Kohr visited the Department of Mathematics of the University of Toronto. She is grateful to the members of this department for their hospitality.

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Special Functions Method for Fractional Analysis and Fractional Modeling S. V. Rogosin and M. V. Dubatovskaya

Dedicated to Professor Heinrich G.W. Begehr on the occasion of his 80th birthday

Abstract This is a survey paper describing the method of special functions for Fractional Calculus. We outline the main properties of special functions which are important for fractional analysis and fractional modeling. Main attention is paid to the functions of the Mittag-Leffler family and close to it the Wright functions. Keywords Special functions · Fractional integrals and derivatives · Fractional equations · Fractional modeling Mathematics Subject Classification (2010) Primary 33E12, 26A33; Secondary 34A08, 34K37, 35R11, 60G22

1 Introduction Fractional Calculus, being born at the end of sixteenth century, becomes now one of the most developed and popular subject of Analysis and its Applications (see [23, 24]). Several important publications in the Fractional Analysis (such as [4, 7, 9, 21]) and in the Fractional Modeling (see, e.g. [1, 11, 18, 22, 27]) have been appeared during last few decades. This direction required new technical tools, in particular, special functions which can be suitable for the study of the problems appeared in analysis. Among the special functions of fractional calculus and fractional modeling we have to point out the functions of the Mittag-Leffler family and related to them functions of the Wright type.

S. V. Rogosin () · M. V. Dubatovskaya Department of Economics, Belarusian State University, Minsk, Belarus © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_13

261

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The most popular in applications are the following functions from the MittagLeffler family of functions (see [7, Chs. 3–5]): the classical Mittag-Leffler function introduced and studied by Magnus Gustaf (Gösta) Mittag-Leffler in the series of 5 consequent notes at the beginning of twentieth century Eα (z) =

∞  n=0

zn , (αn + 1)

Re α > 0,

(1.1)

the two-parametric Mittag-Leffler function, appeared first in the paper by Wiman in 1905 and rediscovered in 1953 by Agarval and Humbert Eα,β (z) =

∞  n=0

zn , (αn + β)

Re α > 0, β ∈ C,

(1.2)

and two Mittag-Leffler functions with three parameters, namely, the Prabhakar function γ Eα,β (z)

=

∞  n=0

(γ )n zn , n!(αn + β)

Re α > 0, β ∈ C, γ > 0,

(1.3)

and the Kilbas-Saigo function Eα,m,l (z) =

∞ 

cn zn , c0 = 1, cn =

n−1  j =0

n=0

(α[j m + l] + 1) , (α[j m + l + 1] + 1)

(1.4)

Re α > 0, m > 0, l ∈ C, α[j m + l] + 1 = 0, −1, −2, . . . . As for the Wright function, the most applicable are (see [7, Appendix F]) the classical Wright function φ(α, β; z) =

∞ 

zn , n!(αn + β)

Re α > 0, β ∈ C,

(1.5)

(−z)n , n!(μn + ν + 1)

μ > −1, ν ∈ C,

(1.6)

n=0

and the Bessel-Wright function Jνμ (z)

=

∞  n=0

Special Functions Method for Fractional Analysis and Fractional Modeling

263

and some special cases of the generalized Wright function (or the Fox-Wright function) 3 p Wq (z)

= p Wq

p 2 (al + αl n) n  4  ∞ z (a1 , α1 ), . . . , (ap , αp )  l=1 z = .  q 2 (b1 , β1 ), . . . , (bq , βq ) n! n=0 (bj + βj n)

(1.7)

j =1

The aim of this paper is to highlight the properties of the above presented functions which make them suitable for fractional analysis and fractional modeling. We separate two possible areas of applications of the above mentioned special functions, namely, analytical applications (Sect. 2) and applications to modeling (Sect. 3).

2 Applications of Special Functions in Fractional Analysis 2.1 The Mittag-Leffler and the Wright Functions as Entire Functions It is known (see [7, Chs. 3–5]) that the two-parametric Mittag-Leffler function Eα,β (z) is an entire function of the complex variable z for all Re α > 0, β ∈ C, and the classical Wright function φ(α, β; z) is an entire function of the complex variable z for all Re α > −1, β ∈ C. From this property one can expect the these functions behave similarly to certain combination of exponential functions what is really correct for some special values of parameters (see [7]). ∞  An entire function representing in the form of power series f (z) = ck zk is k=0

called the function of the finite order if the following limit is finite ρ = ρf = lim sup k→∞

k log k log |c1k |

.

The parameter ρf is called the order of the entire function f . The following characteristic σ = σf of the entire function f of the finite order ρ = ρf is defined by the following relation: 1

1

(σ eρ) ρ = lim sup k ρ

' k

|ck |.

k→∞

σf is called the type of the entire function f of the finite order. An entire function of the finite order ρ and the finite type σ satisfies, in particular, the following

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asymptotic inequality: for any ε > 0 ∃rε > 0 : |f (reiϕ )| ≤ e(σ +ε)r , ∀r > rε . ρ

Such inequality determines the maximal possible growth of an entire function. The functions Eα,β (z) and φ(α, β; z) are functions of the finite order ρE = 1/(Re α), ρφ = 1/(Re α + 1), respectively, and of the finite type σE = 1 and σφ = (Re α + 1)|α|−1/(Re α+1) . Being very close to the exponential functions these functions do not copy the behavior of the latter in a neighborhood of z = ∞. To see this difference one can use so called indicator function characterizing the asymptotics of an entire function f (z) = f (reiϕ ) along fixed rays arg z = ϕ as r → ∞: hf (ϕ) := lim sup r→∞

log |f (reiϕ )| . rρ

The exponential function exp z is an entire function of order ρexp = 1 and has the indicator function hexp (ϕ) = cos ϕ, −π ≤ ϕ ≤ π, though, e.g., the Mittag-Leffler function Eα,β (z) for any α = 1, 0 < α < 2, has the indicator function (see [6])  hEα,β (ϕ) =

cos ϕ/α, 0 ≤ |ϕ| < πα 2 , πα ≤ |ϕ| ≤ π. 0, 2

In particular, for α = 1, 0 < α < 2, and β ∈ R the Mittag-Leffler function Eα,β (z) is either bounded or has power type growth/decay on the negative semi-axes x < 0. The exponential function plays an important role in many classical models, described in terms of ordinary differential equation (see, e.g. [28]). Anyway for many real-life processes exponential growth or exponential decay is too fast. There exists a number of models in which power type behavior is more natural (see, e.g. [26, Ch. 2]). Among these models we have to mention first of all the models involving fractional integrals and derivatives (more details can be found below in Sect. 3.1).

2.2 Zeroes Distribution and Inverse Problems Tough the Mittag-Leffler function and the Wright function are close to exponential, in most cases these functions have infinite collections of zeroes (see [10, 17]). The distribution of zeroes of Mittag-Leffler function plays an important role in the study of inverse problems for differential equations in a Banach space [25]. The above inverse problem is formulated as follows. Let E be a Banach space and A be a closed linear operator in E with domain D(A). Let N ≥ 1 be certain positive integer number and T > 0 be a real number. Consider the differential equation d N u(t) = Au(t) + p, dt N

0 < t < T,

(2.1)

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with an unknown parameter p ∈ E. Problem is to find the function u(t) and the element p satisfying (2.1) as well as the Cauchy conditions u(0) = u0 , . . . , u(N−1) (0) = uN−1 , u0 , . . . , uN−1 ∈ E,

(2.2)

and the terminal condition u(T ) = uN , uN ∈ E.

(2.3)

Solution to the inverse problem (2.1)–(2.3) is a pair (u(t), p), where p ∈ E and u : [0, T ] → E is N times continuously differentiable on (0, T ) and (N − 1) times continuously differentiable on [0, T ]. Unique solvability of the problem depends solely on the distribution of eigenvalues of the operator A. These eigenvalues should be associated (see [25]) with the zeroes λk of the Mittag-Leffler function EN,N+1 (z). Asymptotic distribution of zeroes λk of the function EN,N+1 (z) is given by the following relation (see [17]) 

π λk = − π sin N

N    1 cot (π/N), N = 3, 4, 5, −πkθN ) , θN = k+ + O(e 2 sin(2π/N), N ≥ 6. 2 (2.4)

Moreover, it was shown (see [17]) that all these zeroes λk are real, negative and simple. The uniqueness criterion of the inverse problem reads (see [25]): let (u(t), p) be a solution to the problem (2.1)–(2.3). This solution is unique iff no number λk /T N is an eigenvalue of the operator A.

2.3 Differential Properties and Reduction to Certain Differential Equations Mittag-Leffler and Wright functions are closely related to Fractional Calculus. One of the most attractive property is that these functions are invariant with respect to fractional differentiation and integration. Such property is a consequence of fractional integration and differentiation of a power monomial:   α γ −1 I0+ (x) = t   α γ −1 D0+ (x) = t

(γ ) x γ +α−1 , Re α > 0, (γ + α)

(2.5)

(γ ) x γ −α−1 , Re α ≥ 0. (γ − α)

(2.6)

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As an illustration to the above mentioned invariance we present here two known formulas     x β−1 1 α β−1 I0+ t Eα,β (at α ) (x) = Eα,β (ax α ) − , Re α > 0, Re β > 0, a (β) (2.7)   β−α−1   x α β−1 + ax β−1 Eα,β (ax α ) , D0+ t Eα,β (at α ) (x) = (2.8) (β − α) Re α > 0, Re β > Re α + 1. These two formulas show, in particular, that the Mittag-Leffler function satisfies the fractional integral equation (formula (2.7)) and the fractional differential equation (formula (2.8)). Vice versa, the Kilbas-Saigo function (1.4) was found as a solution to a new class of the fractional differential equation. This follows from the following differential relation (see, e.g., [7, Sec. 5.2])   (α(l − m + 1) + 1) α(l−m) α α(l−m+1) x t Eα,m,l (at αm ) (x) = + D0+ (α(l − m + 1))

(2.9)

  +ax αl Eα,m,l (ax αm ) , α > 0, m > 0, α(im + l) = −1, −2, . . . (i = 0, 1, 2, . . .); l > m − 1 − 1/α. A large collection of the ordinary fractional differential equations is presented in [9]. To show the technique which leads to presentation of the solution via the Mittag-Leffler function we consider here a simplest example. It is the Cauchy type problem for the one-term differential equation with the Riemann-Liouville fractional derivative: 

 α Da+ y (x) − λy(x) = f (x) (a < x ≤ b, α > 0, λ ∈ R),

(2.10)

  α−k Da+ y (a+) = bk (bk ∈ R, k = 1, . . . , n = −[−α]),

(2.11)

where [·] means an integer part of a real number, λ is the real parameter, and the right hand-side of (2.10) is a given Hölder continuous function (f ∈ H γ [a, b], 0 < γ ≤ 1, γ < α). It is known (see, e.g. [9, p. 172]) that the Cauchy type problem for such a fractional differential equation is equivalent in the space H n−α [a, b] to the following Volterra integral equation y(x) =

n  bj (x − a)α−j j =1

λ + (α − j + 1) (α)

x a

y(t)dt 1 + (x − t)1−α (α)

x a

f (t)dt . (x − t)1−α

(2.12)

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Applying to this equation the method of successive approximation with initial term y0 (x) =

n  bj (x − a)α−j j =1

(α − j + 1)

we get the following representation of the solution y(x) =

n  j =1

 ∞  λk (x − a)αk+α−j + bj (αk + α − j + 1)

x

k=0

/

∞  λk (x − a)αk+α−1

(αk + α)

k=0

a

0 f (t)dt. (2.13)

Summation of the above series yields the formula of the solution to (2.10)–(2.11) in the space H n−α [a, b] in terms of the Mittag-Leffler function y(x) =

n 

x bj (x − a)

α−j

Eα,α−j +1 [λ(x − a) ] + α

j =1

a

Eα,α [λ(x − t)α ]f (t)dt . (x − t)1−α (2.14)

2.4 Laplace Transform and Stability of Fractional Order System The method of integral transforms is highly useful in the study of differential equations, including differential equations of fractional order (see [9]). For the last type of equations the Laplace transform of the Mittag-Leffler functions appeared to be important from different aspects. Let us present these formulas for one-, two- and three-parametric Mittag-Leffler functions (see [3, 7]) +∞ s α−1 , L Eα (λt ) (s) := e−t s Eα (λt α )dt = α s −λ %

α

&

(2.15)

0

" # s α−β , L t β−1 Eα,β (λt α ) (s) = α s −λ " # γ L t β−1 Eα,β (λt α ) (s) =

s αγ −β . (s α − λ)γ

(2.16)

(2.17)

The Laplace transform of the classical Wright function is represented in terms of the Mittag-Leffler function L [φ(α, β; t)] (s) =

1 Eα,β s

  1 , α > −1, β ∈ C, Re s > 0, s

(2.18)

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while the Mellin transform of the classical Wright function is represented in terms of Gamma-function M [φ(α, β; t)] (s) =

(s) , Re s > 0. (β − αs)

(2.19)

Formulas (2.15)–(2.17) are similar to the Laplace transform of exponential function. Hence these formulas are helpful for solving of different types of differential equations. Another area, where these formulas are applicable is the stability analysis of the fractional order systems (see, e.g. [14, 19]). The general fractional order system can be described by a fractional differential equation of the form an D αn y (t) + an−1 D αn−1 y (t) + . . . + a0 D α0 y (t) = = bm D βm y (t) + bm−1 D βm−1 y (t) + . . . + b0 D β0 y (t) , γ

(2.20) γ

where D γ = D0+,t denotes the Riemann-Liouville fractional derivative RL D0+,t or γ the Caputo fractional derivative C D0+,t . The study of the general fractional order system can be done by an analysis of the corresponding transfer function (see, e.g. [19]): G (s) =

bm s βm + bm−1 s βm−1 + . . . + b0 s β0 Q (s) . = an s αn + an−1 s αn−1 + . . . + a0 s α0 P (s)

