Fully nonlinear PDEs in real and complex geometry and optics. Cetraro, 2012 9783319009414, 9783319009421

Table of contents :
Preface......Page 6
Contents......Page 10
Contributors......Page 12
1 Introduction......Page 13
2.1 Notation: Topology......Page 15
2.2 Notation: Differentials and Dilation of Mappings......Page 16
2.3 Notation: Complex Analysis......Page 17
3 Conformal Deformations......Page 19
4 Grötzsch Problem and Quasiconformal Deformations......Page 21
4.1 Grötzsch Problem......Page 23
5 Teichmüller Theorem and Extremal Quasiconformal Mappings......Page 25
5.1 Riemann Surfaces......Page 27
5.2 Teichmüller Theorem......Page 28
5.3 Coverings and Group Action......Page 29
5.4 Quadratic Differentials......Page 30
5.5 Ahlfors' Proof of Existence and Uniqueness......Page 31
5.6 Normal Family of Mappings with Integrable Distortion......Page 33
5.7 Teichmüller Mappings in Local Parameters......Page 35
6 A Variation on the Theme: Extremal Mappings of Finite Distortion......Page 37
6.1 The Finite Distortion Version of Grötzsch Problem......Page 38
6.2 Trace Norm vs. Operator Norm......Page 39
6.3 Affine Boundary Data......Page 40
6.4 More General Boundary Data......Page 41
7 Minimal Lipschitz Extensions......Page 43
7.1 Aronsson's Approach in the Scalar Case N=1......Page 44
7.2 Aronsson's Approach in the Vector-Valued Case N> 1......Page 47
7.3 A Refinement of the Aronsson Equation......Page 49
8 Aronsson's Approach for the Extremal Dilation Problem......Page 50
8.1 A Gradient Flow Approach......Page 53
References......Page 56
Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs......Page 59
1 The Steiner Formula and Curvature Measures......Page 60
2 Some Properties of Elementary Symmetric Functions......Page 64
3 Prescribing Curvature Measures......Page 72
3.1 Uniqueness and C1-Estimates......Page 76
3.2 C2-Estimates and the Existence......Page 79
4 Isoperimetric Inequality for Quermassintegrals on Starshaped Domains......Page 83
4.1 Monotonicity Properties......Page 86
4.2 The Harnack Estimate......Page 88
4.3 C2 Estimates......Page 92
5 Appendix......Page 97
6 Notes......Page 102
References......Page 104
1 Introduction......Page 107
2.1 In Vector Form......Page 108
2.1.1 κ1......Page 110
2.2 Derivation of the Snell Law......Page 111
2.3.1 Case κ1......Page 114
2.4 Uniform Refraction: Near Field Case......Page 116
2.5 Case 0

Citation preview

Lecture Notes in Mathematics  2087  CIME Foundation Subseries

Luca Capogna Pengfei Guan Cristian E. Gutiérrez Annamaria Montanari

Fully Nonlinear PDEs in Real and Complex Geometry and Optics Cetraro, Italy 2012

Editors: Cristian E. Gutiérrez Ermanno Lanconelli

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan B. Teissier, Paris

For further volumes: http://www.springer.com/series/304

2087

Fondazione C.I.M.E., Firenze C.I.M.E. stands for Centro Internazionale Matematico Estivo, that is, International Mathematical Summer Centre. Conceived in the early fifties, it was born in 1954 in Florence, Italy, and welcomed by the world mathematical community: it continues successfully, year for year, to this day. Many mathematicians from all over the world have been involved in a way or another in C.I.M.E.’s activities over the years. The main purpose and mode of functioning of the Centre may be summarised as follows: every year, during the summer, sessions on different themes from pure and applied mathematics are offered by application to mathematicians from all countries. A Session is generally based on three or four main courses given by specialists of international renown, plus a certain number of seminars, and is held in an attractive rural location in Italy. The aim of a C.I.M.E. session is to bring to the attention of younger researchers the origins, development, and perspectives of some very active branch of mathematical research. The topics of the courses are generally of international resonance. The full immersion atmosphere of the courses and the daily exchange among participants are thus an initiation to international collaboration in mathematical research. C.I.M.E. Director Pietro ZECCA Dipartimento di Energetica “S. Stecco” Universit`a di Firenze Via S. Marta, 3 50139 Florence Italy e-mail: [email protected]

C.I.M.E. Secretary Elvira MASCOLO Dipartimento di Matematica “U. Dini” Universit`a di Firenze viale G.B. Morgagni 67/A 50134 Florence Italy e-mail: [email protected]

For more information see CIME’s homepage: http://www.cime.unifi.it CIME activity is carried out with the collaboration and financial support of: - INdAM (Istituto Nazionale di Alta Matematica) - MIUR (Ministero dell’Istruzione, dell’Universit`a e della Ricerca) - Ente Cassa di Risparmio di Firenze

Luca Capogna  Pengfei Guan Cristian E. Guti´errez  Annamaria Montanari

Fully Nonlinear PDEs in Real and Complex Geometry and Optics Cetraro, Italy 2012 Editors: Cristian E. Guti´errez Ermanno Lanconelli

123

Luca Capogna Department of Mathematical Sciences Worcester Polytechnic Institute Worcester, MA, USA

Pengfei Guan Mathematics and Statistics McGill University Montreal, QC, Canada

Cristian E. Guti´errez Department of Mathematics Temple University Philadelphia, PA, USA

Annamaria Montanari Dipto. Matematica Universit`a di Bologna Bologna, Italy

ISBN 978-3-319-00941-4 ISBN 978-3-319-00942-1 (eBook) DOI 10.1007/978-3-319-00942-1 Springer Cham Heidelberg New York Dordrecht London Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2013945179 Mathematics Subject Classification (2010): 31-02, 32-T-15, 32-02, 35-02, 35-J-70, 35-J-96, 53-A-10, 53-02, 78-02, 78-A-05, 78-A-25 c Springer International Publishing Switzerland 2014  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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This volume contains the notes from the CIME school “Fully Nonlinear PDEs in Real and Complex Geometry and Optics” held at Cetraro (Cosenza, Italy) during the week of July 9–13, 2012. The school consisted of four courses: Extremal problems for quasiconformal mappings in space by Luca Capogna, Fully nonlinear equations in geometry by Pengfei Guan, Monge-Ampe`re type equations and geometric optics by Cristian E. Guti´errez, and On the Levi Monge–Amp`ere equation by Annamaria Montanari. The purpose of the school was to present current areas of research, arising both in the theoretical and in the applied settings, that involve fully nonlinear partial differential equations. The solution to these problems requires the development of ad hoc techniques arising in the relevant geometrical framework, and in case of the applications, determined by the physical laws governing the studied phenomena. Precisely, the equations presented in the school come from conformal mapping theory, differential geometry, optics, and geometric theory of several complex variables. The following is a quick overview of the contents of each course. Luca Capogna’s lectures provided an introduction to two vector-valued extremal L1 -variational problems involving mappings. The first one concerns a classical problem in geometric function theory that first arose in a work of Grotzsch from 1928: among all orientation preserving quasiconformal homeomorphisms w W  ! 0 whose traces agree with a given mapping u0 W @ ! @0 , find one which   jd uj   minimizes the functional u !   .det d u/ n1  . Variants of this problem occur when, 1 instead of using boundary data, the class of competitors is defined in terms of a fixed homotopy class or by requesting that the traces map quasi-symmetrically @ into @0 . The second problem goes back to two papers from 1934, one by Whitney and the other by MacShane, and leading to the recent theory of absolutely minimal Lipschitz extensions. That is, let   Rn be open, F   be a compact set, and let g 2 Lip.F; Rm /. Among all Lipschitz extensions of g from F to , is there a canonical unique extension that in some sense has the smallest possible Lipschitz norm? The natural questions arising in connection with these problems are v

vi

Preface

existence, uniqueness, and structure of the minimizers. The last few decades have seen intense activity from different communities of mathematicians in the study of both problems. However, at this time there does not seem to be much synergy and communication between these communities, both in terms of shared techniques used in the study of these problems and in terms of common point of views. One of the goals of Capogna’s notes is to foster such synergies by outlining some of the common features in these problems. Pengfei Guan’s lectures considered nonlinear elliptic and parabolic partial differential equations arising from geometric problems for hypersurfaces in RnC1 . The notes are an introduction to geometric analysis. A curvature-type elliptic equation is used to solve the problem of prescribing curvature measures, which is a Minkowskitype problem. Curvature measures are defined using the Steiner formula. An inverse mean curvature-type parabolic equation is employed for the proof of isoperimetrictype inequalities for quermassintegrals of k-convex star-shaped domains. Both types of equations are fully nonlinear geometric PDEs. The emphasis of Guan’s notes is on a priori estimates, a key step in the theory of fully nonlinear PDEs. The presentation is self-contained and requires basic knowledge of PDEs and geometry, namely the standard maximum principles for linear elliptic and parabolic equations, the elementary formulas of Gauss, Codazzi, and Weingarten for hypersurfaces in RnC1 , and the curvature commutator identities. Two basic and deep results are used without proofs: the Evans–Krylov theorem for uniformly fully nonlinear elliptic equations and Krylov’s theorem for uniformly parabolic fully nonlinear PDEs. Guti´errez’s course presented an introduction to Monge–Amp`ere-type equations and its applications to geometric optics. In general, these equations involve the Jacobian determinant of a map and arise in the mathematical description of numerous optical, acoustic, and electromagnetic applications, in particular, in lens and reflector antenna design. The geometric optics problems considered concern refraction, reflection, or both. A typical refraction problem that was considered in the lectures is the following: suppose we have two homogenous media I and II with different refractive indices, a light beam emanates from a point O, surrounded by medium I , and we seek an interface surface separating media I and II described by f.x/x W x 2 g;   SnC1 , and such that all rays are refracted into either a set of prescribed directions or illuminate a given object lying on a surface, say on a plane in medium II. The first type of problem is called far field and the second near field. These two problems are of different mathematical nature: the first one is variational and the second is not. The input and output intensities of radiation are prescribed and so the problem of finding the interface surface is a typical inverse problem. Guti´errez’s notes explain how to solve these problems: in the far field case using mass transport, and in the near field case using the Minkowski method. The physical background underlying these problems is explained using Maxwell equations, and, as a consequence, a deduction of Fresnel formulas is presented. These formulas are finally applied to model the case when there is loss of energy due to internal reflection. Annamaria Montanari’s course focused on the Levi Monge–Amp`ere equation. The equation is related to notions of curvatures associated with pseudoconvexity