(2.21)

where s is the Laplace variable. Here an , . . . , a0 , bm , . . . , b0 are given real constants, and αn , . . . , α0 , βm , . . . , β0 are given real numbers (usually positive). Without loss of generality these sets of parameters can be ordered as αn > . . . > α0 , βm > . . . > β0 . If both sets α-s and β-s constitute an arithmetical progression with the same difference, i.e. αk = kα, k = 0, . . . , n, βk = kα, k = 0, . . . , m then the system (2.20) is called the commensurate order system. Usually it is supposed that parameter α satisfies the inequality 0 < α < 1. In all other cases the system (2.20) is called the incommensurate order system. Anyway, if all parameters αj and βj are rational numbers, then this case can be considered as commensurate one with α = N1 and N being the least common multiple of denominators of fractions αn , . . . , α0 , βm , . . . , β0 . For the commensurate order system its transfer function can be thought as certain branch of the following multi-valued function m 

G (s) =

k=0 n  k=0

bk (s α )k = ak

(s α )k

α) Q(s . P (s α )

(2.22)

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Since the right hand-side of this relation is a rational function of s α , then one can represent G(s) in the form of generalized simple fractions. The most descriptive representation of such a type is that for n > m: G (s) =

p  ri  i=1 j =1

Aij (s α

+ λi )j

,

(2.23)

where −λi is a root of polynomial P (z) of multiplicity ri . In particular, if all roots are simple, then the representation (2.23) has the most simple form G (s) =

n  i=1

sα

Bi . + λi

(2.24)

In this case an analytic solution to the system (2.20) is given by the formula - n 

. Bi · (Lu)(s) = y (t) = L s α + λi i=1 - n .  Bi ∗ u(t) = = L−1 s α + λi i=1  n   α α = Bi t Eα,α (−λi t ) ∗ u(t), −1

(2.25)

i=1

where the symbol “∗” means the Laplace-type convolution and Eμ,ν is the twoparametric Mittag-Leffler function Eμ,ν (z) =

∞  k=0

zk . (μk + ν)

(2.26)

In the case of homogeneous fractional order system an D αn y(t) + an−1 D αn−1 y(t) + . . . + a0 D α0 y(t) = 0

(2.27)

analytical solution is given by the following formula (see, e.g. [14]) y(t) =

∞ 1  (−1)k an k! k=0

 k0 + . . . + kn−2 = k k0 ≥ 0, . . . , kn−2 ≥ 0

(k; k0, . . . , kn−2 )×

(2.28)

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×

n−2 1 i=0

ai an

ki

⎞ n−2  a n−1 Ek ⎝t, − ; an − an−1 , an + (an−1 − aj )kj + 1⎠ , an ⎛

j =0

where (k; k0, . . . , kn−2 ) are the multinomial coefficients and Ek (t, y; μ, ν) is defined by the formula [16] (k) Ek (t, y; μ, ν) = t μk+ν−1 Eμ,ν (yt μ ), (k = 0, 1, 2, . . .),

and (k) (z) Eμ,ν

=

∞  j =0

(j + k)!zj (k = 0, 1, 2, . . .) j !(μj + μk + ν)

is the k-th derivative of two-parametric Mittag-Leffler function Eμ,ν (z) =

∞  j =0

zj (k = 0, 1, 2, . . .). (μj + ν)

Several stability criteria for the fractional order systems were obtained recently (see, e.g. [14]). The method of special functions is one of the leading approaches here.

2.5 Laplace Transform and Fractional Analogue of Green’s Function Another application of the Laplace transform method is related to the representation of the solution to the inhomogeneous fractional ordinary differential equations by using fractional analogue of Green’s function. In some special cases the fractional Green’s function can be represented via special functions of the Mittag-Leffler and the Wright type (see [9]). Let us describe the corresponding scheme in the case of the homogeneous Cauchy type problem for the inhomogeneous multi-term fractional differential equation with constant coefficients and with the Riemann-Liouville derivative: m 

 αk  Ak D0+ y (x) + A0 y(x) = f (x), (x > 0; 0 < α1 < . . . < αm , m ∈ N),

k=1

  αk −jk D0+ y (0+) = 0, jk = 1, . . . , nk = [αk ] + 1, k = 1, . . . , m.

(2.29) (2.30)

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Applying the Laplace transform and using the Laplace convolution formula ⎛ ⎛ x ⎞⎞  ⎝L ⎝ k(x − t)f (t)dt ⎠⎠ (s) = (Lk) (s) (Lf ) (s) 0

we (formally) represent the solution to (2.29) in the following form x y(x) =

Gα1 ,...,αm (x − t)f (t)dt,

(2.31)

0

where Gα1 ,...,αm (·) is the fractional analogue of Green’s function  Gα1 ,...,αm (x) = L−1

1 Pα1 ,...,αm (t)

 (x), Pα1 ,...,αm (t) = A0 +

m 

Am s αk .

k=1

(2.32) It follows from the relation between the Laplace transform and the RiemannLiouville fractional derivative 

n       α−k α LD0+ (t) ϕ (s) = s α (Lϕ) (s) − s α−k D0+ k=1

For instance the solution to the equation  α  D0+ y (x) − λy(x) = f (x) is represented in the form x (x − t)α−1 Eα,α [λ(x − t)α ]f (t)dt

y(x) = 0

since in this case  Gα (x) = L−1

 1 (x) sα − λ

and   1 α−1 α = L[t E (λt )] (s). α,α sα − λ

(0+)

.

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3 Fractional Modeling with Special Functions 3.1 Power Type Behavior and Power Functions in Modeling As it was already mentioned in Sect. 2.1, the Mittag-Leffler function can be bounded or has power growth/decay for negative value of its argument. It leads to possibility to use it at the modeling involving power type functions. Several examples of the use of power type function in the study of physical phenomenon are presented in [26, Ch. 2] (they are expressed in terms of so called standard power function of real μ−1 1 variable μ (t) := Γ (μ) t+ , where a+ = a, if a > 0 and a+ = 0 if a ≤ 0). Let us describe few of them. In particular, power type functions are involved to describe the heredity type processes (or processes with memory). One of the most known example of the hereditary process of power type (or process with slow memory) is the moving of a rigid body in the viscous media. Let the ball of radius r is sleeping up to a moment t0 in a viscous fluid of the density ρ and viscosity η. At t = t0 it gets an impulse and starts the straight and uniform moving with the velocity V . Then the resistance force can be given by the following formula ⎡

A

F (t) = 6πηrV0 ⎣1 +

⎤ ρr 2 ⎦ , t > t0 . πη(t − t0 )

(3.1)

The body is moving with the constant velocity but it remembers when such movement starts. This memory is expressed in the dependence of the resistance on t0 . It was Nutting (see [20]) who first reported that shear stresses in certain deformed viscoelastic materials decay according to the power law σ 1 C#εt −α .

(3.2)

It contradicts to the classical theory of deformation, namely to the exponential decay of the relaxation. Anyway, it was experimentally justified later that certain viscoelastic materials have a slow memory K(t) decaying approximately as t −3/2 . The stress-strain state of the viscoelastic materials with power memory function K(t) =

kα t −α (1 − α)

can be described by the following equation σ (t) = kα



 α D0+ ε (t).

C

(3.3)

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Several constitutive equations of such a type describe the behaviour of the viscoelastic materials. One of the simplest is the so called fractional Maxwell model generalizing the classical Maxwell model. Fractional constitutive equation for this model has the form     α α σ (t) + τ α CD0+ σ (t) = Eτ β CD0+ ε (t), τ > 0. (3.4) The solution to this equation with respect to σ (t) with the given ε(t) reads t σ (t) = cG(t) + Eτ

G(t − x)

β



C α D0+ ε

 (x)dx,

(3.5)

0

where the kernel (fractional Green’s function) G is defined via the Mittag-Leffler function  α t α−1 t G(t) = α Eα,α − α . (3.6) τ τ

3.2 Relaxation Kernel in Integral Models with Memory Volterra in his study of materials and processes with memory creating hereditary theory (see [28]) mainly use the integral equation machinery. The kernel of the equations in the Volterra theory mainly have exponential decay what corresponds, for instance to the materials with a fast memory. In the middle of 1930s it was discovered that there are materials with memory function (relaxation kernel) having power law decay (see, e.g. [20]). In particular, such behavior is characteristic for many viscoelastic materials which possess properties as solids as liquids (see, e.g., [11]). Rabotnov (see extended description in his monograph [18]) presented a general theory of the hereditary solid mechanics using the integral equations approach. Rabotnov introduced an hereditary elastic rheological model with constitutive equation in form of the Volterra integral equation with the weakly singular kernel of a special type ⎡ σ (t) = E ⎣ε(t) − β

ta

⎤ Rα (−β, ta − τ )ε(τ )dτ ⎦ ,

(3.7)

0

where ta is the aging time, α ∈ (−1, 0], β = 0, and the kernel Rα is represented in the form of power series Rα (γ , x) = x

α

∞  k=0

γ k x k(α+1) . ((k + 1)(α + 1))

(3.8)

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This functions is related to the special case of the two-parametric Mittag-Leffler function Rα (γ , x) = x α Eα+1,α+1(γ x α+1 ). It should be noted that in fractional-differential form Rabotnov’s model is equivalent to the four-parametric Bagley-Torvik model (see [26, p. 295]).

3.3 Complete Monotonicity and Fractional Stable Distributions γ

The Mittag-Leffler functions Eα , Eα,β , Eα,β are completely monotonic for some value of parameters (exact statement is given below). Let us repeat basic definitions (see, e.g., [13]). A function f : (0, +∞) → R is said to be completely monotonic, if it possesses derivatives f (n) (x) for all n = 0, 1, 2, 3, . . . and if its derivatives alternate in sign, namely (−1)n f (n) (x) ≥ 0, for all x > 0.

(3.9)

It is known (see, e.g. [13]) that a necessary and sufficient condition for the function f (x) to be completely monotonic is its representability in terms of the LaplaceStiltjes transform ∞ f (x) =

e−xt dα(t)

(3.10)

0

with respect to a non-decreasing density α(t). The classical Mittag-Leffler function Eα (−x) is completely monotonic for 0 < α ≤ 1, the two-parametric Mittag-Leffler function Eα,β (−x) is completely monotonic for 0 < α ≤ 1, β ≥ α (see [13]), and the three-parametric Mittag-Leffler γ function Eα,β (−x) is completely monotonic for 0 < α ≤ 1, 0 < αγ ≤ β ≤ 1 (see [12]) (a list of special functions possessing complete monotonicity property is presented in [13]). An importance of this property is due to its association with certain classes of probability distributions. A real valued random variable X with the cumulative distribution function (cdf) F (·) and the characteristic function (cf) φ is said to be infinitely divisible (or F is infinitely divisible law, or φ is infinitely divisible), if for any n > 1 there exist independent identically distributed random variables X1 , . . . , Xn with cdf say Fn such that X has the same distribution as X1 +. . .+Xn . It is known (see [2]) that if the positive random variable X has a completely monotonic density, then X is infinitely divisible.

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In [15] the Mittag-Leffler distribution was introduced Fα (x) = 1 − Eα (−x α ), 0 < α ≤ 1; x ≥ 0,

(3.11)

where Eα is the classical Mittag-Leffler function. Since the Laplace transform of 1 Fα (x) is equal fα (s) = 1+s α and the latter is the completely monotonic function, then Fα (x) is the probability distribution. It was shown [15] that this distribution is an infinitely divisible for all 0 < α ≤ 1. Moreover (see [15]), the Mittag-Leffler distribution attracted to the stable distribution with exponent α, 0 < α < 1. Due to the infinite divisibility of the Mittag-Leffler distribution, Pillai [15] developed the corresponding stochastic process. He introduced the Mittag-Leffler stochastic process as a stochastic process {X(t), t > 0} with X(0) = 0 and having stationary 1 and independent increments, where X(1) has the Laplace transform 1+s α, 0 < a < 1. Connection between a positive stable stochastic process and the Mittag-Leffler stochastic process with parameter α, 0 < α < 1 was found too (see [15]). Another application of the completely monotonic functions is related to some physical models (see [5]). As discussed by Hanyga [8], complete monotonicity is essential to ensure the monotone decay of the energy in isolated systems (as it appears reasonable from physical considerations); thus, restricting to completely monotonic functions is essential for the physical acceptability and realizability of the dielectric models. Thus, Havriliak and Negami proposed a new model (see, e.g., [5]) with two real powers to take into account, at the same time, both the asymmetry and the broadness observed in the shape of the permittivity spectrum of some polymers. The normalized complex susceptibility proposed in the Havriliak- Negami (HN) model is given by χ EHN (iω) =

1 . (1 + (iτ∗ ω)α )γ

(3.12)

The time-domain response and the time-domain relaxation in the HN model are respectively equal φHN (t) =

(t/τ∗ )αγ −1 γ Eα,αγ (−(t/τ∗ )α ), τ∗ γ

HN (t) = 1 − (t/τ∗ )αγ Eα,αγ +1 (−(t/τ∗ )α ), γ

(3.13) (3.14)

where Eα,β is the three-parametric Mittag-Leffler function. On the basis of the observation of a large amount of experimental data, it was proposed an extension of the range of admissibility of the parameters α and γ to 0 < α, αγ ≤ 1. The completely monotonicity of the relaxation and response functions has been recently proved in (see [5]) also for this extended range of parameters.

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3.4 Applications of the Mittag-Leffler Distribution: Continuous Time Random Walk A continuous time random walk (CTRW) is given by an infinite sequence of spatial positions 0 = x0 , x1 , x2 , · · · , separated by (independent identically distributed) random jumps Xj = xj − xj −1 , whose probability density function w(x) is given as a non-negative function or a generalized function (interpretable as a measure) with support on the real axis −∞ < x < +∞ and normalized by the relation:  ∞

w(x) dx = 1. This random walk being subordinated to a renewal process so that

0

we have a random process x = x(t) on the real axis with the property x(t) = xn for tn ≤ t < tn+1 , n = 0, 1, 2, · · · . By natural probabilistic arguments we arrive at the integral equation for the probability density p(x, t) (a density with respect to the variable x) of the particle being in the point x at the instant t , 

t

p(x, t) = δ(x) (t) +

3

φ(t − t )

0

+∞ −∞





w(x − x ) p(x , t ) dx

4

dt .

(3.15)

We mention here a special choice of the memory function H (t): H (t) =

t −β (s) = s β−1 , , 0 < β < 1 , corresponding to H (1 − β)

giving the Mittag-Leffler waiting time density φ (s) = function φ and relaxation function : φ(t) = −

d Eβ (−t β ) = φ ML (t), dt

(3.16)

1 with response 1 + sβ

(t) = Eβ (−t β ) .

In this case we obtain in the Fourier-Laplace domain % & s β−1 sE p (κ, s) − 1 = [E w (κ) − 1] E p (κ, s) ,

(3.17)

(3.18)

and in the space-time domain the time fractional Kolmogorov-Feller equation β t D∗

 p(x, t) = −p(x, t) +

+∞

−∞

w(x − x ) p(x , t) dx ,

p(x, 0+ ) = δ(x) , (3.19)

β

where t D∗ denotes the fractional derivative of order β in the Caputo sense.

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The time fractional Kolmogorov-Feller equation can be also expressed via the 1−β Riemann-Liouville fractional derivative Dt , that is 3 4  +∞ d 1−β

p(x, t) = t D w(x − x ) p(x , t) dx , −p(x, t) + dt −∞

(3.20)

with p(x, 0+ ) = δ(x). The equivalence of the two forms (3.19) and (3.20) is easily proved in the Fourier-Laplace domain by multiplying both sides of Eq. (3.18) with the factor s 1−β . Further discussion of the application of the Mittag-Leffler waiting time law can be found in [7, Ch. 9]. Acknowledgements The research is partially supported by the Belarusian Fund for Fundamental Scientific Research (Project F17MS-002).

References 1. D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos, vol. 5, 2nd edn. (World Scientific Publishing, Singapore, 2016) 2. A. Bose, A. Dasgupta, H. Rubin, A contemporary review and bibliography of infinitely divisible distributions and processes. Sankhay¯a Indian J. Stat. 64(Ser. A, Pt. 3), 763–819 (2002) 3. L. Debnath, D. Bhatta, Integral Transforms and Their Applications, 3rd edn. (Chapman & Hall/CRC, Boca Raton, 2015) 4. K. Diethelm, The Analysis of Differential Equations of Fractional Order: An ApplicationOriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics, vol. 2004 (Springer, Berlin, 2010) 5. R. Garrappa, F. Mainardi, G. Maione, Models of dielectric relaxation based on completely monotone functions. Fract. Calc. Appl. Anal. 19(5), 1105–1160 (2016) 6. R. Gorenflo, Yu. Luchko, S.V. Rogosin, Mittag-Leffler type functions: notes on growth properties and distribution of zeros. Preprint No. A04-97, Freie Universität Berlin. Serie A. Mathematik, 1997 7. R. Gorenflo, A. Kilbas, F. Mainardi, S. Rogosin, Mittag-Leffler Functions: Related Topics and Applications (Springer, Berlin, 2014) 8. A. Hanyga, Physically acceptable viscoelastic models, in Trends in Applications of Mathematics to Mechanics, ed. by K. Hutter, Y. Wang. Ber. Math. (Shaker Verlag, Aachen, 2005), pp. 125–136 9. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204 (Amsterdam, Elsevier, 2006) 10. Yu. Luchko, On the distribution of zeros of the Wright function. Integr. Transf. Spec. Funct. 11, 195–200 (2001) 11. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity (Imperial College Press, London, 2010) 12. F. Mainardi, R. Garrappa, On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics. J. Comput. Phys. 293, 70–80 (2015) 13. K. Miller, S. Samko, Completely monotonic functions. Integr. Transf. Spec. Funct. 12(4), 389–402 (2001) 14. I. Petráš, Fractional-Order Nonlinear Systems: Modelling, Analysis and Simulation (Springer, Dordrecht, 2011)

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15. R.N. Pillai, On Mittag-Leffler functions and related distributions. Ann. Inst. Stat. Math. 42, 157–161 (1990) 16. I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999) 17. A. Yu. Popov, A.M. Sedletskii, Zeros distribution of Mittag-Leffler functions. Contemp. Math. Fundam. Dir. 40, 3–171 (2011, in Russian). Transl. in J. Math. Sci. 190, 209–409 (2013) 18. Yu.N. Rabotnov, Elements of Hereditary Mechanics of Solids (Nauka, Moscow, 1977, in Russian) 19. M. Rivero, S.V. Rogosin, J.A. Tenreiro Machado, J.J. Trujillo, Stability of fractional order systems. Math. Probl. Eng. 2013, Article ID 356215, 14 pp. http://dx.doi.org/10.1155/2013/ 356215 20. S. Rogosin, F. Mainardi, George William Scott Blair – the pioneer of factional calculus in rheology. Commun. Appl. Ind. Math e-481 (2014). arXiv:1404.3295.v1 21. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, New York, 1993) 22. V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, New York, 2011) 23. J. Tenreiro Machado, V. Kiryakova, F. Mainardi, A poster about the old history of fractional calculus. Fract. Calc. Appl. Anal. 13(4), 447–454 (2010). http://www.math.bas.bg/~fcaa 24. J. Tenreiro Machado, V. Kiryakova, F. Mainardi, A poster about the recent history of fractional calculus. Fract. Calc. Appl. Anal. 13(3), 329–334 (2010). http://www.math.bas.bg/~fcaa 25. I.V. Tikhonov, Yu.S. Éidel’man, Inverse scattering transform for differential equations in Banach space and the distribution of zeros of an entire Mittag-Leffler type function. Differentsial’nye Uravneniya [Diff. Equ]. 38(5), 637–644 (2002) 26. V.V. Uchaikin, Method of Fractional Derivatives (Artishock, Ulyanovsk, 2008, in Russian) 27. V.V. Uchaikin, Fractional Derivatives for Physicists and Engineers, vols. I, II (Springer, Berlin; Higher Education Press, Beijing, 2013) 28. V. Volterra, Opere matematiche: memorie e note, vols. I–V. Accademia Nazionale dei Lincei, Roma, Cremonese (1954/1962)

On Elliptic Systems of Two Equations on the Plane A. P. Soldatov

Dedicated to Professor Heinrich G. W. Begher on the occasion of his 80th birthday

Abstract We considered an elliptic second order system on the plane consisting of two equations with constant (and only leading) coefficients. An explicit representation of the general solution of this system is given via the so-called J -analytic functions. A classification of systems with respect to the Dirichlet problem is given. Explicit expressions for the generalized potentials of a double layer are derived and their applications to solution of the Dirichlet problem are described. The results are illustrated by the example of the Lamé system of plane elasticity theory. Keywords Elliptic systems · Analytic functions · Bitsadze representation · Dirichlet problem · Generalized potentials of a double layer · Lamé system Mathematics Subject Classification (2010) Primary 35J47, 35J57; Secondary 31A10

1 Representation of Solutions Consider a second order elliptic system a0

∂ 2u ∂ 2u ∂ 2u + a2 2 = 0 + 2a1 2 ∂x ∂x∂y ∂y

(1.1)

A. P. Soldatov () Federal Research Center “Computer Science and Control” of Russian Academic of Sciences, Moscow, Russia © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_14

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with real constant 2 × 2-matrix-valued coefficients aj ∈ R2×2 . The associated characteristic matrix-valued and scalar-valued polynomials read respectively  2

p(z) = a0 + 2a1 z + a2 z =

p1 p3 p4 p2

 (z),

χ(z) = det p(z).

(1.2)

The ellipticity condition translates to the requirements that the matrix a2 is should be invertible and that the polynomial χ(z) should haves no real roots. We denote by σ a set of these roots on the upper half-plane, this set consisting of one or two points. There are only the following three possibility: (i) σ = {ν1 , ν2 }, ν1 = ν2 ;

(ii) σ = {ν}, p(ν) = 0;

(iii) σ = {ν}, p(ν) = 0.

If two scalar second order polynomials have a common root ν on the upper halfplane, then they are linearly dependent more precisely, they are proportional to (z − ν)(z − ν). ¯ So the class (iii) can be described by the matrix-valued polynomial p(z) = (z2 − 2(Re ν)z + |ν|2 )a2 . Accordingly for each of these classes we introduce the matrixes  (i) J =

ν1 0 0 ν2



 ,

(ii) J =

ν1 0ν

 ,

(iii) J = ν,

(1.3)

where from now on, a scalar matrix is identified with scalar. The following lemma[1] defines another importing matrix b closely related to with the system (1.1). Lemma 1.1 There exists a matrix b ∈ C2×2 , such that 

2

a0 b + 2a1 bJ + a2 bJ = 0,

b bJ det b bJ

 = 0.

(1.4)

If a matrix b˜ satisfies the same relations then b˜ = bd where d is an invertible matrix of the form  (i) d =

λ0 0μ



 ;

(ii) d =

 λμ ; 0 λ

(iii) d ∈ R2×2 .

(1.5)

Obviously, in case (iii) we can take as b whatever invertible matrix; more specifically we can set b = 1. The matrix J defines the simplest first order elliptic system ∂φ ∂φ −J = 0, ∂y ∂x

(1.6)

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which is generalizes the classic Cauchy–Riemann system (the latter corresponds to the case J = i). The solutions φ = (φ1 , φ2 ) of the system (1.6) are connected with the pair ψ = (ψ1 , ψ2 ) of analytic functions by the mutually invertible relations (i) φj (z) = ψj (x + νj y), j = 1, 2; (ii) φ1 (z) = ψ1 (x + νy) + yψ2 (x + νy), φ2 (z) = ψ2 (x + νy); (iii) φj (z) = ψj (x + νy), j = 1, 2.

(1.7)

In sequel, and to lighten notation, solutions of (1.6) will be referred to as are shortly J -analytic functions because the main notions of analytic functions theory have their counterparts with respect to the matrix zJ = x + yJ

(1.8)

For example if the vector-valued function φ is continuous in a closed domain D, bounded by the smooth contour , then the following Cauchy formula holds: φ(z) =

1 2πi

 (t − z)J dtJ φ(t),

z ∈ D,



where the  is positively oriented with respect to D. Here we put dtJ = dt1 + dt2 J analogously to (1.8). In particularly we have the Taylor series φ(z) =

 (z − z0 )kJ φ k (z0 ), k≥0

φk =

∂kφ , ∂x k

which uniformly converges in the neighborhood of the point z0 . Note that first order elliptic systems theory was developed by many authors (for example [2–8]). Solutions of the system (1.6) with respect to (1.1) play the same role as analytic functions play with respect to the Laplace equation. Theorem 1.2 A general solution of the system (1.1) is described through J -analytic function by the formula u = Re bφ,

(1.9)

where φ is uniquely defined to an arbitrary additional vector ξ ∈ R2 , but its derivative φ = ∂φ/∂x is single-valued (in multiple connected domains). This theorem (in more general context of elliptic l × l-systems) was established by Bitsadze [9] (see also [1, 10]). In fact A.V. Bitsadze uses the representation based on analytic functions, which he obtains by substituting (1.6) into (1.9). But an investigation of boundary value problems for elliptic system of the second and higher order is considerably simplified when we use the J -analytic functions.

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The matrix b the significance of which is demonstrated by Theorem 1.2 can be calculated implicitly in terms of polynomials pj in (1.2) and the roots ν ∈ σ of the characteristic polynomial χ. By virtue of the second part of Lemma 1.1 it is surely sufficient to point out only one such matrixes. Theorem 1.3 (a) Let one of the polynomial pairs {p2 , p4 } and {p1 , p3 } be linearly independent. Then we have to deal with one of the cases (i), (ii) and depending on the case, the matrix b satisfying the conditions of Lemma 1.1 reads  (i) b =

 p2 (ν1 ) p2 (ν2 ) , −p4 (ν1 ) −p4 (ν2 )

 (ii) b =

p2 (ν) p2 (ν) −p4 (ν) −p4 (ν)

 ,

(1.10a)

,

(1.10b)

if pair {p2 , p4 } is linearly independent and the matrix  (i) b =

 −p3 (ν1 ) −p3 (ν2 ) , p1 (ν1 ) p1 (ν2 )

 (ii) b =

−p3 (ν) −p3 (ν) p1 (ν) p1 (ν)



if the pair {p1 , p3 } is linearly independent. (b) Let both pairs {p2 , p4 } and {p1 , p3 } be linearly dependent. Then we are led to one of the cases (i), (iii) and in the first of these cases, if we enumerate the roots in such a way as to have: p1 (ν1 ) = p3 (ν1 ) = 0,

p2 (ν2 ) = p4 (ν2 ) = 0,

(1.11)

conditions of Lemma 1.1 are satisfied by the matrix  b=

δ2 −δ3 −δ4 δ1

 ,

(1.12)

where δj is stands for the leading coefficient of pj (z). Proof Let b(j ) , j = 1, 2 be denote the columns of the matrix b, considered as elements of C2 . If the case at hand is (i) then the matrix J is diagonal and it follows from (1.4) that ej = b(j ) is the eigenvector corresponding to eigenvalue νj of the matrix-polynomial p i.e. p(νj )ej = 0. The second condition in (1.4) shows that this vector has to be different from zero. So with accurate to proportionality it is uniquely defined. In case at hand is the (ii) it follows from the definition of the matrix J that e = b(1) is eigenvector but e0 = b(2) is an adjoined eigenvector of p, i.e. p(ν)e = 0, p(ν)e0 p (ν)e = 0, where prime denotes the derivative p (z) = 2(a1 +a2 z). Here as above the second condition in (1.4) shows that the vector e has to be different from zero despite the fact that but the adjoined vector e is defined with accuracy up to λe, λ ∈ R. The indeterminacy of in the choice of e and e0 is exactly described by the second part of Lemma 1.1.