Preface

vii

and the Levi form, in a way similar to how the classical Gauss and mean curvatures are related to the convexity and the Hessian matrix. Given a nonnegative function k, the Levi Monge–Amp`ere equation for the graph of a function u W R2nC1 ! R is det ƒ D k.xI u/.1 C jDuj2 /

nC1 2

;

where ƒ is the Levi form of the graph of u and Du is the Euclidean gradient of u. More generally, in her notes, Montanari considers elementary symmetric functions of the eigenvalues of the Levi form ƒ and shows that these curvature equations contain geometric information about hypersurface considered. Next, the notes show that the curvature operator leads to a new class of second order fully nonlinear equations whose characteristic form, when computed on generalized pseudoconvex functions, is nonnegative definite with a kernel of dimension one. Thus, the equations are not elliptic at any point. However, they have the following redeeming feature: the missing ellipticity direction can be recovered by suitable commutation relations. Using this property, existence, uniqueness, and regularity of viscosity solutions of the Dirichlet problem for graphs with prescribed Levi curvature are proved. Basic notions and results from the theory of functions of several complex variables, and from the theory of viscosity solutions for fully nonlinear degenerate elliptic equations, are also described in the notes. It is a great pleasure to thank the speakers for their very interesting lectures and for the useful lectures notes they have prepared for this volume. We also want to thank all the participants of the school for their interest in these subjects. Finally, we would like to warmly thank the CIME foundation for giving us the opportunity to organize and finance this school. We give our special gratitude to Pietro Zecca, Elvira Mascolo, and all the CIME staff for their invaluable help and support. Also, a special thanks to Mrs. Ute McCrory at Springer for her assistance in preparing this volume. Worcester, MA QC, Canada Philadelphia, PA Bologna, Italy Bologna, Italy April 5, 2013

Luca Capogna Pengfei Guan Cristian E. Guti´errez Annamaria Montanari Ermanno Lanconelli

Contents

L1 -Extremal Mappings in AMLE and Teichmuller ¨ Theory . . . . . . . . . . . . . . . Luca Capogna

1

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Pengfei Guan

47

Refraction Problems in Geometric Optics . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Cristian E. Guti´errez

95

On the Levi Monge-Amp`ere Equation . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 151 Annamaria Montanari List of Participants .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 209

ix

Contributors

Luca Capogna Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA, USA Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN, USA Pengfei Guan Department of Mathematics, McGill University, Montreal, QC, Canada Cristian E. Guti´errez Department of Mathematics, Temple University, Philadelphia, PA, USA Annamaria Montanari Dipartimento di Matematica, Universit`a di Bologna, Bologna, Italy

xi

L1 -Extremal Mappings in AMLE ¨ and Teichmuller Theory Luca Capogna

Abstract These lecture focus on two vector-valued extremal problems which have a common feature in that the corresponding energy functionals involve L1 norm of an energy density rather than the more familiar Lp norms. Specifically, we will address (a) the problem of extremal quasiconformal mappings and (b) the problem of absolutely minimizing Lipschitz extensions.

1 Introduction These notes originate from a C. I. M. E. mini course held by the author in July 2012 in Cetraro, Italy. They are meant to provide a quick introduction to two model L1 variational problems involving mappings, i.e. where the set of competitors is not scalar but vector-valued. The first concerns a classical problem in geometric function theory that first arose in 1928 in the work of Grotzsch [32]. Problem 1. Among all orientation preserving quasiconformal homeomorphisms w W ˝ ! ˝ 0 whose traces agree with a given mapping u0 W @˝ ! @˝ 0 , find one which minimizes the functional    jduj    u!  :  . det du/ n1  1

L. Capogna () Department of Mathematical Sciences, Stratton Hall, Worcester Polytechnic Institute 100 Institute Road Worcester, MA 01609, USA e-mail: [email protected] L. Capogna et al., Fully Nonlinear PDEs in Real and Complex Geometry and Optics, Lecture Notes in Mathematics 2087, DOI 10.1007/978-3-319-00942-1 1, © Springer International Publishing Switzerland 2014

1

2

L. Capogna

Variants of this problem occur when, instead of using boundary data, the class of competitors is defined in terms of a fixed homotopy class or by requesting that the traces map quasi-symmetrically boundary into boundary. The second problem has also a classical flavor. It goes back to the work of Whitney [69] and the work of MacShane [49] in 1934, and leads to the recent theory of absolutely minimal Lipschitz extensions (AMLE) [5]. Problem 2. Let ˝  Rn , F  ˝N be a compact set and let g 2 Lip.F; Rm /. Among all Lipschitz extensions of F to ˝ is there a canonical unique extension that in some sense has the smallest possible Lipschitz norm? Some natural questions arise in connection to these problems • Do minimizers exist? • Are minimizers unique? • What is the structure of the minimizers? In which norm there is continuity with respect to the data? The last few decades have seen intense activity from different communities of mathematicians in the study of both problems. However at this time there does not seem to be much synergy and communication between these communities, both in terms of shared techniques used in the study of these problems and in terms of common point of views. One of the goals of these notes is to foster such synergies by outlining some of the common features in these problems. The notes (as well as the lectures) are mainly addressed to graduate students and because of this we have included some very basic material and maintained throughout an informal style of exposition. Since there are no original results in this survey, all proofs are merely sketched, and references to the detailed arguments are provided. There are several other sources that discuss more extensively either extremal quasiconformal mappings or vector valued AMLE, but we are not aware of a reference striving for a unified point of view. The (possibly too optimistic) goal of this set of notes is to provide such perspective. Regarding other pertinent references: For classical extremal quasiconformal mappings we recommend the surveys of Strebel [64,65]. Two very clear and extremely well-written accounts of the classical Teichm¨uller theory can be found in [1,14]. The paper of Gr¨otzsch [32] is at the origin of the subject and Hamilton’s dissertation [33] provided an interesting development. The reader will also benefit from reading the classic monograph [3] as well as the more recent [8]. For the higher dimensional theory of quasiconformal mappings and the corresponding extremal problems I recommend the following fundamental contributions by Gehring and Vaisala [29, 30, 67], as well as the more recent comprehensive book by Iwaniec and Martin [39]. Various aspects of the extremal problem can be found in [6, 7, 9, 23, 60–62]. There is considerably less literature on the vector valued extremal Lipschitz extension problem: A good introduction is in the papers of Barron et al. [11, 12]. More recent developments can be found in the work of Naor and Sheffield [52], Sheffield and Smart [63], Katzourakis [47], Ou et al. [53]. We also want to point out two relevant references that, in our opinion,

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

3

have great potential for applications to the problems discussed here: Dacorogna and Gangbo [20] and Evans et al. [21]. Although these notes do not involve specific applications, the topic of L1 variational problems arises naturally in mathematical models of several real-world phenomena. To this regard, we conclude this introduction with a quote from Robert Jensen’s seminal paper [44] The importance of variational problems in L1 is due to their frequent appearance in applications. The following examples give just a small sample of these. In the engineering of a load-bearing column it is preferable to minimize the maximal stress (i.e., the L1 norm of the stress) in the column rather than some average of the stress. When constructing a rocket, the maximal acceleration applied to the payload is an important factor in the design. Optimal operation of a heating-cooling system for an office building requires control of the maximal and minimal temperature within the building rather than the average temperature. Windows on airplanes are made without corners to prevent high pointwise stress concentrations. These considerations motivate the study of the issues of existence, uniqueness, and regularity etc. etc.

2 Notation and Preliminaries In this section we set the notation for the rest of the notes and include some basic, elementary definitions and results that will be needed later on.

2.1

Notation: Topology

• An homeomorphism between two topological spaces is a continuous bijection whose inverse is also continuous. • A topological manifold of dimension n 2 N is a topological space for which every point has a neighborhood homeomorphic to Rn . • A smooth manifold of dimension n is an n-dimensional topological space along with a collection of charts .U˛ ; f˛ /˛2A with U˛  M open and such that they cover M , f˛ W U˛ ! f˛ .U˛ /  Rn homeomorphism and such that f˛ ı fˇ1 is smooth on its domain. • An homotopy between two continuous functions f; g between two topological spaces X and Y is a continuous function H W X Œ0; 1 ! Y such that H.x; 0/ D f .x/ and H.x; 1/ D g.x/ for all x 2 X . • The fundamental group 1 .M; p/ of a topological manifold M with p 2 M is the quotient of the space of loops at p through the equivalence relation    if and only if  ı 1 is homotopic to the identity. If 1 .M; p/ D 0 then M is simply connected. • A triangle T in a surface S is a closed set obtained as the homeomorphic image of a planar triangle. The image of vertices and edges of the planar triangle are also called vertices and edges. A triangulation of a compact surface S is a finite

4

L. Capogna

set of triangles T1 ; : : : ; Tm such that [m i D1 Ti D S and every pair Ti ; Tj is either disjoint or intersects at a single point (vertex) or a shared edge. • The Euler characteristic of a triangulated compact surface S is given by  D v  e C f where v is the number of vertices of the triangulation, e is the number of edges and f the number of triangles. This number does not depend on the specific triangulation of S . The genus of S is the number g obtained from the identity  D 2  2g. Example 1. The sphere has genus zero, as does the unit disc. The torus has genus 1. Roughly, for general orientable surfaces, the genus is the number of handles in the surface.