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In the case (iii) both columns b(j ) are eigenvectors corresponding to eigenvalue νj of the matrix-polynomial p as all C2 consists of these eigenvectors. Let us consider the adjoined matrix p∗ =



p2 −p3 −p4 p1

 (1.13)

∗ , k = 1, 2 be its columns. and let p(k) We will use the identity

p(z)p∗ (z) = χ(z),

(1.14)

where χ = det p is considered as a scalar matrix. It shows that the nonzero column ∗ (ν) is eigenvector with eigenvalue ν ∈ σ . So in the case (i) we can put e = p(k) ∗ b(j ) = p(s (ν ), j) j

j = 1, 2,

(1.15)

where sj is the number of a nonzero column of the matrix p∗ (νj ). The case (ii) can be considered analogously. In this case we have equality χ(ν) = χ (ν) = 0 and it follows from (1.14) alongside with differentiating the identity, that p (ν)p∗ (ν) + p(ν)(p∗ ) (ν) = 0. Thus we arrive at the relations ∗ (ν) = 0, p(ν)p(r)

∗ p(ν)(p∗ ) (r) (ν) + p (ν)p(r) (ν) = 0.

∗ (ν) and e ∗ So e = p(s) 0 = (p )(s) (ν) are eigenvector and adjoined vector respectively. In this case we can put ∗ b(1) = p(s) (ν),

b(2) = (p∗ ) (s) (ν),

(1.16)

where s is the number of a nonzero columns of the matrix p∗ (ν). ∗ (ν) is equal to zero on ν ∈ σ , then obviously the pair of If a column of p(s) its elements is linearly dependent. Conversely for example let elements of the first column of the matrix (1.13) be linearly dependent. Then p4 = λp2 , λ = 0 or p2 = 0. In the first case we have the equality χ = p2 (p1 − λp3 ) which shows that p2 (ν) = 0 for some ν ∈ σ . But then p4 (ν) = λp2 (ν) and, therefore, p1∗ (ν) = 0. The case p2 = 0 may be considered analogously. ∗ (ν) = 0 for some ν ∈ σ is equivalent to linear dependence of So the equality p(s) the elements of the s—the column of the matrix p∗ . Therefore if the pair {p2 , p4 } is linearly independent then in (1.15) and (1.16) we can put sj = s = 1 which gives

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the formulas (1.10a). Analogously if the pair {p1 , p3 } is linearly independent then in (1.15) and (1.16) we can put sj = s = 2, which leads to (1.10b). Let finally both pairs {p2 , p4 } and {p1 , p3 } be linearly dependent. Then there are ∗ (ν), p∗ (ν) are equal to zero in the point ν ∈ σ and two possibilities: columns p(1) (2) then we obtain the case (iii), or we have the case (i) and in accordance with (1.11) ∗ (ν ) = p∗ (ν ) = 0, p∗ (ν ) = 0, p∗ (ν ) = 0. In this case we have relations p(1) 2 (2) 1 (1) 1 (2) 2 the matrix   p2 (ν1 ) −p3 (ν2 ) b= −p4 (ν1 ) p1 (ν2 ) satisfies the conditions of Lemma 1.1. By virtue of (1.11) the polynomials pj take the form pj (z) = δj (z − ν1 )(z − ν 1 ), pj (z) = δj (z − ν2 )(z − ν 2 ),

j = 1, 3, j = 2, 4,

so taking into account the last part of Lemma 1.1 we can consider the matrix (1.12) as the matrix b.   Note that the det b = 0 in (1.12) because det b coincides with the leading coefficients of the characteristic polynomial χ = p1 p2 − p3 p4 . We more specifically consider the case p3 = p4 = 0, when the system (1.1) reduces to two scalar equations. For this system both coefficients β1 and β2 are not equal to zero and we obtain the cases (i) or (iii). In both cases we can put b = 1 (in the first case we should use the second part of Lemma 1.1). Let us illustrate the theorem in the case of example of the system (1.1) with coefficients   αj −βj aj = , 0 ≤ j ≤ 2. (1.17) βj αj With respect to w = u1 + iu2 we can write this system in the form of one C-linear equation γ0

∂2 ∂2 ∂2 + γ + 2γ =0 1 2 ∂x 2 ∂x∂y ∂y 2

(1.18)

with coefficients γj = αj + iβj . By the way coefficients of a general system (1.1) can be uniquely represented in the form      j αj −βj α −β 1 0 aj = + j . βj βj αj αj 0 −1

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It follows that we can cast this system into the form of a R-linear equation lw+ l˜w¯ = 0, where lw is the left hand part-side of (1.18) and l˜ is defined analogously with j . respect to γj = αj + i β For the system (1.1), (1.17) the polynomials pj in (1.2) satisfy the relation p4 = −p3 , p2 = p1 , so χ = p12 + p32 = q q¯ with polynomials q = p1 + ip3 and q¯ = p1 − ip3 . The first of these is naturally called the characteristic polynomial of Eq. (1.18). Obviously case (iii) corresponds to p1 (ν) = p3 (ν) = 0, in which we can put b = 1 in Lemma 1.1. Lemma 1.4 In the cases (i) and (ii) the matrices  (i) b =

1 1 ε(ν1 )i ε(ν2 )i



 ,

(ii) b =

1 1 ε(ν)i ε(ν)i

 ,

(1.19)

satisfy the conditions of Lemma 1.1 respectively for the system (1.1), (1.17), where ε(ν) = 1, if q(ν) = 0, and ε(ν) = −1, if q(¯ν ) = 0. In particular det b = 0 if and only if the roots of the characteristic polynomial q lie either side of the real line. Proof It is obvious that exactly one of the equalities q(ν) = 0 or q(¯ν ) = 0 holds in the cases (i) and (ii), and p1 (ν)p3 (ν) = 0 at that. Thus by virtue of Theorem 1.3 we can use in the last two case for the matrix b the following matrix resp.:  (i) b =

 p1 (ν1 ) p1 (ν2 ) , p3 (ν1 ) p3 (ν2 )

 (ii) b =

p1 (ν) p1 (ν) p3 (ν) p3 (ν)

 (1.20)

Additionally in for the case (ii) we have either q(ν) = q (ν) = 0, or q(¯ν ) = q (¯ν ) = 0. In fact let for example q(ν) = 0, q(ν) ¯ = 0. Then the equality q (ν)q(ν) ¯ + q(ν)q¯ (ν) = 0 implies q (ν) = 0. It follows then that p1 (ν)p3 (ν) = 0, otherwise the polynomials p1 and p3 have to be constants and respectively p1 and p3 have to be polynomials of the first degree. But this is impossible. So, if q(ν) = 0, then p1 (ν) = ip3 (ν) and in the case (ii) also p1 (ν) = ip3 (ν). Analogously p(¯ν ) = 0 implies p1 (ν) = −ip3 (ν), and in the case (ii) also p1 (ν) = −ip3 (ν). In accordance with the second part of Lemma 1.1, this implies that the matrices (1.19) also satisfy the conditions of this lemma together with the matrices (1.20).  

2 The Dirichlet Problem Let us consider for the solution of equation (1.1) the Dirichlet problem  u  = f

(2.1)

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for a domain D bounded by the smooth contour . The domain D may be either finite or infinite. In the latter case, the following condition is imposed on the function u(z) as z → ∞      ∂u   ∂u    +   = O(|z|−2). (2.2)  ∂x   ∂x  It is well known that this problem is of the Fredholm type in the case of a single elliptic equation. In 1948 A.V. Bitsadze established that in the case of systems this property can be violated. In order to provide support for this discovery of his, he cited two simple examples of elliptic systems   ∂ 2u ∂ 2u 0 −1 ∂ 2 u − + 2 =0 1 0 ∂y∂y ∂x 2 ∂y 2

(2.3a)

  ∂ 2 u √ 0 −1 ∂ 2 u ∂ 2u − − 2 = 0, 1 0 ∂y∂y ∂x 2 ∂y 2

(2.3b)

and

for which the homogeneous problem (2.1) has an infinite number of linearly independent solutions in the unite circle |z| < 1. It is obviously that for the first system we obtain the case √(ii) with ν = i for the first system, but we obtain the case (i) with ν1,2 = (i ± 1)/ 2 for the second one. Each of them can be written in the form (1.18) with the characteristic polynomial q(z) = 1 − 2iz − z2 in the first case and the polynomial q(z) = 1 − 2iz − z2 in the second one. The polynomial q √ has respectively the multiple root ν¯ = −i and two distinct roots ν¯ 12 = (−i ± 1)/ 2 in the lower half-plane. In particulary by virtue of Lemma 1.4 det b = 0 for both systems. In this connection A.V. Bitsadze introduced the following notion: the system (1.1) is called weakly connected, if in the notation of Lemma 1.1 the matrix b is invertible and strongly connected otherwise. By the second assertion of Lemma 1.1, this definition does not depend on the choice of the matrix b. Using Bitsadze’s representation based on analytic functions, Tovmasyan [11] found that in the class of functions that satisfy the Holder condition in the closed domain D, the Dirichlet problem for a weakly connected system is Fredholm and its index is zero. The proof was based on the reduction of this problem to a system of singular integral equations with a Cauchy kernel on the contour  to which the results of the classical theory [12] apply. In this case the contour  was assumed to be of Lyapunov type. One can show [13] that the property of weak connectedness of a system can be expressed directly in terms of the polynomial p by the condition  det

R

p

−1

 (t)dt

= 0,

(2.4)

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without resorting to the matrix b. It follows from this that together with (1.1) the system a0&

2˜ 2˜ ∂ 2 u˜ & ∂ u &∂ u + a + 2a =0 1 2 ∂x 2 ∂y∂y ∂y 2

F (1.1)

is conjugate with respect to Lagrange. Often the system (1.1) is written in the form  i,j =1,2

aij

∂ 2u =0 ∂xi ∂xj

(2.5)

with respect to the coefficients a11 = a0 , a12 + a21 = 2a1 , a22 = a2 and to the variables x1 = x, x2 = y, and then the conjugated system takes the form  i,j =1,2

a˜ ij

∂ 2 u˜ = 0, ∂xi ∂xj

a˜ ij = aj&i .

F (2.5)

The indicated choice of the coefficients a˜ ij ensures the Green’s identity 

       ∂u ∂ u˜ u ni a˜ ij ni aij ud ˜ 1x = d1 x ∂xj ∂xj 

 i,j =1,2

(2.6)

i,j =1,2

F respectively from the holds for any solutions u and u˜ of the systems (2.5) and (2.5) class C 1 (D). Here d1 x denotes an element of arc lengths and n is the unit outer normal. From the point of view of the general elliptic theory [14] the Fredholm property of the Dirichlet problem in the space C 2,μ (D) and the Schauder a priori estimates satisfy the so-called Shapiro–Lopatinskii condition. As applied to the Dirichlet problem with respect to each point t ∈ , it can be formulated as follows. We set p(n, z) = (n2 − n1 z)2 A0 + 2(n2 − n1 z)(n1 + n2 z)A1 + (n1 + n2 z)2 A2 , where n = n1 + in2 is the unit outer normal at the point t ∈ . Then if for ξ ∈ C2 the meromorphic vector-valued function (p& )−1 (n, z)ξ has no poles in the upper half-plane, then ξ = 0. Therefore, it is natural to expect that this condition should describe the class of weakly connected systems. Actually, one can establish this by the sole use of linear algebra [13]. The result of N.E. Tovmasyan on the Fredholm property of the Dirichlet problem in the Holder class can be extended to a whole series of spaces[13], of which the Hardy space hp (D), 1 < p < ∞ is the widest. This space is introduced for solutions of weakly connected systems in the domain D, bounded by the Lyapunov

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contour  of the class C 1, nu , 0 < nu < 1. Recall that  ∈ C 1,ν if for any point t0 ∈  there is a rectangle P = I1 × I2 ⊆ R2 with center at this point, that  ∩ P is in the Cartesian coordinates by the graph y = h(x), x ∈ I1 , or x = h(y), y ∈ I2 , of a function h belonging to the class C 1,ν on the corresponding interval. By definition, the sequence of contours n ⊆ D, n = 1, 2, . . ., converges to  in the class C 1,ν if for any point t0 ∈  in the previous notation, the intersection n ∩ P for sufficiently large n is the corresponding graph of the function hn , with hn → h in the class C 1,ν . Given this sequence, the space hp (D) of solutions u(z) of weakly connected elliptic systems is determined by the condition  |u(t)|p d1 t < ∞.

sup n

(2.7)

n

An analogous space for J -analytic functions is denoted by H p (D). These definitions are usually introduced [15, 16] for harmonic and analytic functions, the corresponding spaces are called Hardy–Smirnov and are denoted by ep (D) and E p (D) resp., and the symbols hp and H p are left for the case when the domain D is the unit circle. The following theorem [13, 17] is an analogue of the corresponding results [15, 16] for harmonic and analytic functions. Theorem 2.1 Suppose that a domain D is bounded by a contour of class C 1,ν . Then the following propositions hold. (a) J -analytic function φ belongs to the class H p (D) if and only if there exist angular boundary values φ + (t0 ) for almost all points t0 in, the boundary function φ + ∈ Lp () and the Cauchy formula holds. In this case, with respect to the norm |φ| = |φ + |Lp , the space H p is a Banach space. (b) The integral of Cauchy type (I ϕ) =

1 2πi

 (t − z)J dtJ ϕ(t),

z ∈ D,



defines the operator I , bounded Lp () → H p (D) and C μ () → C μ (D). (c) A solution u of a weakly connected system (1.1) belongs to the class hp (D) or C μ (D) if and only if the function φ in the representation (1.9) of Theorem 1.2 belongs to the corresponding class H p (D) or C μ (D). This theorem shows, in particular, that the definition (2.7) of the classes hp and does not depend on the choice of the sequence n converging to . We now state the main result of [13] on the Dirichlet problem.

Hp

Theorem 2.2 Let the contour  ∈ C 1,ν and the system (1.1) be weakly connected. (a) For the Dirichlet problem with the right-hand side f ∈ Lp (), 1 < p < ∞, the following Fredholm alternatives hold:

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1) the homogeneous Dirichlet problem (i.e. with the right-hand side f = 0) has finite number u1 , . . . , us linearly independent solutions that are continuously differentiable in the closed domain D; F has the same 2) the homogeneous Dirichlet problem for the system (1.1) number of linearly independent solutions u˜ 1 , . . . , u˜ s ; 3) the inhomogeneous Dirichlet problem is solvable in the class hp (D) if and only if the following orthogonality conditions hold: 

  i,j =1,2

f

  ∂ u˜ k d1 x = 0, ni a˜ ij ∂xj

k = 1, . . . , s.

(b) Any solution u ∈ hp (D) of the nonhomogeneous problem with right-hand side f , belonging to one of the classes C(), C μ (), 0 < μ < ν, C 1,μ (), also belong to the corresponding classes C(D), C μ (D), C 1,μ (D). An important class of weakly connected systems is formed by strongly elliptic systems, introduced by Vishik [18] in 1951. With respect to the matrix polynomial p(z), they are determined by the following condition for the scalar product that for all non-zero ξ ∈ R2 and t ∈ R [p(t)ξ ]ξ = 0.

(2.8)

By virtue of criterion (2.4), systems of this type are really weakly connected. Passing from p to −p, without loss of generality we can assume that the scalar product in this condition is positive. Therefore, it is equivalent to the positive definiteness of the symmetric part (p + p& )/2 of the matrix p. In particular, if the matrix p(t) is symmetric, then by the Sylvester criterion it reduces to the inequalities p1 (t) > 0 and det p(t) > 0 for any t ∈ R. As it was noted Vishik[18], if the system (1.1) is strongly elliptic, then any solution of the homogeneous Dirichlet problem from the Sobolev class W12 (D) is equal to zero. In fact, when writing the system in question in the form (2.5), for this solution we have     ∂u ∂u aij d2 x = 0. ∂xj ∂xi D i,j =1,2

Continuing the function u with zero to the whole plane (preserving the notation) and noting that the extended function belongs to W12 (R2 ), in the previous equality we can replace D by R2 . But then for the Fourier transform uˆ of this function this equation leads to 



D i,j =1,2

(aij ξi u)ξ ˆ j ud ˆ 2 ξ = 0.

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One more thing to note is that condition (2.8) can be rewritten in the form of inequality " i,j

 # aij ξi ξj η η > 0,

which is valid for any nonzero ξ, η ∈ R2 and, consequently, the integrand of the preceding equality is nonnegative. In connection to Theorem 2.2, this fact leads to the following result. Theorem 2.3 If the system (1.1) is strongly elliptic, then under the hypotheses of Theorem 2.2 the number s = 0, that is, the Dirichlet problem for this system is uniquely solvable. We agree to say that the system (1.1) is equivalent to an analogous system with coefficients a˜ i if invertible matrices c and d can be found such that a˜ i = cai d, i = 0, 1, 2. It is clear that for equivalent systems the dimension of s in Theorem 2.2 is the same. The following result is derived in [19]. Theorem 2.4 The following statements are equivalent. (a) System (1.1) is equivalent to strongly elliptic one. (b) For this system det(a0 + 2βa1 + γ a2 ) = 0 and β 2 ≤ γ .

(2.9)

(c) The elliptic system (1.1) is such that in any domain D, bounded by Lyapunov contour, the homogeneous Dirichlet problem has only trivial solution. The condition (2.9) was proposed by Dean-Shia [20] and also called by him a condition of strong ellipticity. The implication (c) ⇒ (b) is derived as follows: let condition (2.9) be violated, so that there are β, γ with the property β 2 ≤ γ and a nonzero vector ξ such that (a0 + 2βa1 + γ2 a2 )ξ = 0. But then the vector-valued function u with components uj (x, y) = (x 2 + βxy + γ y 2 − 1)ξj ,

j = 1, 1

satisfies (1.1) and vanishes on the ellipse x 2 + βxy + γ y 2 = 1. Nevertheless, there exist domains in which the Dirichlet problem is uniquely solvable for any weakly connected system. One such domain is the upper half-plane μ (D) the class of functions u(z) in this half-plane, D = {y > 0}. We denote by C which together with the function u(1/¯z) belong to C μ (K) on any compact subset K ⊆ D and vanish at infinity. As was shown in [21], for this class the Dirichlet problem for any weakly connected elliptic system is uniquely solvable and in the notation (1.6) its solution is given by the analogue of Schwartz’s formula u(z) =

1 π

 R

# " −1 f (t)dt, Im b(t − z)−1 J b

Im z > 0.

On Elliptic Systems of Two Equations on the Plane

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If the system (1.1) is strongly connected, i.e. bη = 0 for some nonzero vector η, then μ (D), analytic in the half-plane D, vector function for any scalar function χ ∈ C u(z) = Re [bχ(zJ )η], where the matrix χ(zJ ) is the value of the analytic function χ of zJ , is a solution of the homogeneous Dirichlet problem. In particular, the homogeneous problem for this system has an infinite number of linearly independent solutions. As shown in [1], associated to strongly elliptic systems are elliptic systems of the form (2.5) for which (a11 ξ + a12η)ξ + (a21 ξ + a22 η)ξ ≥ 0

(2.10)

for any ξ, η ∈ R2 . In the case of strict inequality (for nonzero ξ, η) this class of systems was introduced by Somigliana [22]. In the case when, in addition to (2.9), any nonzero solution of the homogeneous system a11ξ + a12η = 0,

a21 ξ + a22 η = 0

(2.11)

is such that the vectors ξ and η are linearly independent, the ellipticity condition is automatically satisfied. An important example of a strongly elliptic system of the form (2.5) is the Lamé system of the plane theory of elasticity with coefficients  a11 =  a21 =

α1 α6 α6 α3 α6 α3 α4 α5



 ,

a12 =



 ,

a22 =

α6 α4 α3 α5 α3 α5 α5 α2

 , (2.12)

 .

Here, the constants αj , called elasticity moduli, satisfy the requirement of positive definiteness of the matrix ⎛

⎞ α1 α4 α6 α = ⎝ α4 α2 α5 ⎠ . α6 α5 α3 According to the Sylvester criterion this fact can be expressed by means of the inequalities αj > 0, j = 1, 2, 3; α1 α2 − α42 > 0; det α = α1 α2 α3 + 2α4 α5 α6 − α1 α52 − α2 α62 − α3 α42 > 0.

(2.13)

It is easy to see that condition (2.10) is satisfied for these coefficients, and the vectors ξ, η, which constitute the non-zero solution of system (2.11), are linearly

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independent (and the space of these solutions is one-dimensional). Therefore, the Lamé system (2.5), (2.12) is indeed strongly elliptic. For this system the polynomials pj in (1.2) have the form p1 (z) = α1 + 2α6 z + α3 z2 , p2 (z) = α3 + 2α5 z + α2 z2 , p4 (z) = p3 (z) = α6 + (α3 + α4 )z + α5 z2 .

(2.14)

It is easy to show that polynomials p2 and p3 are linearly dependent if and only if α3 α5 = α2 α6 , α2 (α3 + α4 ) = 2α52 .

(2.15a)

Similarly, the linear dependence of the polynomials p1 and p3 is equivalent to the conditions α1 α5 = α3 α6 , α1 (α3 + α4 ) = 2α62 .

(2.15b)

Lemma 2.5 The simultaneous fulfillment of both groups of relations (2.15) is possible if and only if the polynomial p3 = 0. When this condition is fulfilled, the Lamé system belongs to the case (i) with roots @ ν1 = i

α1 , α3

@ ν2 = i

α3 . α2

(2.16)

In particular, the case (iii) is impossible for the Lamé system. In the case (ii), both pairs {p1 , p3 } and {p2 , p3 } are linearly independent. Proof Suppose that both pairs {p2 , p3 } and {p1 , p3 } are linearly dependent, i.e. all the relations (2.15) are satisfied. Then, obviously, either α3 + α4 = 0, or p3 = 0, i.e. the following relations hold: α5 = α6 = α3 + α4 = 0. We first consider the latter case. In this case χ = p1 p2 , and p1 (z) = α1 + α3 z2 and p2 (z) = α3 + α2 z2 . In particular, the expressions (2.16) are the roots of the polynomials pj . These roots are distinct, since α32 = α42 and taking into account (2.13), the equality α32 = α1 α2 , necessary for the roots in (2.16) to coincide, is impossible. Now let α3 +α4 = 0 in the relations (2.15). Then α5 α6 = 0 and the first equalities in (2.15) are possible only for α1 α2 = α32 . In particular, the last two equalities (2.14) give the relation 2α5 α6 = α32 (α3 + α4 ). Substitution of all these relations into the last expression (2.13) yields the equality det α = 0, which is impossible. Thus, (2.15) is equivalent to p3 = 0, and since the roots of (2.16) are distinct, the case (iii) is impossible (non-feasible) for the Lamé system. We now turn to the case of (ii) of a multiple root and suppose, for example, that the pair {p2 , p3 } is linearly dependent; p3 = λp2 with some λ = 0. Then p2 (ν) = p3 (ν) = 0 and χ = p1 p2 − p32 = p2 (p1 − λ2 p2 ). Since the root ν of the polynomial p2 is simple, then (p1 − λ2 p2 )(ν) = 0, so that pi (ν) = 0 for all i = 1, 2, 3. But then we have the case (iii), which, as shown above, is non-feasible.  

On Elliptic Systems of Two Equations on the Plane

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It follows from Lemma and (2.14) that for the Lamé system in Theorem 1.3 it is necessary to use relations (1.10a) if the polynomials p2 and p3 are linearly independent, and by relations (1.10b) if the polynomials p1 and p3 are linearly independent, if necessary, put p4 = p3 in them. In the case when (2.15) is satisfied, in accordance with the remark to Theorem 1.3, we can take as b the identity matrix. All these expressions for the matrix b (in a slightly different formulation) are listed in [23]. In the case of an orthotropic medium case which that corresponds to α5 = α6 = 0, the characteristic equation χ(z) = 0 is biquadratic and its roots ν are explicitly written out, and the expressions for the matrix b are somewhat simplified [23]. We have an even greater simplification in the isotropic case, when in addition α1 = α2 = 2α3 + α4 . In this case we can put  b=

1 0 i −æ



with the constant æ = (α1 + α3 )/(α1 − α3 ) > 1. We note that when it comes to the study of boundary value problems plain elasticity by classical methods, two main directions. The first of them consists in the use of analytical functions on analogy with the Kolosov–Muskhelishvili formulas [12] in the isotropic case. This direction is represented by the works of S.G. Lechnitsky G.N. Savina, S.G. Mikhlin and others (see, for example, [12, 24]). The second direction relies in using instead of analytic functions the solutions of certain first order elliptic systems (see, for example, [25–27]). The approach considered below adheres to this second direction and is based on the system (2.2), more precisely, on the representation of the general solution of the Lamé system in terms of J -analytic functions.

3 Generalized Double Layer Potentials Since the boundary value problems for elliptic systems reduce to systems of singular equations on the boundary, these problems are usually considered in the Holder classes C μ . In the case of using the spaces hp (D) this reduction leads to systems of singular equations in the class Lp (), which is also well adapted for these systems. In a similar way, there are a priori Schauder estimates [14], which are also established in Holder or Sobolev spaces with summability degree p > 1. The consideration of the spaces C(D) was made due thanks to the use of generalized potentials of the double layer, which act according to the formula 1 (P ϕ)(z) = π

 

Re [n(t)(t − z)] H (t − z)ϕ(t)d1 t, |t − z|2

z ∈ D,

(3.1)

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where H (ξ ) is a homogeneous of degree zero (2 × 2)-matrix depending only on the coefficients of system (1.1), n = n1 + in2 is the unit outer normal and the vectorvalued function ϕ ∈ Lp () is real. We note that for H = 1 this formula becomes an equality that determines the classical potential of the double layer for the Laplace operator. In the notation (1.8) and Lemma 1.1, the matrix H is defined by the equality H (ξ ) = Im [b(iξ )J ξJ−1 b−1 ],

ξ = ξ1 + iξ2 ,

(3.2)

by virtue of the second part of Lemma 1.1 this definition doesn’t depend on the choice of the matrix b. The matrix (iξ )J ξJ−l can be considered as a value of the matrix J analytic function h(ξ, z) =

−ξ2 + ξ1 z , ξ1 + ξ2 z

Imz > 0.

(3.3)

In an explicit form, and in accordance with the three cases in (1.3), it is given by  (i) h(ξ, J ) =  (ii) h(ξ, J ) =

0 h(ξ, ν1 ) 0 h(ξ, ν2 )

h(ξ, ν) |ξ |2 (ξ1 + νξ2 )−2 0 h(ξ, ν)



 ,

(iii) h(ξ, J ) = h(ξ, ν).

One can show [13] that 3 (P ϕ)(z) = Re

1 πi

 b(t 

−1 − z)−1 J dtJ b ϕ(t)

4 ,

(3.4)

therefore the expression in square brackets is an J -analytic function and, on the basis of Theorem 1.2, the function u = P ϕ is indeed a solution of system (1.1). The Sochocki–Plemel formula for Cauchy type integrals holds, therefore, under the conditions of Theorem 2.1, for the boundary values of the function u = P ϕ the formula (P ϕ)+ = ϕ + Kϕ,

(3.5)

holds, where the operator (Kϕ)(t0 ) is obtained from (3.4) by replacing z ∈ D by t0 ∈ . By virtue of Theorem 2.1(b), the operator P is bounded Lp () tohp (). Moreover, the operator K is compact in space in Lp (), since its kernel has a weak singularity. In fact, the generalized potential P ϕ has the same boundary properties as the classical double layer integral for the Laplace operator. Namely, the following result is established in the same way as in Lemma 6.1 from [23].

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Lemma 3.1 Let  ∈ C 1,ν , 0 < μ < ν, and X denote any of the symbols C, C μ , C 1,μ . Then the operator P is bounded X() toX(D), and the operator K is compact in X(). The question as to the representation of an arbitrary solution u ∈ hp (D) of a weakly connected system (1.1) with a generalized generalized potential P ϕ then arises. This question is solved in [13] for the general case of (l timesl)-systems. We first note that according to (3.4), only those solutions u ∈ hp (D) of the system (1.1) can be represented as a generalized potential P ϕ, in the representation (1.9) for which J -analytic function φ is single-valued (and vanishes on ∞ if p the domain D is infinite). Denote by h0 (D) the class of all such solutions. There exists such a finite dimensional space U0 (D) ⊆ C ∞ (D) such that  h (D) = U0 (D) p

p ⊕ h0 (D),

dim U0 (D) =

2(m − 1), D, 2m, D,

(3.6)

where m is the number of connected components of . This subspace is constructed as follows: [23]. Consider the matrix Y (z) = ln zJ as the value of the analytic function ln w from the matrix zJ . In explicit form and, in accordance with the three types of matrix J , it is given by equalities  (i) Y (z) =  (ii) Y (z) =

 0 ln(x + ν1 y) , 0 ln(x + ν2 y)

ln(x + νy) y(x + νy)−1 0 ln(x + νy)

 ,

(iii) Y (z) = ln(x + νy).