2.2 Notation: Differentials and Dilation of Mappings The background for fine properties of mappings, their dilation and much more can be found in the monograph [39]. Let ˝  Rn and denote by u D .u1 ; : : : ; un / W ˝ ! Rn

(1)

1;n a Wloc .˝/ orientation-preserving homeomorphism. At points x 2 ˝ of differentiability of u we denote by du.x/ W Rn ! Rn the differential of u. In coordinates one has that for v 2 Rn the action of the differential is1 Œdu.x/.v/i D duij vj , i D 1; : : : ; n where we have let .du/ij D @xj ui . Set jduj2 D trace.duT du/ D duij duij : At points of differentiability, the pull-back du .x/gE of the Euclidean metric gE is given by d.u .x/gE /ij D ŒduT duij D @xi uk @xj uk , for i; j D 1; : : : ; n: If n D 2 it is convenient to use complex notation: Set u D u1 C iu2 , and

@z u D

1 1 .@x u  i@y u/ and @zN u D .@x u C i@y u/: 2 2

Note @zN u D @Nz uN . We also let dz D dx C idy and note that dz.@z / D 1 while dz.@zN / D 0. Similarly d zN D dx  idy and d zN.@z / D 0 and d zN.@zN / D 1. Next we introduce different ways in which one can quantify how a differentiable homeomorphism u W ˝ ! Rn can distort the ambient geometry. We start by considering linear bijections A W Rn ! Rn expressed in coordinates as y i D Aij x j for i D 1; : : : ; n. Denote by jAjO WD maxjV jD1 jAVj the operator norm of A and consider the following quantities • the linear dilation of A is H.A/ D

1

maxjhjD1 jAhj : minjhjD1 jAhj

Implicit summation on repeated indices is used throughout the paper.

(2)

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

5

• the outer dilation of A is Ho .A/ D • the trace dilation of A is K.A/ D n

j

jAjnO : jdet Aj

P ij

A2ij jn=2

jdet Aj

(3)

:

(4)

If u is as in (1) then we set Ku .x/ D K.du.x//.

2.3 Notation: Complex Analysis Basic references for the complex analysis background are the classical book of Ahlfors [2] and Jost’s monograph [45]. • A C 1 function w D u C iv W C ! C is holomorphic if @zN w D ux C iuy C ivx  vy D 0 Equivalently w must satisfy the Cauchy-Riemann equations ux D vy and uy D vx . • Conformal invariance of harmonic functions. If w D h.z/ is a holomorphic function and f W C ! C is smooth then   @ @f @h @f @hN @2 f ı h.z/ D C @z@Nz @z @w @Nz @wN @Nz     @2 @hN @ @f N jh.z/ D f jh.z/ @z h@zN h: D @z @wN @Nz @w@wN • A holomorphic map u W U  C ! C is called conformal if @z u ¤ 0 at every point in U . Example 2. Set D D fz 2 Cj jzj < 1g and H D fz D x C iyj y > 0g These are conformally equivalent under the map H ! D given by z !

zz0 . zNz0

Theorem 1. Every f W D ! D (or f W H ! H ) which is biholomorphic (i.e., conformal and bijective) is a M¨obius transformation, i.e. there are a; b; c; d 2 C such that f .z/ D

az C b : cz C d

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L. Capogna

For any ring R define the group  SL.2; R/ D f

 ab jad  cb D 1g c d

while PSL.2; R/ denotes its quotient by the sub-group generated by ˙Id. Every element in PSL.2; R/ defines a M¨obius transformation H ! H . Definition 1. A group G acts as a transformation group on a manifold M if there is a map G  M ! M denoted as .g; x/ ! gx with .g1 g2 /.x/ D g1 .g2 x/ and ex D x. The isotropy group of x 2 M is a subgroup of G which fixes x. Example 3. The group PSL.2; R/ acts as a transformation group of H . The isotropy group of each element is isomorphic to SO.2/. Both D and H can be given a (non-Euclidean) metric structure through the hyperbolic metric 1 dzd zN on H y2

and

1 dzd zN on D: .1  jzj2 /2

An isometry between two Riemannian manifolds u W .M; g/ ! .M 0 ; g 0 / is a map such that 0 .dx uV; dx uW/ D gx .V; W / gu.x/

for any x 2 M and V; W 2 Tx M . Theorem 2. All isometries between the hyperbolic H and D are M¨obius transformations. The isometry group of H is PSL.2; R/. Definition 2. A group action G on M is properly discontinuous if every x 2 M has a neighborhood U such that fg 2 GjgU \ U ¤ 0g is finite and if x; y are not in the same orbit then they have neighborhoods Ux ; Uy such that gUx \ Uy D ; for all g 2 G. Definition 3. Let  PSL.2; R/ be properly discontinuous subgroup and z1 ; z2 2 H . We say that z1 and z2 are equivalent if there exists g 2 such that gz1 D z2 . Consider H= the space of quotient classes equipped with the quotient topology. Proposition 1. Let  PSL.2; R/. If the action of on H properly discontinuous and does not fix points (gx ¤ x for all x 2 H and all g ¤ id) then the quotient H= can be given a Riemann surface structure.

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

7

3 Conformal Deformations An a.e. differentiable homeomorphism u W Rn ! Rn is conformal if there exists a scalar function such that at every point of differentiability duT du D Id

(5)

The pull-back of the Euclidean metric dx2 is a scalar multiple of dx2 , i.e. angles are preserved. Equivalently, a.e. in ˝, one must have g WD The function

duT du 2

. det du/ n

D Id:

(6)

p trace.g/ D jduj=.det du/1=n is called dilation of u.

Remark 1. At every point of differentiability for u one has Ku D trace.g/  n, with the equality being achieved if and only if g D Id. Definition 4. Following Ahlfors (see also [39]) we define the distortion tensor of u at a point of differentiability x 2 Rn S.g/ WD

g C gT trace.g/ trace.g/  Id D g  Id; 2 n n

(7)

and denote by p K.u; ˝/ D jjKu jjL1 .˝/ D jj trace.g/jjL1 .˝/ ;

(8)

the maximal dilation of u in ˝. Proposition 2. With the notation above, one has that a diffeomorphism u is conformal if and only if S.g/ D 0 and if and only if K.u; ˝/2 D K2u D trace.g/ D n identically in ˝. p Remark 2. It is not difficult to show that if Ku D K0 > n, then there exists " D ".K0 / > 0 so that  1 "  jS.g/j2  K4u 1  : n When n D 2, if we denote by 0  1  2 be the eigenvalues of g, then one can find an explicit lower bound. In this case, 1 2 D 1 and 1 1 1 jS.g/j2 D 21 C 22  . 1 C 2 /2 D . 1 C 2 /2  2 1 2 D .K4u  4/: 2 2 2

8

L. Capogna

Remark 3. Denote by COC .n/ the space of differentials of orientation preserving conformal mappings, then its tangent space TCOC .n/ at the identity is A 2 Rnn s:t: S.A/ D

trace.A/ A C AT  Id D 0 : 2 n

Accordingly we have that the distance of a matrix A from COC .n/ satisfies d 2 .A; COC .n// D cjS.A  I /j2 C O.jA  I j4 /: This shows that the operator S arises naturally when considering the linearization of the distance of a deformation from being conformal. For more results from this point of view, including a remarkable geometric rigidity result in the spirit of Frieseke et al. [24], see the work of Faraco and Zhong [22]. Three remarkable properties of conformal deformations 1;n • Conformal implies smooth. If an homeomorphism u 2 Wloc .˝; Rn / satisfies p K.u; ˝/ D n then a result of F. Gehring [28] implies that u 2 C 1 .˝/. The proof is based on regularity of weak solutions to the n-Laplacian, via the De Giorgi-Nash-Moser theorem. See the discussion below on Liouville theorem for more details. Moreover, if f is a weak solution to the n-Laplacian and u is conformal then f ıu is also n-harmonic. This is the so-called morphism property. • For n D 2; Conformal transformation are holomorphic diffeomorphism and viceversa. In particular the space of conformal planar deformations is infinite dimensional. • Riemann mapping theorem. Any non-empty, simply connected open planar set can be mapped conformally to the disc (uniquely if one prescribes target for three points).

Rigidity of conformal deformations. Despite the flexibility of the Riemann mapping theorem and the usefulness in changes of variables arguments, conformal mappings exhibit aspects of rigidity that make it too restrictive for many applications. • Liouville theorem. For n  3 conformal deformations are compositions of translations, rotations, dilations and inversions. The theorem was proved originally by J. Liouville [48] with the hypothesis that the fourth order derivatives of the maps be continuous. Gehring [28] and Reshtnyak [57] established remarkable generalizations respectively to quasiconformal and to quasiregular mappings in 1;n Wlot . For n D 2l a sharp form of the Liouville theorem was established by Iwaniec and Martin in [38]. In this paper, among other things, it is proved that 1;l for l > 1, every u 2 Wloc .˝; R2l / with det du  0 (or det du  0) a.e. and such that H.du/ D 1, i.e. jjdujjO  minjvjD1 jduvj a.e. is either constant or the restriction of a M¨obius transformation to ˝. The Sobolev exponent l is optimal 1;p in the sense that there are weak Wloc solutions of the Cauchy-Riemann equations with p < l, which are not M¨obius.