In every simply connected subdomain D0 ⊆ C \ 0, the vector-valued function Y (z)η, η ∈ C2 , is J -analytic, and by traversing the point z = 0 it gets an increment of 2πiη, in particular, for η ∈ R2 the real vector-function u = Re Y η is singlevalued. Let 1 , . . . , m be simple contours that form the boundary contour  of the domain Dj , 1 ≤ j ≤ m, with boundary ∂Dj = j and let these form a complement

be an infinite component of D = C \ D. In the case of a finite domain D, let Dm

for definiteness. In each domain, we choose Dj , 1 ≤ j ≤ m, at the point aj . If the domain D is finite, then in this notation the class U0 (D) is defined as the class of all solutions of the system (1.1) of the form u(z) = Re

m−1 j =1

Y (z − aj )ξj ,

ξj ∈ R2 , 1 ≤ j ≤ m − 1.

If the domain D is infinite, then then this expansion is replaced by u(z) = ξm + Re

m−1 j =1

[Y (z − aj ) − Y (z − am )]ξj ,

ξj ∈ R2 , 1 ≤ j ≤ m.

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Here it is taken into account that the matrix-valued functions Y (z − aj ) − Y (z − am ) are single valued in a neighborhood of ∞ and tend to zero as z → ∞. It is not difficult to see that the so-defined space U0 satisfies conditions (3.6). We also introduce the space V0 () of real 2-vector-valued functions ϕ that are constant on the contours j , and in the case of a finite domain D they vanish on the outer contour m . It is clear that the dimensionality k0 of this space coincides with dim U0 . Theorem 3.2 Suppose that the contour  = ∂D belongs to the class C 1,ν and consists of the connected components 1 , . . . , m . Let the domains Dj , 1 ≤ j ≤ m with boundary ∂Dj = j constitute the complement D = C \ D. To ensure

is infinite. Finally, definiteness, in the case of a finite domain D, the component Dm let sj be the dimension of the solution space of the weakly connected system (1.1) of the homogeneous Dirichlet problem in the domain Dj and k = k0 + s1 + . . . + sm , where k0 = dim U0 = dim V0 . Then there exist linearly independent systems of real 2-vector-valued functions g1 , . . . , gk ∈ C() and solutions u1 , . . . , uk inC 1,μ (D), 0 < μ < ν, the systems (1.1), the first k0 of which form the bases, respectively, U0 and V0 , such that any solution u ∈ hp (D) of this system is uniquely representable in the form u = Pϕ +

k 1

 (ϕ, gj )uj ,

(ϕ, g) =

ϕ(t)g(t)d1 t,

(3.7)



with some real l-vector-valued function ϕ ∈ Lp (). If in this representation the function u belongs to the class X(D), where X means any of the symbols C, C μ , C 1,μ , then also ϕ ∈ X(). It follows immediately from this theorem together with (3.5) that the Dirichlet problem (1.1), (2.1) is reduced to an equivalent system of integral Fredholm equations ϕ + Kϕ +

k 1

(ϕ, gj )u+ j = f.

The connection between the solutions of the problem and this equation is established by the equality (3.7). We also note that in the case of strongly elliptic systems or elliptic systems equivalent to them, and on the basis of Theorem 2.2 the spaces U and V in Theorem 2.3 can be replaced by, respectively, U0 and V0 . Using Theorem 1.3 for the matrix H , we can give unified explicit expressions that depend only on the coefficients of the polynomials pj and combinations s = ν1 + ν2 ,

t = ν1 ν2 ,

(3.8)

of roots ν ∈ σ . Obviously, in the case of multiple roots s = 2ν and t = ν 2 . To this end, we introduce the following quadratic forms ω(ξ ) = ξ12 + sξ1 ξ2 + tξ22 ,

2ω1 (ξ ) = sξ12 + 2tξ1 ξ2 + s¯tξ22 .

(3.9)

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Note that ω(ξ ) = (ξ1 + ν1 ξ2 )(ξ1 + ν2 ξ2 ), and for ν1 = ν2 = ν the function |ω(ξ )| is also quadratic form ξ12 + 2(Re ν)ξ1 ξ2 + |ν|2 ξ22 . It is convenient to associate with each pair of polynomials f, g the bilinear forms [f, g] =

f (ν1 )g(ν2 ) − f (ν2 )g(ν1 ) , ν1 − ν2

{f, g} =

f (ν1 )g(ν2 ) + f (ν2 )g(ν1 ) , 2 (3.10)

which for νj → ν readily transforms into [f, g] = f (ν)g(ν) − f (ν)g (ν),

{f, g} = f (ν)g(ν).

These forms are expressed in terms of the coefficients of the polynomials f =   i and g = i by the formulas α z β z i i i≥0 i≥0 [f, g] = {f, g} =





i>j (αi βj

− αj βi )[zi , zj ],

i j i>j (αi βj + αj βi ){z , z } +



(3.11) i i i αi βi {z , z },

where it is taken into account that the first of them is skew-symmetric, and the second one is symmetric. As regards the quantities [zi , zj ] and {zi , zj }, they are symmetric as functions of νj and, therefore, are explicitly expressed in terms of combinations (3.8). For example, [z, 1] = 1, [z2 , 1] = s, [z2 , z] = t, {1, 1} = 1, {z, 1} = s/2, {z, z} = t, 2 {z , 1} = (s 2 − 2t)/2, {z2 , z} = st 2 /2, {z2 , z2 } = t 2 .

(3.12)

We note that in the notation for the determinant of the matrix b, which appears in (1.10), we have the corresponding expression (i) det b = (ν1 − ν2 )[p4 , p2 ],

(ii) det b = [p4 , p2 ],

if the polynomial pair {p4 , p2 } is linearly independent, and the expression (i) det b = (ν1 − ν2 )[p1 , p3 ],

(ii) det b = [p1 , p3 ],

if the pair {p1 , p3 } is linearly independent. If both these pairs are linearly dependent and in the relations (1.11) the roots ν1 = ν2 , then, according to (1.12), the determinant det b = δ1 δ2 − δ3 δ4 . Finally, in the case when all pj (ν) = 0, that is, in the case of (iii), we can put b = 1. We recall that by hypothesis the system (1.1) is weakly connected, so that all these determinants are nonzero.

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Lemma 3.3 In the notation of (3.9), the matrix homogeneous of degree 2 is defined by the equality   1 −{p2 , p2 }ω [p4 , p2 ]ω1 − {p4 , p2 }ω , G = Im [p4 , p2 ]ω1 + {p4 , p2 }ω {p4 , p4 }ω [p4 , p2 ] (3.13a) if the pair {p4 , p2 } is linearly independent and by the equality   1 −{p3 , p3 }ω [p1 , p3 ]ω1 − {p1 , p3 }ω G = Im , [p1 , p3 ]ω1 + {p1 , p3 }ω {p1 , p1 }ω [p1 , p3 ] (3.13b) if the pair {p1 , p3 } is linearly independent. If both these pairs are linearly dependent, and in the relations (1.11) the roots ν1 = ν2 , then   1 γ1 ω1 + γ2 (ν1 − ν2 )ω γ3 (ν1 − ν2 )ω , (3.14) G = Im −γ4 (ν1 − ν2 )ω γ1 ω1 − γ2 (ν1 − ν2 )ω γ1 where γ1 = δ1 δ2 − δ3 δ4 , γ2 = (δ1 δ2 + δ3 δ4 )/2, γ3 = δ2 δ3 , γ4 = δ1 δ4 . Finally in the case (iii) G(ξ ) = (Im ν)|ω(ξ )| = (Im ν)[ξ12 + 2(Re ν)ξ1 ξ2 + |ν|2 ξ22 ].

(3.15)

Proof In the case (iii), the matrix b = 1 and formula (3.15) follow immediately from H (ξ ) = Im h(ξ, ν) =

|ξ |2 Im ν . |ξ1 + νξ2 |2

In the cases (i) and (ii), by Theorem 8.1 from [23] we have the equality G(ξ ) = Im [ω1 (ξ ) + ω(ξ )(bb−1)],

(3.16)

where parallel to the two cases (i) and (ii) we let (i)  =

ν1 − ν2 2



 1 0 , 0 −1

 (ii)  =

 01 . 00

Substituting the expressions (1.10) here, we arrive at the expressions (3.13) in the same way as in Lemma 8.1 from [23]. Further, let both pairs {p4 , p2 } and {p1 , p3 } be linearly dependent, and in the relations (1.11) the roots ν1 = ν2 . Then the substitution of (1.12) into (3.16) yields equality (3.14).  

On Elliptic Systems of Two Equations on the Plane

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In terms of the matrix G, the generalized potential (3.1) takes the form 1 (P ϕ)(z) = π

 

Re [n(t)(t − z)] G(t − z)ϕ(t)d1 t, |ω(t − z)|2

z ∈ D.

(3.17)

Formulas (3.11), (3.12) for the values of bilinear forms of the polynomials pj appearing in (3.13) lead, in general, to cumbersome expressions. For example, for the Lamé plane theory of elasticity, for which the polynomials pj are defined (2.14), the indicated values are given by the equalities [23] [p3 , p2 ] = α3 (α3 + α4 ) − 2α5 α6 + (α3 α5 − α2 α6 )s + [2α52 − α2 (α3 + α4 )]t, {p2 , p2 } = α32 + 4α52 t + α22 t 2 + 2α3 α5 s + α2 α3 (s 2 − 2t) + 2α2 α5 st, {p3 , p3 } = α62 + (α3 + α4 )2 t + α52 t 2 + (α3 + α4 )α6 s + α5 α6 (s 2 − 2t)+ +(α3 + α4 )α5 st, 2{p2 , p3 } = 2α3 α6 + [2α5 α6 + α3 (α3 + α4 )]s + 4(α3 + α4 )α5 t+ +(α2 α6 + α3 α5 )(s 2 − 2t) + [α2 (α3 + α4 ) + 2α52 ]st + 2α2 α5 t 2 . In the case of an orthotropic medium, when α5 = α6 = 0, for the numbers (3.8) we have the expressions s = iρ0 , t = −ρ 2 with positive constants @ ρ = 2

α1 , α2

ρ02

√ α1 α2 − α42 + 2α3 ( α1 α2 − α4 ) = . α2 α3

Accordingly, the above formulas for the matrix G( xi) are substantially simplified [23]: ρ0 G(ξ ) = √ α3 + α1 α2



ρ 2 (α2 ξ12 + α3 ξ22 ) (α3 + α4 )ξ1 ξ2 ρ 2 (α3 + α4 )ξ1 ξ2 α3 ξ12 + α1 ξ22

 .

This formula is further simplified in the case of an isotropic medium, when the relations α1 = α2 = 2α3 + α4 hold. In this case, ρ = 1, ρ0 = 2, we have a multiple root ν = i and the inequality α1 > α3 . As a result, we arrive at the equality 1 G(ξ ) = æ



2ξ1 ξ2 æ|ξ |2 + ξ12 − ξ22 2 2ξ1 ξ2 æ|ξ | + ξ22 − ξ12



with constant æ = (α1 + α3 )/(α1 − α3 ). Since in the considered case ω(z) = |z|2 , the formula (3.17) becomes 1 (P ϕ)(z) = π

 

with the indicated matrix G.

Re [n(t)(t − z)] G(t − z)ϕ(t)d1 t, |t − z|4

z ∈ D,

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References 1. A.P. Soldatov, On the first and second boundary value problems for elliptic systems on the plane. Differentcial’nye Uravneniya 39, 674–686 (2003). English transl. in Differ. Equ. 39 (2003) 2. A. Douglis, Function theoretic approach to elliptic systems of equations in two variables. Commun. Pure Appl. Math. 6, 259–289 (1953) 3. J. Horvath, A generalization of the Cauchy–Riemann equations. Contrib. Differ. Equ. 1, 39–57 (1961) 4. B. Bojarski, Theory of generalized analytic vector. Ann. Polon. Math. 17, 281–320 (1966) 5. R. Gilbert, J. Buchanan, First Order Elliptic Systems: A Function Theoretic Approach (Academic, New York, 1983) 6. R. Gilbert, G. Hile, Generalized hypercomplex function theory. Trans. Am. Math. Soc. 195, 1–29 (1974) 7. G. Hile, Elliptic systems in the plane with order term and constant coefficients. Commun. Pure Appl. Math. 3, 949–977 (1978) 8. H. Begehr, R. Gilbert, Pseudohyperanalytic functions. Complex Variables: Theory Appl. 9, 343–357 (1988) 9. A.V. Bitsadze, Boundary Value Problems for Second Order Elliptic Equations (Nauka, Moskow, 1966) 10. A.P. Soldatov, A function theory method in boundary value problems 1n the plane. I: the smooth case. Izv. Akad. Nauk SSSR Ser. Mat. 55, 1070–1100 (1991) 11. N.E. Tovmasyan, The Dirichlet problem for elliptic differential equations of the second order. Sov. Math. Dokl. 160, 1275–1278 (1964) 12. N.I. Muschelishvili, Singular Integral Equations (Noordhoff, Groningen, 1953) 13. A.P. Soldatov, The Dirichlet problem for weakly connected elliptic systems on the plane. Differentcial’nye Uravneniya 49, 674–686 (2013) 14. S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Commun. Pure Appl. Math. 17, 35–92 (1964) 15. G.M. Goluzin, Geometric Theory of Functions of a Complex Variable (American Mathematical Society, Providence, 1969) 16. V.I. Vlasov, A.V. Rachkov, On weighted spaces of Hardy type. Dokl. RAN 328, 281–284 (1993) 17. A.P. Soldatov, Hardy space of solutions of elliptic systems on the plane. J. Math. Sci. 160, 103–115 (2009) 18. M.I. Visik, On strongly elliptic systems of differential equations. Matem. sbornik 29, 615–676 (1951) 19. H.L. Keng, L. Wey, W. CiQuian, Second Order Systems of Partial Differential Equations in the Plane (Pitman (Advanced Publishing Program), Boston, 1985), 292 pp. 20. D.S. Kuai, In the definition of the second order elliptic systems of partial differential equations with constant coefficients. Acta Math. Sin. 10, 276–287 (1960) 21. A.P. Soldatov, Second order elliptic systems in half-plane. Izv. RAN. (Seriya Matem.) 70, 161– 192 (2006) 22. C. Somigliana, Sui sisteme simmetrici di equazioni a derivate parziali. Ann. Math. Pure Appl. II 22, 143–156 (1894) 23. A.P. Soldatov, Generalized potentials of double layer in plane theory of elasticity. Evrasian Math. J. 5, 78–125 (2014) 24. A.H. England, Complex Variable Methods in Elasticity (Wiley-Interscience, London, 1971) 25. R.P. Gilbert, W.L. Wendland, Analytic, generalized, hyperanalytic function theory and an application to elasticity. Proc. R. Soc. Edinb. 73A, 317–371 (1975)

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Real Variable Inverse Laplace Transform Vu Kim Tuan, A. Boumenir, and Dinh Thanh Duc

Dedicated to the 80th anniversary of Prof. Heinrich Begehr

Abstract The aim of this work is to provide a review of authors’ contributions to the field of the Laplace transform in the last 20 years. Keywords Inverse Laplace transform · Dirichlet series · Hardy space Mathematics Subject Classification (2010) Primary 44A10; Secondary 30B50, 30H10, 65M32

1 Bromwich Inverse Formula Let f be an arbitrary function defined on the interval R+ = (0, ∞). The integral 

∞

Lf (s) = F (s) =

e−st f (t)dt,

(1.1)

0

if it exists, is called the Laplace transform of f , [15, 40]. If f is locally integrable and does not grow faster than an exponential function at infinity, then there exists a number d0 , −∞ ≤ d0 < ∞, such that the integral (1.1) converges for all s with '(s) > d0 , and diverges for all s with '(s) < d0 . The number d0 is called the minimal abscissa of convergence. The function F (s) is an analytic function in the right half-plane '(s) > d0 , of the order o(2(s)) as s → ∞ in any half-plane '(s) ≥ d > d0 , and has at least one singular point on the line '(s) = d0 . V. K. Tuan () · A. Boumenir Department of Mathematics, University of West Georgia, Carrollton, GA, USA e-mail: [email protected]; [email protected] D. T. Duc Department of Mathematics, Quy Nhon University, Quy Nhon, Binh Dinh, Viet Nam © Springer Nature Switzerland AG 2019 S. Rogosin, A. O. Çelebi (eds.), Analysis as a Life, Trends in Mathematics, https://doi.org/10.1007/978-3-030-02650-9_15

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The Laplace transform occurs frequently in applications of mathematics, especially in those areas involving differential or integral equations, and in many other problems. The original f can be recovered by the Bromwich contour integral [15, 40] 1 f (t) = 2πi



d+i∞

est F (s)ds,

d > d0 ,

(1.2)

d−i∞

if f is continuous at t. Such a d is called an abscissa of convergence. The problem of computing the inverse Laplace transform of a function is crucial in pure and in applied mathematics as well. Since the function est is oscillatory on the contour (d − i∞, d + i∞), special numerical methods are required to compute the integral (1.2). Many computer codes to approximate (1.2) have been developed [1, 3, 5, 13, 17, 18, 20, 23, 24, 28, 38]. For a comparison of many of these methods see [14, 16]. However, none of these codes is suitable for automatic inversion. Indeed, an automatic code is intended to run without the help of the user. However all of these codes require an abscissa of convergence d to be supplied by the users a priori, and this in general is not an easy task. Take, for example, f (t) =

    at 1 at cosh sin . t 2 2

Then [26], F (s) = arctan

 a  + 1 + arctan −1 , s s

a

and d0 =

1 (|'(a)| + |2(a)|). 2

As one can see in this example d0 cannot be easily predicted by users without a priori analysis. In reality many users overlook the importance of abscissae of convergence d and d0 . If f is absolutely integrable, or square integrable on (0, ∞), then d0 ≤ 0, and therefore, d can be chosen to be any small positive number. But if f grows exponentially fast at infinity, then d0 > 0, and the problem of automatic determination of a suitable d has been looked into in [32]. An algorithm to find an abscissa of convergence d > d0 , but close to d0 , is proposed in [32]. Once an abscissa of convergence d that is a good approximation to the exact abscissa of convergence d0 is found, one can apply one of the codes described in the cited papers to evaluate the inverse Laplace transform (1.2). The algorithm is based on the following observation:

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For any d > d0 we have [26] 1 2πi



d+i∞

d−i∞

F (s) st 1 e ds = sk (k)



t

(t − y)k−1 f (y)dy,

k ≥ 1.

(1.3)

0

Putting t = 0 in (1.3), we get 1 I (d) := 2πi



d+i∞ d−i∞

F (s) ds = 0, sk

k ≥ 1.

(1.4)

Hence, if d > d0 the integral (1.4) equals 0. If d ≤ d0 , then in the strip d ≤ '(s) ≤ d0 there should be some singular points of F (s), otherwise d0 is not the minimal abscissa of convergence. Therefore, the integral (1.4) in this case is in general not equal to 0. Thus, equality of the integral in (1.4) to zero is necessary for d to be an abscissa of convergence. But it is not a sufficient condition. If d is not an abscissa of convergence, it may still happen that the integral (1.4) vanishes. It is the case when the sum of the residues of all the poles in the strip d ≤ '(s) ≤ d0 is zero. To get a numerically sufficient condition, we consider instead of the integral (1.4) the modified integral  J (d) :=

d+i∞ d−i∞

s 2 (s

F (s) ds, + ih)F (d + ih)

(1.5)

where h is a real number, randomly selected by the program. The role of the divisor F (s) F (d +ih) will be explained later. Since s+ih is the Laplace transform of the function + t e−iht 0 f (y)eihy dy (see [26]), the integral (1.5) equals zero if d is an abscissa of convergence. Because h is chosen randomly, the probability that the sum of the F (s) residues of all the poles of the function s 2 (s+ih) in the strip d ≤ '(s) ≤ d0 equals 0 is practically zero. Therefore, the equality of the integral (1.5) to zero practically guarantees that d is an abscissa of convergence. The Bromwich contour integral (1.2) may converge slowly due to the oscillatory nature of the term est , and therefore requires special numerical integration techniques to speed up convergence. The integral (1.5) does not have the oscillatory term est , and since the function F (s), as an image of the Laplace transform of a function, is of the order o(2(s)) as 2(s) tends to ±∞, the integral (1.5) converges faster. Hence, numerically speaking, the integral (1.5) is therefore easy to compute (for example, by a Gaussian quadrature). Observe that J (d) is a piecewise constant function, with J (d) = 0 for d < d0 , and J (d) = 0 for d > d0 . To find d close to d0 , say d − d0 < q for some tolerance error q, we find d1 with 0 < d − d1 < q, such that J (d) = 0, but J (d1 ) = 0. In that case d1 ≤ d0 < d, and therefore, d − d0 < q. It can be done by using, say, interval bisection. Because of floating-point arithmetic, in order to compare J (d) with zero, the normalizing factor F −1 (d + ih) is introduced, and some error tolerance * should be specified: J (d) is assumed to be zero if |J (d)| ≤ *. This tolerance error * depends both on the computer’s precision and on the numerical integration code being used.

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Algorithm Input parameters: The user provides parameters Bound, q, * > 0. An abscissa of convergence d is sought in the interval (0, Bound). Bound—the maximum range of abscissa of convergence allowed: d ≤ Bound. q—the tolerance error for d: d − d0 < q if d0 ≥ 0, and d < q if d0 < 0. For most of applications we can select q = 1. *—the tolerance error for numerical integration. The algorithm consists of the following steps: Step 1. Select h ∈ (1, 2) randomly. Put d = Bound and d1 = 0. Step 2. If |J (Bound)| > *, exit: either the abscissa of convergence exceeds the maximum range Bound allowed, or F is not a Laplace transform of any function. Step 3. Put d2 = (d + d1)/2. If |J (d2)| ≤ *, put d = d2. Otherwise, put d1 = d2. Step 4. If d − d1 ≥ q, go to Step 3. Otherwise, exit: d is an abscissa of convergence in the interval (d0 , d0 + q), if d0 ≥ 0, and in the interval (0, q), if d0 < 0. As it can be seen, if d0 is large, the integral (1.5) has to be evaluated for several d, and the algorithm becomes expensive. Still, the main cost of the computation is due to the evaluation of the inverse Laplace integral (1.2), since integral (1.5) can be computed with machine accuracy, while integral (1.2) for large s0 should be computed with much higher accuracy (for example, with double precision), and several times for different t ∈ [0, T ].

2 Hardy Space Let H2 (C+ ) denote the Hardy space in the right-half plane [21, 31], i.e. F ∈ H2 (C+ ) if, and only if, F is analytic in the right-half plane '(s) > 0, and A 1 2π

F H2 (C+ ) := sup

x>0



∞ −∞

|F (x + iy)|2 dy < ∞.

It is well-known [21, 31] that F ∈ H2 (C+ ) if, and only if, F (s) is the Laplace transform (1.1) of a function f ∈ L2 (R+ ). Moreover, F H2 (C+ ) = f L2 (R+ ) . Bounded periodic and almost periodic functions do not belong to L2 (R+ ), but have a bounded averages property 1 sup T >0 T

 0

T

|f (t)|2 dt < ∞.

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In [30] a similar description of the Laplace transform of functions of bounded averages has been obtained Theorem 2.1 F is the Laplace transform of a function f of bounded averages if and only if F (s) is analytic in the right-half plane '(s) > 0, and  sup x x>0

∞

−∞

|F (x + iy)|2 dy < ∞.

In [9] the question whether the sequence {s0 + an}n≥0 , ' (s0 ) > 0, a > 0, is a set of uniqueness in the sense that F (s) ∈ H2 (C+ ) can be uniquely recovered from its values at these points has been answered affirmatively. Namely, it was proved that if F1 , F2 ∈ H2 (C+ ) , and F1 (s) = F2 (s) at {s0 + an}n≥0 , then F1 (s) = F2 (s) for '(s) > 0. The next question is: What would be the sampling formula? In other words, can we find a sequence of sampling functions Cn (s) such that a similar Shannon sampling formula as in [31] holds for any F ∈ H2 (C+ ) , F (s) =



F (s0 + an)Cn (s),

(2.1)

n≥0

where the series converges uniformly on any compact domain in the right half plane. This would extend the celebrated Shannon sampling theorem from the Paley-Wiener space to the Hardy space. Unfortunately, the answer to this question is negative [9]. However, the following theorem provides not only a sampling formula for the Hardy space, but also a characterization of the Hardy space, has been obtained in [9, 31]. Theorem 2.2 Let F ∈ H2 (C+ ). Then F (s) =

∞  k=0

    k (2k + 1) 12 − s  1 (−k)n (k + 1)n k   F n+ , (n!)2 2 s + 12 n=0

(2.2)

k+1

where the series converges uniformly on any compact subset of the right half plane, and  k  2 ∞   1  (−k)n (k + 1)n  F n+   < ∞.  (n!)2 2  k=0 n=0

Conversely, if {fn } is a sequence of complex numbers such that  k 2 ∞   (−k)n (k + 1)n   fn  < ∞,    (n!)2 k=0 n=0

(2.3)

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then the series   k ∞ (2k + 1) 1 − s   2 (−k)n (k + 1)n k   fn , 1 (n!)2 s+2 k=0 n=0 k+1

converges uniformly on any compact subset of C+ to a function F (s) ∈ H2 (C+ ), and moreover   1 F n+ = fn 2 for any n ∈ N0 . above formula can be extended to Here (a) : k := a(a + 1) · · · (a + k − 1). The  ? Ea := F : f ∈ L1,loc (R+ ) and f (t) = O eat , where a ≥ 0 [9]. Theorem 2.3 For any F ∈ Ea , we have uniform convergence in any compact domain contained in '(s) > b > a + 12 , F (s) =

∞ k  (2k + 1) (b − s)k  (−k)n (k + 1)n F (n + b) . (n!)2 (s + 1 − b)k+1 k=0

(2.4)

n=0

For truncation errors we have [9] Theorem 2.4 If f ∈ L1 (R+ ) then the truncation error for ' (s) >

1 2

is given by

       k N (2k + 1) 1 − s    2 1  (−k)n (k + 1)n  k   F n+ F (s) −  1  n!n! 2  s+2 k=0 n=0   k+1

≤

c N s−1

f 1 .

If f ∈ L∞ (R+ ) then for ' (s) > b > 12 ,   k N    (2k + 1) (b − s)k  (−k)n (k + 1)n   F (n + b) F (s) −   n!n! (s + 1 − b)k+1 k=0

c ≤ f ∞ . bN s−1

n=0

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If f ∈ L2 (R+ ) then for ' (s) > b > 12 ,   k N    (2k + 1) (b − s)k  (−k)n (k + 1)n   F (n + b) F (s) −   n!n! (s + 1 − b)k+1 k=0

≤√

c 2bN s−1

n=0

f 2 .