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• Rigidity with respect to boundary data. Even in the plane, despite the Riemann mapping theorem one cannot prescribe boundary data (more than three points) when mapping conformally one domain into the other. For instance, in mapping one rectangular box into another, sending sides to sides, one can achieve this through a conformal deformation only if the boxes are similar (see the next section). The intrinsic rigidity of conformal mappings provided a motivation for the extension to a larger family, that of quasiconformal mappings. Quoting F. Gehring [29], . . . quasiconformal mappings constitute a closed class of mappings interpolating between homeomorphisms and diffeomorphisms for which many results of geometric topology hold regardless of dimension.

In the next section we will see how at the genesis of the theory of quasiconformal mappings lies a L1 variational problem.

4 Gr¨otzsch Problem and Quasiconformal Deformations Let R and R0 be two rectangles with sides a; b and a0 ; b 0 , that are not similar (i.w. a=b ¤ a0 =b 0 ). It is then easy to see that there is no conformal deformation mapping R to R0 sending edges to edges. In connection to this observation, in 1928 H. Gr¨otzsch [32] posed the following question Problem 3. Is there a most nearly conformal mapping between R and R0 ? Quoting L. Ahlfors [1] in relation to this problem This calls for a measure of approximate conformality, and in supplying such a measure Gr¨otzsch took the first step toward the creation of a theory of q.c. mappings.

To address Gr¨otzsch’s question one would need to identify a quantitative way of determining how non-conformal a mapping can be and then find an extremal point for this quantity in a suitable class of competitors. For such a general scheme to work it is of paramount importance to have good compactness properties for the class of competitors. Such considerations hint at the need of introducing a more general class of deformations that are less rigid, yet retain some of the useful features of conformal mappings. One also would like to have an instrument to quantify how far a given deformation is from being conformal. 1;n .˝; Rn / is an homeomorDefinition 5. Let ˝  Rn be an open set. If u 2 Wloc phism then we say it is quasiconformal if

  p   jduj   < 1: K.u; ˝/ D jj trace.g/jjL1 D  1=n . det du/ L1 We say u is K-quasiconformal if K D kHO .du/k1 D kjdujO = det du1=n k1 .

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Example 4. In the following we list some simple examples of quasiconformal mappings. • Linear bijections x ! Ax with A 2 Rnn . • Diffeomorphisms with non-vanishing Jacobians are locally quasiconformal. • For a ¤ 0 consider the family of quasiconformal mappings u.x/ D jxja1 x. For a D 1 this is the conformal inversion. • In cylindrical coordinates .r; ; z/ set D˛ D f 2 .0; ˛/g and define f W D˛ ! Dˇ as f .r; ; z/ D .r; ˇ=˛; z/ (folding map). • In spherical coordinates .R; ; / define C˛ a cone of angle ˛ by 0  < ˛. Set f W C˛ ! Cˇ as f .R; ; / D .R; ; ˇ =˛/. The map is quasiconformal for ˇ <  and but fails to be quasiconformal for ˇ D . Definition 5 seems to require a-priori information on a.e. differentiability of the mapping which are counterintuitive in relation to the need for compactness of the class of competitors we referred to. There are in fact previous equivalent definitions for quasiconformality which do not require any a priori differentiability. Definition 6. Geometric definition Let r > 0 and x 2 ˝. Set L.x; r/ D

sup

ju.x/  u.y/j

y2˝j jxyjr

and l.x; r/ D

inf

y2˝j jxyjr

ju.x/  u.y/j

The homeomorphism u is quasiconformal if there exists H  1 such that for every x 2 ˝ the linear dilation satisfies H(x,u) WD lim sup r!0

L.x; r/  H < 1: l.x; r/

(9)

Remark 4. If A W Rn ! Rn is a bijection then H.x; A/ D H.A/ with H.A/ defined as in (2). If u W Rn ! Rn is differentiable at the point x with non-vanishing Jacobian determinant then H.x; u/ D H.du.x//. Remark 5. The differential du.x/ transforms circles centered at the origin into similar ellipses. The quantity H.du/ is the ratio of the axis of such ellipse. Thus quasiconformal deformations map infinitesimal circles into ellipses with a bounded ratio of the axis. Homemorphism for which (9) holds with lim sup substituted by sup are called quasisymmetric. It was F. Gehring [27] who first proved that quasiconformal implies quasisymmetric if n  2. See also [34] for quantitative estimates and extensions to more general metric spaces. Theorem 3. Consider an homeomorphism u W ˝ ! ˝ 0 , then the quantity 1;n jjH.x; u/jjL1 is finite if and only if u 2 Wlot .˝; Rn / and K.u; ˝/ is finite.

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Theorem 4 (Gehring, 1962). For every K  1 and n 2 N; n  2 there exits n;K W .0; 1/ ! R increasing, with limr!0 n;K .r/ D 0 and limr!1 n;K .r/ D 1 such that for every f W ˝ ! ˝ 0 K-quasiconformal one has   d.f .x/; f .y// d.x; y/ ;  n;K d.f .x/; @˝ 0 / d.x; @˝/ for all distinct x; y 2 ˝ sufficiently close. Moreover for r sufficiently small, one can choose n;K .r/ D cn r ˛ with ˛ D K 1=.1n/ . In complex notation one denotes the map as z D x C iy 2 ˝  C ! .z/ D  C i; and set p D @z and q D @zN , so that d D pdz C qdNz. The mapping d is affine and satisfies ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇjpj  jqjˇjdzj  jd j  ˇjpj  jqjˇjdzj ˇ ˇ ˇ ˇ From the latter we see that the ratio of the axes of the ellipse obtained as image of a circle under d is given by the maximal dilation K D sup ˝

jpj C jqj : jpj  jqj

jqj K1 We also define the maximal excentricity  D KC1 D sup jpj : Note that is conformal iff K D 1,  D 0. The Jacobian determinant of the map is J D jpj2  jqj2 .

Remark 6. Since the derivatives of the inverse map ! z. / are given by p 0 D J 1 pN and q 0 D J 1 q then the mapping D .z/ and z D z. / have the same dilation at corresponding points, hence the same maximal dilation. Moreover, the dilation is invariant by conformal deformation in both the z and the planes.

4.1 Gr¨otzsch Problem Let us return to Gr¨otzsch original question. Consider two rectangles R; R0 with sides parallel to the axis and with one vertex at the origin, as illustrated in Fig. 1. Remark 7. The affine transformation mapping R ! R0 that maps edges to edges is 0 0 the anisotropic dilation  D aa x and  D bb y i.e.,   b0 a0 b 0 1 a0 Œ C z C Œ  Nz : (10)

D 2 a b a b

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R R b

b a a Fig. 1 Gr¨otzsch problem

The dilation of the affine map is a constant ( KD

a0 b b0 a ab0 ba0

if if

a0 b0 a0 b0

>
3) and where the constraint defining the class of competitors is given by membership in the same homotopy class. In this setting one has existence, uniqueness and some amount of regularity for the minimizers. For n > 2 less is known. The fundamental reference by Gehring and Vaisala [30] establishes the problem in a more general setting and provides some existence results. The higher dimensional analogue of Gr¨otzsch problem was solved by Fehlmann [23]. A great amount of recent literature focuses on the Lp variational problems, which we will briefly describe through the work of Astala et al. [7]. We also recall related work of Balogh et al. [10] and Astala et al. [9]. In a (rough) comparison with similar problems in elasticity, conformal deformations correspond to isometries. Accordingly, the variational problems stated above corresponds to finding deformations closest to isometries in given classes of competitors. Some of the main obstacles in studying this L1 problem are • Lack of convexity. • L1 functionals are not sensitive to deformations of functions away from their maximum. Unlike Lp averages they are not “local”. This makes uniqueness unlikely. • The problem is vector-valued, and as such not approachable through the established techniques from game theory or viscosity solutions. • There is a topological constraint.

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5.1 Riemann Surfaces A Conformal Atlas on a two dimensional smooth manifold is an atlas .U˛ ; z˛ /˛2A with z˛ W U˛ ! C local charts such that the transition maps zˇ ız1 ˛ W z˛ .U˛ \Uˇ / ! zˇ .U˛ \ Uˇ / are holomorphic. An atlas .U˛ ; z˛ / is compatible to another atlas .Vˇ ; wˇ / if their union is still a conformal atlas. A conformal structure is the union of an atlas with all other compatible charts. Definition 7. A Riemann surface is a two dimensional smooth manifold with a conformal structure. Example 5. The Riemann sphere S 2 D fx12 C x22 C x32 D 1g  R3 . To show that S 2 is a Riemann surface we consider an atlas with open sets U1 ; U2 obtained from the whole sphere minus respectively the north pole .0; 0; 1/ and the south pole .0; 0; 1/. Define the stereographic projections charts z1 .x/ D

x1 C ix2 x1  ix2 on U1 and z2 .x/ D on U2 1  x3 1 C x3

Note that z1 .U1 \ U2 / D z2 .U1 \ U2 / D C n f0g and that z2 ı z1 1 .a C ib/ D

a C ib 1 D a2 C b 2 a C ib

Example 6. The Riemann Torus is defined as follows: Set w1 ; w2 2 C two non-zero vectors and define an equivalence relation in C by saying that a C ib  x C iy if there exists two rational numbers m; n such that a C ib D x C iy C mw1 C nw2 . The discrete Abelian subgroup M D fmw1 C nw2 g is called a lattice. If we let  be the projection to the quotient space then T D .C/ is a Riemann surface. To define an atlas we consider open sets O  C containing to equivalent pairs (for instance a subset of a fundamental domain) and define the chart U D .O/ and z D j1 O . Since z˛ ı z1 is a translation then this is a conformal atlas. ˇ We have already seen that if  PSL.2; R/ and the action of on H is properly discontinuous and does not fix points then the quotient H= can be given a Riemann surface structure. Viceversa one has the following Theorem 5 (Uniformization theorem). Let ˙ be a compact Riemann surface of genus p. There exists a conformal diffeomorphism f W ˙ ! ˙ 0 where ˙ 0 is either (i) of the form H= if p  2; (ii) A torus C=M if p D 1; (iii) the Riemann sphere if p D 0. As corollary, the universal cover of a compact Riemann surface is conformally equivalent to S 2 , C or D.