The constant c is the same for all of the above formulas. The sampling formula (2.2) has also been used to compute eigenvalues of singular Sturm-Liouville problems and to interpolate the Titchmarsh-Weyl function [6–8, 27]. Another interpolation formula for Hardy functions at n arbitrary points pk , k = 1, 2, · · · , n, follows from [12, 35]. Theorem 2.5 Let n % 2

Fn (s)=

n  k,l=1

j =1

F (pk )

(pk +pl )

(pk + p j )(p l + pj )

n 2 j =1,j =k

(pk −pj )

&

n 2 j =1,j =l

(p l −pj )

1 , s + pl

(2.5)

Then Fn (s) ∈ H2 (C+ ), and Fn (pk ) = F (pk ) for k = 1, · · · , n. Moreover, Fn (s) is a minimal approximation of F (s) in the sense that among functions of the Hardy space H2 (C+ ) that attain the values F (pk ) at pk , k = 1, 2, · · · , n, it has the minimal norm in H2 (C+ ). The following theorem deals with the convergence of this interpolation formula [35]. Theorem 2.6 Let {pk }∞ k=1 be a sequence of distinct numbers on C+ , that is either convergent to p ∈ C+ , or pk = αk + β for some α, β > 0, and any k > 0. Then lim Fn (s) = F (s),

n→∞

'(s) > 0,

(2.6)

where the convergence is both pointwise and in H2 (C+ ) norm. As first example take pk = k − 12 . Then

lim

n→∞

n 

(−1)k+l (k + l − 1)

k,l=1

    n+k−1 n+l−1 k+l−2 2 n−l n−k k−1   F k − 12 × = F (s), '(s) > 0, s + l − 12

where the convergence is both pointwise and in H2 (C+ ) norm.

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Take now pk = k + α/2 with α > −2. Then    n+k+α n+l+α lim (−1) (k + l + α) n→∞ n−l n−k k,l=1    k + l + α − 1 k + l + α − 1 F (k + α/2) = F (s), × s + l + α/2 k−1 l−1 n 

k+l

'(s) > 0,

where the convergence is both pointwise and in H2 (C+ ) norm. 1 α Choose now pk = + , then 2 k n 2

lim

n→∞

n 

(−1)k+l

k,l=1

j =1



(αj k + j + k)(αj l + j + l)

(αkl + k + l)k!(n − k)!l!(n − l)!

·

F s



1 α k + 2 + 1l + α2

= F (s),

where the convergence is both pointwise and in H& 2 (C+ ) norm. The last formula recovers F from data on a finite interval α2 , α2 + 1 .

3 Inversion by Pseudo-Differential Operators The Laplace transform can also be seen as a self-adjoint integral operator acting in the Hilbert % √ space √ L&2 (R+ ) . In fact its spectrum is simple and continuous, and is given by − π, π . Thus although the Laplace transform is a bounded operator, the presence of zero in its spectrum makes its inverse an unbounded and densely defined operator. Since f ∈ L2 (R+ ) if and only if F ∈ H2 (C+ ) , we deduce that the domain of the inverse Laplace transform operator is the restriction of H2 (C+ ) on R+ . Using a simple transformation one can express the Laplace transform in term of differential operators. To this end recall that 

∞

L (f ) (s) = 2

0

1 f (t)dt, s+t

(3.1)

and using the isometry V : L2 (R+ ) → L2 (R) defined by   V y(t) = et /2 y e2t , we obtain a convolution operator acting in L2 (R) VL V 2

−1

 g(x) =

∞ −∞

k (x − t) g(t)dt,

(3.2)

Real Variable Inverse Laplace Transform

311

where 1

k(x) =

2 cosh

x , 2

π and whose Fourier transform is . Thus L2 is similar to the multiplication cosh (πλ) π , and is a bounded operator. Its inverse is then given by by cosh (πλ) V L−2 V −1 =

1 cos (πD) , π

where D is the differentiation operator which yields an expression for L−1 . In [4] the following formula was proved.  Theorem 3.1 Let F ∈ H2 (C+ )R+ , then f =

1 −1 V cos (πD) V LF. π

4 Post-Widder Inverse Formula If the Laplace transform F is given only on R+ , that is the case when the Laplace transform (1.1) is considered as an integral equation of the first kind, one should use an inverse formula involving only the values of F on R+ . A first formula of this kind was introduced by Post [25] and Widder [39, 40] f (t) = lim f˜n (t), n→∞

(−1)n  n n+1 (n)  n  . f˜n (t) = F n! t t

(4.1)

Using (4.1) Widder [40] obtained the following characterization of the Laplace transform on R+ : A function F on R+ is the Laplace transform of f ∈ L∞ (R+ ) if and only if F is infinitely differentiable and satisfies    1 n+1 (n)   F (s) < ∞. sup  s s>0, n∈N0 n! In [33] it was shown that Theorem 4.1 If f is differentiable, f (t), tf (t) ∈ L2 (R+ ), then f˜n converges to f

in L2 (R+ ) norm with the rate O(n−1/2 ). If, moreover, t 2 f (t) ∈ L2 (R+ ), then the −1 error has the order O(n ).

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Another real variable inverse formula for the Laplace transform, namely f (t) = lim fˆn (t), n→∞

fˆn (t) =

31 43  4 n  n n t d F , 1+ k dt t t

(4.2)

k=1

has been considered in [34], and a real variable inverse formula for the bilateral Laplace transform has been studied in [41]. Formulas (4.1) and (4.2) reconstruct the original f by means of the derivatives of high order of its Laplace transform F on the positive real axis. In [34] it was shown that if f is differentiable, f (t), tf (t) ∈ L2 (R+ ), then fˆn converges to f in L2 (R+ ) norm with the rate

O(n−1/2 ). If, moreover, t 2 f (t) ∈ L2 (R+ ), then the error has the order O(n−1 ). We recall now a space of functions M−1 c,γ (L2 ) ⊂ L2 (R+ ) [29, 37]. Let −1 2 signc + signγ ≥ 0. By Mc,γ (L2 ) we denote the subset of L2 (R+ ), consisting of all functions f such that f (t) =

1 2πi



1/2+i∞

f ∗ (s)t −s ds,

1/2−i∞

where f ∗ (s)s γ eπc|s| ∈ L2 (1/2 − i∞, 1/2 + i∞). The space M−1 c,γ (L2 ), equipped with the norm f (t)M−1 = f ∗ (s)L2 ((1/2−i∞,1/2+i∞);|s|2γ e2π c|s| ) c,γ (L2 ) is a Banach space. −1 If c = γ = 0, then M−1 0,0 (L2 ) = L2 (R+ ). If c = 0, γ > 0, then f ∈ M0,γ (L2 ) γ if and only if t γ D0 f (t) ∈ L2 (R+ ). If c > 0, then f ∈ M−1 c,γ (L2 ) if and only if f is infinitely differentiable, and moreover, 5 52 n ∞  5 (2πc)2n 5 γ γ 5 td t D0+ f (t)5 5 5 < ∞. (2n)! dt 2

n=0

γ

Here D0+ is the Riemann-Liouville fractional derivative of order γ , if '(γ ) > 0, and the Riemann-Liouville fractional integral of order −γ , if '(γ ) ≤ 0. In [29, 37] it was shown that Theorem 4.2 The Laplace transform is a bounded operator from M−1 c,γ (L2 ) onto (L ). M−1 1 (L2 ). In particular, it maps L2 (R+ ) onto M−1 2 1 c+ 2 ,γ

In

other words, M−1 1 (L2 ) = 2 ,0 −1 The space Mc,γ (L2 ) is

 H2 (C+ )

2 ,0

R+

.

also very important in studying convolution integral transforms. For example, the Stieltjes transform (3.1) maps M−1 c,γ (L2 ) onto −1 −1 Mc+1,γ (L2 ). In particular, it maps L2 (R+ ) onto M1,0 (L2 ).

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5 Peng-Chung Inverse Formula Formulas (4.1) and (4.2) involve derivatives of infinite order, so they are not very suitable for numerical purpose, when there are some noises in data. In [22] Peng and Chung discovered another real-variable inverse formula for the Laplace transform free of derivatives f (t) = lim f σ (t),

f σ (t) =

σ →∞

∞  (−1)j −1 j =1

(j − 1)!

e

jσ

σ F t



jσ t

 .

(5.1)

They showed that if f ∈ L∞ (R+ ), then f σ converges in weak*-topology to f . No convergence rate is given for formula (5.1). Still, formula (5.1) requires the values of F on the whole R+ , that also restricts its applications. It would be interesting to have an inverse formula for the Laplace transform using the samples of F only at a sequence of points. One formula of such kind was known to Amério [2] 

t

f (x)dx = lim

σ →∞

0

∞  (−1)j j =1

j!

ej σ t F (j σ ).

(5.2)

Notice that formula (5.2) does not recover the function f , but only its prime integral. In [36] a real-variable inverse formula for the Laplace transform f (t) = lim n n→∞

∞  (−1)j −1 j =1

(j − 1)!

enj t F (nj ) ,

(5.3)

has been introduced. This formula combines the advantages of both formulas (5.2) and (5.1). It recovers the function f , using the samples of F only at a sequence of points, that makes it very useful computationally. Moreover, convergence rate of the formula under very weak restrictions on f is also obtained. Theorem 5.1 Let f ∈ L∞ (R+ ). If f has a jump discontinuity at t, then lim n

n→∞

∞  (−1)j −1 j =1

(j − 1)!

enj t F (nj ) = (1 − e−1 )f (t + 0) + e−1 f (t − 0).

If f is continuous at t, then lim n

n→∞

∞  (−1)j −1 j =1

(j − 1)!

enj t F (nj ) = f (t).

Let f satisfy a Hölder condition at t |f (x) − f (t)| ≤ C|x − t|λ ,

0 < λ ≤ 1, if |x − t| < δ.

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Then [36] n

∞  (−1)j −1 j =1

(j − 1)!

 e

nj t

F (nj ) − f (t) = O

 lnλ n . nλ

In particular, if f satisfies a Lipschitz condition in a neighborhood of t with a Lipschitz constant C: |f (x) − f (t)| ≤ C|x − t|,

if

|x − t| < δ,

that is the case, for example, if f is differentiable at t, then n

∞  (−1)j −1 j =1

(j − 1)!

 e

nj t

F (nj ) − f (t) = O

 ln n . n

6 Inverse Formula on Bounded Domain All previous Laplace inverse formula require the knowledge of F on unbounded set. In practice, it is very important to have an inverse Laplace formula that uses the data F only on a bounded set. In [35] we established the following Theorem 6.1 Let F ∈ H2 (C+ ), and {pk }∞ k=1 be a sequence of distinct numbers on C+ , that is either convergent to p ∈ C+ , or pk = αk +β for some α, β > 0, and any k > 0. Then the Laplace inverse f (t) of F (s) can be determined from {F (pk )}∞ k=1 by the formula n % 2

f (t) = lim

n→∞

n 

j =1

F (pk )

k,l=1

(pk + p l )

(pk + p j )(p l + pj )

n 2 j =1,j =k

(pk − pj )

&

n 2 j =1,j =l

e−pl t , (p l − p j ) (6.1)

where the convergence is in L2 (R+ ) norm. As first example take pk = k − 12 . Then the real variable inverse Laplace transform has the form f (t) = lim

n→∞

n 

(−1)

k,l=1

k+l

    n+k−1 n+l−1 k+l−2 2 (k + l − 1) n−l n−k k−1

  1 −(l− 1 )t 2 . e ×F k − 2

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Take now pk = k + α/2 with α > −2. Then the real variable inverse Laplace transform has the form    n  n+k+α n+l+α k+l (−1) (k + l + α) f (t) = lim n→∞ n−l n−k k,l=1    k+l+α−1 k+l+α−1 × F (k + α/2)e−(l+α/2)t . k−1 l−1 1 α Choose now pk = + , then the real variable inverse Laplace transform has the 2 k form n 2

f (t) = lim

n→∞

n 

(−1)

k+l j =1

k,l=1

(αj k + j + k)(αj l + j + l)



(αkl + k + l)k!(n − k)!l!(n − l)!

F

 1 α −( 1 + α )t + e l 2 . k 2

The recovers the Laplace inverse from data on a finite interval &  α αlast formula , + 1 . 2 2

7 Dirichlet Series When solving inverse heat equations [10, 11], the observation at one point in time can be expressed through a Dirichlet series S(t) =

∞ 

an e−λn t ,

(7.1)

n=0

where {λn }n≥0 are the eigenvalues of the problem, and one needs to identify {an , λn }n≥0 from the observation S(t) in order to use the Gelfand-Levitan inverse spectral theory [19]. If S(t) is observed for infinite period of time, then one can use the method of limits [10, 11]. For engineering purposes, and in control theory, practical identification means measurement should take place in finite time. So here we discuss the problem of recovering the coefficients an and exponents λn , n ∈ N0 , of a signal S(t) under the assumptions   an = 0, an = O nk ,

−Q ≤ λ0 < λ1 < λ2 < · · · ,

λn > Knδ , k, δ > 0, (7.2)

from the given signal S(t), t ∈ (T , T1 ) where 0 < T < T1 < ∞. Denote the Heaviside step function by  H (x) =

1, x ≥ 0 . 0, x < 0

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Rewriting (7.1) in the form e

−Q(t +*)

∞

∞

S(t + *)  e−(λn +Q)(t +*)  = = an an t +* t +* n=0

=

∞ 

n=0



λn +Q

e−λ(t +*)dλ

0



∞

=

∞

e−λ(t +*)H (λ − λn − Q)dλ

an

n=0



∞



e

−λt

e

∞ 

−*λ

0

 an H (λ − λn − Q) dλ,

* > 0,

n=0

(7.3) we see that under the condition (7.2), for any * > 0, f (λ) := e−*λ

∞ 

an H (λ − λn − Q)

n=0

is a finite sum, that for each λ > 0 has O(λ1/δ ) terms, and each term has the order O(e−*λ λk/δ ), hence f (λ) ∈ L2 (R+ ), and S(t + *)/ (t + *) is its Laplace transform. We can apply (6.1) to obtain ∞ 

an (t − λn )+ = lim

n→∞

n=0 n % 2

×

j =1

n  S(tk ) tk

(7.4)

k,l=1

(tk + tj − 2*)(tl + tj − 2*)

(tk + tl − 2*)

n 2 j =1,j =k

(tk − tj )

n 2 j =1,j =l

&

(tl − tj )

1 − e−tl Q−tl t , tl

t > −Q, (7.5)

where the convergence on the right hand side is pointwise, and  λ+ =

λ, λ > 0 . 0, λ ≤ 0

The left hand side of (7.5) is a piecewise linear function, with slope jump an at λn . Thus, we can read off all the an and λn from the sequence {S(tk )}k≥1 . Acknowledgement The work of the third author was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2017.310.

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