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Definition 8. A continuous map u W S ! S 0 is holomorphic if it is so when expressed (locally) through conformal charts. If these local expression have non vanishing @z u derivative then u is conformal. Let us recall the topological classification of compact Riemann surfaces Theorem 6. Two differentiable, orientable, compact triangulated surfaces2 are homeomorphic if and only if they have the same genus. Every Riemannian metric gij dxi dxj on a oriented surface can be written locally in complex coordinates as .z/jdz C .z/d zNj2 D .z/.dz C d zN/.Nz C dz/ N where  > 0 (real) and jj < 1. Theorem 7. Every oriented Riemannian surface admits a conformal structure and a conformal Riemann metric dzd zN. A local system of holomorphic coordinates is given by the solutions of the equation @zN u D @z u and D @z@ . u@Nz uN

5.2 Teichmuller Theorem ¨ The focus of Teichm¨uller theorem is on a classification of all possible conformal structures of a given Riemann surface S , and on establishing a structure theorem for such a moduli space. The natural candidate for space of all conformal structures is Definition 9. Given a compact Riemann surface S with genus p, we define the moduli space Mp of conformal structures on S where .S; g1 / and .S; g2 / are identified if there exists a conformal diffeomorphism between them. However this moduli space does not have a manifold structure and its topology is very complicated. To somewhat simplify the structure Teichm¨uller proposed a new notion of moduli space of conformal structures, known today as Teichm¨uller space. Definition 10. Given a compact Riemann surface S with genus p, we define the moduli space Tp of conformal structures on S where .S; g1 / and .S; g2 / are identified if there exists a conformal diffeomorphism between them which is homotopic to the identity. The first step in studying the structure of this space is given by the following existence theorem

2

Recall that every Riemann surface is orientable and any conformal atlas yields a triangulation.

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Theorem 8 (Existence Theorem). Given S; S 0 closed Riemann surfaces of same genus and ˛ W S ! S 0 an homeomorphism, there exists a quasiconformal mapping

W S ! S 0 homotopic to ˛ and minimizing the maximal dilation in the homotopy class of ˛. What is needed next is a uniqueness result for such minimizers as well as an algebraic characterization that would allow to define a manifold structure on the moduli space. In the next sections we will state such uniqueness results and then proceed to sketch Ahlfors’ proof of this remarkable characterization.

5.3 Coverings and Group Action If S; S 0 are Riemann surfaces of same genus g > 1 realized as D=G; D 0 =G 0 . The quotient map p W D ! D=G D S is a covering map and the group G, which acts on D is called a Fuchsian group. The disc D is the universal cover of S . Theorem 9. Any homeomorphism map W S ! S 0 can be lifted to a family of mappings W D ! D 0 with the property that for every g 2 G there exists a unique g 0 WD ˛.g/ 2 G 0 such that

.g.z// D g 0 . .z//: Viceversa, any homeomorphism W D ! D 0 which satisfies the identity above induces an homeomorphism W S ! S 0 . Lifts of quasiconformal mappings are quasiconformal with the same dilation. The maps g ! ˛.g/ are group isomorphisms. Any two lifts are related by a inner automorphism of G or G 0 . Theorem 10. Any two homeomorphism maps S ! S 0 are homotopic if and only if the determine isomorphisms G ! G 0 which differ only by an inner automorphism. So essentially, modulo renormalization, there exists a one-to-one correspondence between homotopy classes of homeomorphisms and isomorphisms G ! G 0 . This result allows to reframe Teichm¨uller theorem and the variational problem only in terms of mappings W D ! D 0 which satisfy the functional equation

.g.z// D g 0 . .z//: Theorem 11 (Existence Theorem reformulated). Given 0 W D ! D 0 a fixed homeomorphism, let ˛ W G ! G 0 denote the induced isomorphism. There exists a quasiconformal mapping W D ! D 0 minimizing the maximal dilation in the class of all homeomorphisms satisfying the function equation

.g.z// D ˛.g/. .z//: Consider the set of all quasiconformal mappings satisfying the identity (this is not empty since the surfaces are diffeomorphic) and with dilation less than a fixed

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L. Capogna

number K. Gehring’s theorem implies that this is a normal family, and hence any minimizing sequence will converge to either a quasiconformal mapping satisfying the same functional identity or a constant. Constant are ruled out by the functional identity and the fact that no element of D 0 is fixed by every g 0 2 G 0 . Uniqueness of the representative is provided by a deep connection between extremal quasiconformal mappings and quadratic differentials.

5.4 Quadratic Differentials Consider a 1-form dz D dx C idy in C. If F W C ! C is a conformal map and we denote z.w/ D F .w/ then for any .a; b/ 2 R2 we can compute the action of dF dF W R2 ! R2 in complex notation as dF .a; b/ D dw .a C ib/. In view of this then dz  0 the pull-back F dz is given by F .w/dw D dw dw, in fact the action of F  dz on any complex tangent vector a C ib can be computed through F  dz.a C ib/ D dz.dF.a C ib// D dz.F 0 .w/.a C ib// D F 0 .w/dw.a C ib/ Through a similar computation one sees that the symmetric 2-tensor .z/dz2 dz 2 pulls back to .z.w//. dw / dw2 : These computations motivate the following Definition 11. Let S be a Riemann surface and f.U˛ ; h˛ /g denote its conformal structure. A meromorphic (resp. holomorphic) quadratic differential h in S is a set of meromorphic (res. holomorphic) functions f˛ in the local coordinates given by z˛ D h˛ .p/ with p 2 S satisfying the transformation law 

@zˇ f˛ .z˛ / D fˇ .zˇ / @z˛

2 ;

for all charts .U˛ ; z˛ / and .Uˇ ; zˇ / around a point p 2 S . Recalling the formula for the pull back of a complex 2-tensor described earlier we can write the definition above as h˛ .z˛ /dz2˛ D hˇ .zˇ /dz2ˇ Observe that quadratic differentials are holomorphic sections of the bundle of holomorphic symmetric tensors. Let Q.S / denote the space of all quadratic differentials on a given compact Riemann surface S . Since the sum of two quadratic differentials as well as the multiplication by a scalar of a quadratic differential are still elements of Q.S / then the latter is a complex vector space. The following is a consequence of the Riemann-Roch Theorem

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Theorem 12. The space Q.S / has finite dimension (over R) 6p  6. Theorem 13 (Structure of minimizers). Given S; S 0 closed Riemann surfaces of same genus p > 1 and ˛ W S ! S 0 an homeomorphism. Denote by W S ! S 0 a quasiconformal mapping homotopic to ˛ and minimizing the maximal dilation in the homotopy class of ˛. Either is analytic or there exists a quadratic differential fdz2 on S and a constant  2 .0; 1/, such that is differentiable away from the zero set of f (with non-vanishing complex derivatives q; p) and satisfies fN q D : p jf j The quadratic differential is uniquely represented up to a positive constant factor and  represents the (constant) eccentricity of the extremal mapping. Theorem 14 (Uniqueness). Every map whose complex derivatives satisfy q fN D p jf j has a maximal dilation which is strictly smaller than the dilation of any other mapping (not conformally equivalent to ). These result yield that for every homotopy class one has existence of a unique minimizer for the maximal dilation and associated to this minimizer there is a unique pair of quadratic differentials. This correspondence gives a manifold structure to the Teichm¨uller space, with the same dimension 6p  6 as the space of quadratic differentials.

5.5 Ahlfors’ Proof of Existence and Uniqueness In [1], Ahlfors considers the following m-mean distortion functional: For every a. e. differentiable map D  C i W D ! D 0 and m  1, 1 Im . / D 

Z Z D0

jpj2 C jqj2 jpj2  jqj2

!m ˇ ˇ ˇ ˇ ˇ

d d:

1 .Ci/

The customary 1-parameter deformations used in the calculus of variations to derive the Euler-Lagrange equations are of the form s D C s where s 2 ."; "/ and  2 C01 .D; D 0 / serves as a test function. However if is merely quasiconformal, in particular not a C 1 diffeomorphism then the deformation

s D C s may fail to be an homeomorphism and hence be outside of the set of competitors, making it useless for the purpose of deriving a PDE which describes the behavior of minimizers.

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To circumvent this problem one may choose to do a different set of perturbations, acting on the domain of the map, rather than one the image, thus setting: z D H.z0 ; "/ WD s 0 C "h.z0 / C o."/; yielding @z H D 1 C "@z h C o."/I and @zN H D "@zN h C o."/: If G is a Fuchsian group acting on a Riemann surface S then for H to determine a deformation of the surface one needs H.gz; "/ D gH.z; "/ for every g 2 G. This eventually yields h.gz/ D @z g h.z/; which characterizes all infinitesimal deformations of S . Remark 8. A brief digression: If one considers the Dirichlet energy Z Z j@z wj2 C j@zN wj2 dxdy D

then the usual exterior deformations lead to the Laplacian @z @zN w D 0. If instead one proceeds as in Ahlfors (and Hopf, Morrey, and many others) and carries out inner variations as described earlier then one obtains the PDE   @zN @z w @zN w D 0; which is of a very different nature from Laplace’s equation. To the best of our knowledge, currently the sharpest regularity result known for such PDE is Lipschitz continuity, see Iwaniec et al. [43]. See also earlier work of Bauman et al. [13]. In Ahlfors’ argument the inner variation produces the equation in weak form Z Z Re

D0

jpj2 C jqj2 jpj2  jqj2

!m

p qN @zN hd d: jpj2 C jqj2

Changing variables z0 D .z/ we obtain Z Z Re D\fjpj>jqjg

jpj2 C jqj2 jpj2  jqj2

!m1 p q@ N zN hdz ^ d zN:

Set !m1

(

jpj2 Cjqj2 jpj2 jqj2

Um D 0

if jpj > jqj otherwise:

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P 0 and  D g 0 2G 0 j@z g j one obtains a reformulation of this PDE in terms of integration over the original surface Z Z Um p q@ N zN h D 0: S

Z Z Um p q@ N zN h D 0: S

in particular this yields Lemma 1. The function fm D Um p qN is holomorphic and so describes a holomorphic quadratic differential fm .z/dz2 in D. We let Cm > 0 be constants defined so that Um p qN D cm fm .z/ with

R S

jfm jdxdy D 1

5.6 Normal Family of Mappings with Integrable Distortion Theorem 15. If f 2 W 1;n .˝; RN / is a mapping whose distortion is m-integrable with m > n  1, then f is continuous and the modulus of continuity depends only on the Lm norm of the distortion. In this form and in this setting, this result is due to Ahlfors [1]. It is also a consequence of work of Iwaniec and Sverak [40] and of Manfredi and Villamor [68]. See also the work of Koskela et al. [41, 42]. Given any diffeomorphism ˛ W D ! D 0 , then in view of Ascoli-Arzela and Theorem 15 one has that for every m there exists (a possibly not unique)

m W D ! D 0 in the same homotopy class as ˛ and which minimizes Im . We denote by pm ; qm its complex differentials and set Z Z min Im . / D Im . m / D

D0

jpm j2 C jqm j2 jpm j2  jqm j2

!m :

Denote the quantity above by Kmm . In view of H¨older inequality Km is monotone increasing and bounded (by the dilation of ˛) hence Km ! K < 1. Let 0   < 1 be defined by

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L. Capogna

KD

1 C 2 : 1  2

Normality and a diagonalization argument yields: Lemma 2. For a subsequence one has m ! ; as m ! 1, uniformly on compact sets, with quasiconformal. Lemma 3. For a subsequence one has 1

.Cm / m ! K as m ! 1. Lemma 4. For a subsequence one has ˇ Z Z ˇˇ ˇ ˇ ˇ ˇjqm j  jpm jˇdz ^ d zN ! 0; ˇ ˇ D as m ! 1. Note that the relation Cm fm D

jpj2 C jqj2 jpj2  jqj2

!m1 . m /pm qNm

yields fm pm qN m D jfm j jpm jjqm j and consequently ˇ ˇ ˇ ˇ ˇ fm ˇ ˇ pm ˇ ˇ ˇ ˇ ˇ ˇ jf j qm  pm ˇ D ˇ jp j jqm j  pm ˇ D jjqm j  jpm jj m

m

Passing to a subsequence then we can assume that fm tend to a limit f and that

m ! uniformly on compact sets. The limit mapping has complex derivatives p; q which are limit of pm ; qm and thus satisfy f q D p jf j

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

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5.7 Teichmuller Mappings in Local Parameters ¨ A homeomorphism z ! .z/ W D ! D whose complex derivatives satisfy 

f p D q jf j

is called a Teichm¨uller mapping. We show that there exists local parameters (i.e. a local set of conformal coordinates)  ; z such that in this coordinates the map reads as the composition of two conformal transformations conjugating an affine mapping (just as in Gr¨otzsch’s problem). Denote by ! z the inverse mapping and by p 0 ; q 0 its complex derivatives. Differentiating the formula z D z. .z// along z and zN one can see that p; q; p 0 ; q 0 are related by the formula p0 D

jpj2

pN q and q 0 D  2 : 2  jqj jpj  jqj2

If .z/ is quasiconformal extremal then so is z. / and its associated quadratic differential . / satisfies: p0  D 0: jj q Consequently it follows that if z ! is Teichm¨uller then  qN D p jj Next we introduce two new local charts Z p Z p z WD f dz and  WD

d This can be done in a sufficiently small neighborhood of a point where f;  do not vanish and with fixed branches of the square roots and arbitrary integration constants. Keeping in mind the expression  . .z.z /// then in terms of these new variables one has

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L. Capogna

p d  d  d dz  p D  D  D p p dz d dz dz f 

text and similarly p  q D q q fN Next, observe that p p p N f fN    f q D q q D p q D p q D p  : jf j jf j N N f f fN 

Similarly, if we use  qN D p jj then we obtain qN  D p  : Since q  D qN  then q  is real and so is p  . Next, observe that d .    N  / D q   p  D 0 d zN Hence    N  is holomorphic and its complex derivative is d  .   N  / D p   q  D p   p  D p  .1   2 /: dz Since derivatives of a holomorphic functions are also holomorphic then p  is both real and holomorphic, hence it must be constant. Consequently one has that

 .z / D p  z C q  zN D A.z C Nz / C B for some constants A; B 2 C, proving our statement on the local structure of Teichm¨uller mappings.

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25

5.8 Uniqueness (Rough Idea) The uniqueness part follows from a Gr¨otzsch-like argument (more complicated in view of possible singularities). The analogues of the rectangular regions arise in the following way: We consider a Riemannian metric ds2 D  2 d dN where  is the quadratic differential in the target region associated to a Teichm¨uller mapping. This metric is complete and non-positively curved, thus yielding unique p geodesics between any pair of points. By the geodesic equation the quantity d is constant along geodesics. p We call horizontal arcs those geodesics along which the argument of d is zero. Likewise we call vertical those for which the argument is . The local charts z ;  we have introduced earlier transforms rectangular boxes in D defined by horizontal and vertical arcs into actual rectangles in the complex plane, while at the same time the Teichm¨uller mapping is affine, transforming one rectangle into the other, when read in those coordinates. An argument similar to the one we have used for Gr¨otzsch problem yields the uniqueness and the extremality of the Teichm¨uller mapping.

6 A Variation on the Theme: Extremal Mappings of Finite Distortion The integral version of the extremal mapping problem, as well as the notion of map with integrable power of the distortion have appeared in the work of Ahlfors in 1954 [1] and later in several papers from the Russian school, in particular Semenov [60–62] and references therein. As we have seen, in Ahlfors’s approach to the extremal mapping problems he used a relaxation of the L1 variational problem, where the interest is shifted to minimizers of the Lp norm of the dilation, rather than to the L1 norm. Problem 5. Lp variational problem Let u0 W ˝ ! ˝ 0 be a homeomorphism of finite distortion. Among all homeomorphisms u W ˝ ! ˝ 0 whose extension to @˝ coincide with u0 find one minimizing 

Z ˝

where

 jduj dx; .det.du//1=n

W Œ1; 1/ ! Œ1; 1/ strictly increasing convex function with

.1/ D 1.

This problem, along with generalizations to more general boundary data, has recently been studied in a sequence of papers by Astala, Iwaniecz, Martin, Onninen

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L. Capogna

and several collaborators, see [7, 9]. In the following we give a quick survey of their work. Remark 9. From a Calculus of Variations point of view, one can see that following [39, Sect. 8.8.2] the functional 

Z F .du; ˝/ D ˝

 jduj dx; .det.du//1=n

although not convex, is indeed quasiconvex, i.e. for every constant differential A 2 Rnn and for any  2 C01 .˝/ one has F .A; ˝/  F .A C d; ˝/: This notion was introduced by Morrey in 1952, see [50, 51]. Quasiconvex energy densities are those for which affine deformations are minimizers with respect to their own boundary conditions. We recall that quasiconvexity plus some growth estimates are roughly equivalent to lower semicontinuity, see Giaquinta’s book [31] for a more detailed statement. Hence quasi-convexity is used often to prove existence of minimizers (as well as regularity of the extremals). In the case at hand there are two problems: • The growth conditions are not satisfied. • There is a topological constraint: The space of competitors is not a vector space. In conclusion, the results currently available from Calculus of Variations are not sufficient to attack the problem and new techniques are needed.

6.1 The Finite Distortion Version of Gr¨otzsch Problem Let R D Œ0; 1  Œ0; 1 and R0 D Œ0; 2  Œ0; 1. The same argument holds for any pair of rectangles. Consider the set F =f all homeomorphisms u W R ! R0 such that 1;1 u 2 Wloc .R; R2 / taking vertices into verticesg. Theorem 16. [7] There is a unique minimizer for the L1 variational problem: ˇ Z ˇˇ 2 ˇ ˇ jduj ˇ min ˇ ˇ u2F R ˇ det.du/ ˇ Remark 10. The affine map .x; y/ ! .2x; y/ sends R to R0 by mapping vertices to vertices and has distortion 5 jduj2 D : det du 2

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

27

If we were to measure distortion using the operator norm we would have obtained jdujO D 2 and hence jduj2O D 2: det du Proof. The proof is very similar to the proof of Gr¨otzsch problem we presented earlier: Setting u D ˛ C iˇ and arguing as we did then yields Z

Z ˛dy D 2 and

ˇdx D 1

@R

@R

Using Stokes Theorem yields Z

Z ˛x dxdy D 2 and R

ˇy dxdy D 1 R

and Z .2˛x C ˇy /dxdy D 5: R

Z 5D R

Z p Z q 2 .2˛x C ˇy /dxdy  2 C1 ˛x2 C ˇy2 dxdy R

R

p p Z p Z  5 jjdujjdxdy D 5 K.u; z/ det.du/dxdy R

R

s sZ sZ p Z p  5 K 2 .u; z/dxdy det.du/dxdy D 10 K 2 .u; z/dxdy: R

R

R

This shows that the minimum of the functional is 5=2, which is achieved by the linear map .x; y/ ! .2x; y/. An examination of the case when the inequalities above are equalities yields that the minimum can only be achieved by this linear map. t u

6.2 Trace Norm vs. Operator Norm In [7], Astala et al. show that if the above problem one substitutes the operator norm jAjO D maxjvjD1 jAvj to the Hilbert-Schmidt norm, i.e. one studies minimizers of

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L. Capogna

ˇ Z ˇˇ 2 ˇ ˇ jdujO ˇ ˇ ˇ R ˇ det.du/ ˇ then the situation changes completely and one can find infinitely many minimizers. To see this first one uses the argument above to show that ˇ Z ˇˇ 2 ˇ ˇ jdujO ˇ 2 ˇ: ˇ R ˇ det.du/ ˇ Next we observe that there is a 1-parameter family of minimizers for a 2 Œ0; 1/, ( U.x; y/ D

x C iy 2a 1a x 

a 1a

for x C iy 2 Œ0; a  Œ0; 1 C iy for x C iy 2 Œa; 1  Œ0; 1:

6.3 Affine Boundary Data Following Astala, Iwaniecz, Martin and Onninen [7] we consider more general affine boundary data in higher dimension. Let us start with the case of affine orientation preserving data u0 W Rn ! Rn prescribed on a domain ˝ with .n  1/ rectifiable boundary. Theorem 17. Given any homeomorphism of finite distortion u W ˝N ! ˝N 0 such that u D u0 on @˝ then ! ! Z Z jdu0 jn jdujn dx  dx det du0 det du ˝ ˝ with equality if and only if u D u0 in ˝. Sketch of the proof. We first recall two basic estimates 1. The sub-gradient inequality .t/ 

.t0 / 

0

.t0 /.t  t0 /

valid for a.e. t; t0 2 Œ1; 1/. 2. The function .x; y/ ! x ˛ =y ˇ defined for x; y 2 R and ˛  ˇ C 1  1 is convex. In particular

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

29

x˛ a˛ a˛1 a˛   ˛ .x  a/  ˇ .y  ˇ/ yˇ bˇ bˇ b ˇC1 Using these estimates one can easily prove that jdujn det du

!

jdu0 jn det du0



! 

0

C

0

! jdu0 jn jdu0 jn2 hdu0 ; du  du0 i det du0 det du0 ! jdu0 jn jdu0 jn .det du0  det du/ det du0 .det du0 /2

Integrating the latter over ˝ yields Z "

jdujn det du

˝

Z "  ˝

C

0

! 

jdu0 jn det du0

!# dx

! jdu0 jn jdu0 jn2 hd u0 ; du  du0 i det du0 det du0 # ! jdu0 jn jdu0 jn .det du0  det du/ dx det du0 .det du0 /2 0

Observe that since du0 D const and u D Ru0 on @˝ then R the first term on the LHS vanishes. As for the second term, note that ˝ det du D ˝ det du0 D j˝ 0 j. Thus the LHS has non-negative integral proving the first assertion. Uniqueness follow from a careful analysis of the consequences of having an identity in the above argument. t u

6.4 More General Boundary Data The case of more general boundary data is still open. In [7], Astala et al. prove the following remarkable theorem: Theorem 18. Let ˝  R2 be a convex domain and set C D fu 2 W 1;2 .˝; R2 / homeomorphism of finite distortion for which Z jduj2 is finiteg: (12) ˝ det du Let u0 2 C .

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L. Capogna

There exists a unique smooth diffeomorphism solution to the minimization problem Z min

u2C ;uDu0 in @˝

˝

jduj2 dx det du

The key idea in the proof is to put in relation the extremal problem above with the classical Dirichlet problem Problem 6 (n-harmonic mappings). Given h0 2 W 1;n .˝ 0 ; Rn /, minimize the nenergy Z ˝0

jdhjnO dy

over the class h 2 h0 C W01;n .˝ 0 ; Rn /. The link between the two problems rests on the following theorem in [7]. 1;n Theorem 19. Let u 2 Wlot .˝; ˝ 0 / be a homeomorphism of finite distortion with

Z

jdu1 jnO ˝

det du1

.x/dx < 1

The inverse map h W ˝ 0 ! ˝ belongs to W 1;n .˝ 0 ; ˝/ and moreover satisfies Z

Z ˝0

jdh.y/jnO dy D

jdu1 jnO ˝

det du1

.x/dx:

See also recent developments by Hencl and Koskela [35], Hencl et al. [36, 37] and by Fusco et al. [25] and by Cs¨ornyei et al. [19]. The proof of Theorem 19 is based on the a.e. differentiability result of Vaisala [66] and on a change of variable formula for homeomorphisms in Sobolev spaces due to Reshetnyak [58]. The two previous theorems state that the minimization problem for the n-energy Z ˝0

jdhj2O dy

of h W ˝ 0 ! ˝ in h0 C W01;2 .˝ 0 ; ˝/ is equivalent to a minimization problem (with corresponding boundary data) for the inner distortion Z ˝

jdu1 j2O .x/dx: det du1

However, when n D 2 one has that inner and outer distortion agree, so that

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

31

jdu1 j2O jduj2O D ; det du det du1 hence minimizing the Dirichlet energy Z ˝0

jdhj2O dy

is equivalent to minimizing the mean dilation Z ˝

jduj2O dx: det du

To return to the Hilbert-Schmidt norm from the operator norm we observe that in n D 2   jAj2O det A jAj2 D C det A det A jAj2O and that the mapping K!KC

1 K

is monotone. Consequently     2    jduj2    D  jdujO  C   det du   det du   1 1  



1

 jduj2O   det du 

1

then the minimization problem for the operator norm has the same solution as the one for the Hilbert-Schmidt norm. Other generalization of Gr¨otzsch problem in higher dimension have appeared in the work of Fehlmann [23].

7 Minimal Lipschitz Extensions We start by looking at a (relatively) simpler functional, which has been extensively studied in the last few decades. Problem 7. Consider two sufficiently smooth bounded open sets ˝  Rn and ˝ 0  RN . Among all Lipschitz mappings u 2 Lip.˝; ˝ 0 / with prescribed trace, find one which minimizes the functional      u !  du   : 1

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L. Capogna

The problem is related to that of finding a canonical (unique) Lipschitz extension of the boundary map. We are interested in questions of existence, uniqueness and continuous dependence from the data.

7.1 Aronsson’s Approach in the Scalar Case N D 1 In the following we describe the N D 1 scalar case for this L1 variational problem. Since jjrujj1 is equivalent to the Lipschitz norm of the scalar function u W ˝ ! R this leads to the following Definition 12. A minimizing Lipschitz extension is an extension of a Lipschitz scalar function f W @˝ ! R to u W ˝ ! R with u D f on @˝ and Lip.u; ˝/ D

sup x¤y;x;y2˝

ju.x/  u.y/j D Lip.f; @˝/ jx  yj

In 1934, independently E. J. MacShane [49] and H. Whitney [69] noted the following: Theorem 20. Such extensions always exist but are not unique. The proof of existence is based on the following observation: Assume that an extension u exists and let D Lip.f; @˝/. Since Lip.u; ˝/ D Lip.f; @˝/ then for all x 2 @˝ and all y 2 ˝ one must have  

ju.y/  f .x/j  ; d.x; y/

and hence f .x/  d.x; y/  u.y/  f .x/ C d.x; y/: Since for all x 2 @˝ and y 2 ˝ f .x/  d.x; y/  u.y/  f .x/ C d.x; y/: if we define the upper and lower functions L.y/ D sup .f .x/  d.x; y// and U.y/ D inf .f .x/ C d.x; y// x2@˝

x2@˝

then these are minimizing Lipschitz extensions of f and so is any u such that L  u  U in ˝.

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

33

Remark 11. We recall an example due to Jensen [44], showing that the problem of minimal Lipschitz extension does not have a unique solution. Let ˝ D B.0; 1/ and f .x; y/ D 2xy. For every 0  ˛  1=2 set (

for x 2 C y 2  ˛ 2

0 p

u .x; y/ D ˛

x 2 Cy 2 ˛/

2xy. .1˛/.x 2 Cy 2 /

for ˛ 2  x 2 C y 2  1:

Note that for x 2 C y 2 D 1 we have u˛ .x; y/ D 2xy and more over Lip.u˛ ; ˝/ D Lip.f; @˝/ So there are infinitely many distinct minimal Lipschitz extensions. Problem 8. Is there a special class of canonical extensions for which uniqueness holds? In 1967 G. Aronsson (see [4]) proposed a way to localize the functional by introducing the formal approximation scheme: • Consider minimzers up of Z jrujp They are p-harmonic, i.e. weak solutions of the equation p up D div.jrup jp2 rup / D 0 • In case u 2 C 2 then we can rewrite this PDE in non-divergence form   jruj2 u D 0 .p  2/jrujp4 ui uj uij C p2 • Let p ! 1 and formally obtain the 1-Laplacian 1 u D uij ui uj D

1 hrjruj2 ; rui D 0: 2

Remark 12. Note that a priori there is no link between solutions of the non-linear, degenerate elliptic PDE ui uj uij D 0 and the problem of minimal Lipschitz extensions. The previous computation is purely formal.

34

L. Capogna

In [4], Aronsson established a link between sufficiently smooth solutions of the infinity Laplacian and correspondingly smooth minimizers of the L1 variational problem. Theorem 21 (1-harmonic implies AMLE [4]). C 2 solutions of 1 u D 0 are Absolute Minimizing Lipschitz Extensions (AMLE), i.e. they minimize Lip.u; D/ on every subdomain D  ˝ Lip.u; D/ D Lip.u; @D/ for every D  ˝ In some sense, the localization built-in in the notion of AMLE is inherited from the Lp problem. The key observation in the proof is that for C 2 solutions one has jruj is constant along integral curves of ru in D  ˝. Aronsson proved that such curves cannot vanish in the interior of the domain and cannot wind up infinitely many times within the domain, hence they have to reach the boundary. As a converse to the previous theorem, Aronsson also proved Theorem 22. Every C 2 AMLE is 1-harmonic. Regarding existence of AMLE, Aronsson established the following Theorem 23 (Existence of AMLE). Given any ˝  Rn and f 2 Lip.@˝/ there exists always a AMLE. We say that a minimal Lipschitz extension u 2 Lip.˝/, with boundary values f 2 Lip.@˝/, has the property A , if for every D 0  D one has u  U 0 in D 0 where U 0 is the upper function in D 0 with respect to the boundary value u. The AMLE corresponding to f 2 Lip.@˝/ is then defined as u.x/ WD inf g.x/ g

where the inf is taken over all functions with the A property with respect to f . N In 1968 Aronsson proved that there can be at most one u 2 C 2 .˝/ \ C.˝/ 2 solution of the 1-Laplacian. Thus showing that there can be at most one C AMLE. Remark 13. Aronsson shows that C 2 solutions have nowhere vanishing gradient, however any C 2 solution with boundary data 2xy must have a critical point. So there may not be C 2 solutions of the 1-Laplacian for this data. The C 2 hypothesis in Aronsson’s work was a severe limitation until in 1993 Jensen (see [44]) removed it using the theory of viscosity solutions, and eventually proving uniqueness of AMLE. Definition 13 (Viscosity solution). A continuous function u is 1-subharmonic in the viscosity sense if for any point x 2 ˝ and  2 C 2 .˝/ such that   u has a minimum at x one has ij i j  0. Supersolutions and solutions are defined in a similar fashion. See [5, 17, 18] for a broad exposition and a detailed list of references.

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

35

Theorem 24 (Bhattacharya et al. [15]). Given fixed boundary data, p-harmonic functions converge to viscosity 1-harmonic functions as p ! 1. Theorem 25 (Jensen [44]). AMLE are viscosity 1-harmonic functions. Theorem 26 (Jensen [44]). The Dirichlet problem for viscosity 1-harmonic functions has a unique solution. At present, thanks to the work of Armstrong, Barron, Champion, Crandall, De Pascale, Evans, Gariepy, Jensen, Juutinen, Manfredi, Oberman, Parviainen, Rossi, Smart, Wang, Yu (to quote just a few) as well as the more recent approach of Naor, Peres, Sheffield, Schramm and Wilson one can prove the uniqueness of AMLE in a variety of ways. In particular, this can be achieved without directly using the 1-Laplacian operator and viscosity solutions for PDE. See [5, 17] for a detailed account of these developments. However, at present, out of this multitude of approaches there is no method that can be immediately extended to approach uniqueness in the vector valued case.

7.2 Aronsson’s Approach in the Vector-Valued Case N > 1 Existence of minimizing Lipschitz extensions for mappings follows from the classical Theorem 27 (Kirszbraun’s Theorem). Let X; Y be two Hilbert spaces and U  X and open set. If f W U ! Y is a Lipnschitz mapping then there exits an extension F W X ! Y with the same Lipschitz constant. However, as we have seen, such extensions may not be unique. Generalizing Aronsson’s approach beyond the real-valued mappings setting, and in particular to the vector-valued case u W ˝ ! RN , is very challenging but, aside from being an important problem in its own right, may have several potential applications in image processing (specifically image inpainting and surface reconstruction). A first step in this direction was taken by Naor and Sheffield in [52] where the focus is on absolutely minimizing Lipschitz extensions in the context of tree-valued mappings. Their main result in [52] consists in existence of a unique AMLE of any prescribed Lipschitz mappings from a subset of a locally compact length metric space to a metric theory. Among other things, the authors also introduce a general definition of discrete infinity harmonic function and prove existence of infinity harmonic extensions. Shortly afterwards, in [63], Sheffield and Smart considered minimizing extensions of the Lipschitz norm Lip.u; ˝/ D sup x;y2˝

d.u.x/; u.y// D sup jdujO : d.x; y/ ˝

36

L. Capogna

as well as its discrete analogue for mappings u W G ! RN where G D .E; X; Y / is a finite graph Su.x/ WD sup d.u.x/; u.y// yx

and two vertices x; y 2 E are in relation if they are separated by an edge. The subset of vertices Y  X here plays the role of the domain for the mapping to be extended. Definition 14. A mapping is said to be discrete 1-harmonic at x 2 X n Y if there is no way to decrease Su.x/ by changing the value of u at x. Peres, Schramm, Sheffield and Wilson have shown that for any Lipschitz f W Y ! R there exists a unique Lipschitz extension u W X ! R which is 1-harmonic. In [63], Sheffield and Smart prove that the uniqueness fails for the vector valued case. To recover uniqueness in [63] Sheffield and Smart introduce a new notion, that of tight extension that is stronger than discrete 1-harmonic: Definition 15 (Tightness). Consider mappings u; v W X ! RN that agree on Y . The mapping v is tighter that u on G if supfsuj su > Svg > supfSvj Sv > sug: The mapping u is tight on G if there is no tighter v. Theorem 28 (Sheffield and Smart [63]). Let G D .E; X; Y / be a finite connected graph. • Every Lipschitz f W Y ! RN has a unique tight extension u W X ! RN . Moreover u is tighter than every other extension of f . P • For every p > 0 consider a minimizer up W X ! Y of Ip .w/ WD x .Sw.x//p , with up D f on Y . As p ! 1 the mappings up ! u pointwise, where u is the tight extension of f . Motivated by this result, Sheffield and Smart introduced a notion of tight extension in the continuous setting: First one sets Lu.x/ D infr>0 Lip.u; ˝ \ B.x; r//: N RN / be two Lipschitz function which agree on @˝. Definition 16. Let u; v 2 C.˝; We say that v is tighter that u if supfLuj Lv < Lug > fLvj Lv > Lug: A mapping u is called tight if there is no tighter v. Definition 17. A principal direction for a mapping u 2 C 1 .˝; RN / is a continuous, unit vector field in ˝ such that at each point it spans the principal eigenspace of

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

37

duT du. If N D 1 then the field is ru=jruj. Note that the existence of a principal direction field implies that the largest eigenvalue for duT du is simple. Recall that the linear transformation y ! du.x/y sends spheres into ellipsoids. The principal direction corresponds to the largest axis of such ellipsoid. Theorem 29 (Sheffield and Smart [63]). Let u 2 C 3 .˝; RN / have a principal direction field a 2 C 2 .˝; Rn /. The mapping u is tight if and only if .uij aj /k ak D 0: Theorem 30 (Sheffield and Smart [63]). Let ˝  C be a bounded open set and N The mapping u is tight if and only u W ˝ ! C be analytic in a neighborhood of ˝. if either (i) @z @z u D 0 in ˝ (i.e., u is affine); or (ii) The meromorphic function

uz uzz Re .uzzz /2

!  2;

in the set where uzz ¤ 0. If u is a diffeomorphism and uzz never vanishes then part (ii) can be rewritten as 1 .  1 / log ju1 z j  0: In other words, the level sets of juz j are convex.

7.3 A Refinement of the Aronsson Equation If we use Aronsson’s scheme in the scalar case then we have seen as (with sufficient regularity) the approximating p-harmonic functions satisfy   jruj2 u D 0 .p  2/jrujp4 ui uj uij C p2 In the vector case, using the Euclidean norm this time, it is easy to see that one obtains instead dulk @jk ul @j u C

jduj2 u D 0: p1

If one lets p ! 1 then formally we obtain the 1-Laplacian system uljk ulk uij D 0 for i D 1; : : : ; N:

(13)

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L. Capogna

uljk ulk uij D 0 for i D 1; : : : ; N: Theorem 31 (Katzourakis [47]). There exists 1-Laplacian above with the same boundary data.

distinct

solutions

of

the

The explicit counterexamples are all 1-dimensional, with ˝  R. In view of such examples it appears that the 1-Laplacian analogue may not an appropriate PDE to characterize unique extremals. In [47], Katzourakis observed that one can recover more information, leading to an augmented (formal) Aronsson system: Recall dulk @jk ul @j u C

jduj2 u D 0: p1

Notice that the term dulk @jk ul @j u lies in the image of du. Consequently for (13) to hold we must also have N.du/ u D 0 where N.du/ D fv 2 RN j duv D 0g is the null-space of the linear application v ! duv and N denotes the orthogonal projection in RN onto such space. Thus a more complete choice for the Aronsson system would be the coupled system uljk ulk uij D 0 and N.du/ u D 0;

(14)

As noted in [47], this system may have discontinuous coefficients even for smooth du, since the rank of du may change from point to point. Although the previous derivation is purely formal one has the following variational interpretation Theorem 32 (Katzourakis [47]). Let ˝  Rn and u 2 C 2 .˝; Rn / be diffeomorphism with non-vanishing Jacobian. The mapping u solves (14) if and only for every subdomain D  ˝ and for every g 2 Lip0 .D; R/ and  2 Rn one has jjrujjL1.D/  jjr.u C g/jjL1 .D/ The actual result is more general and involves C 2 mappings u W ˝ ! RN and an additional variational characterization.

8 Aronsson’s Approach for the Extremal Dilation Problem In this final section we return to the extremal problem for quasiconformal mappings and recall recent results by Raich and the author [16] in which the Aronsson’s approximation scheme is used to introduce a notion of absolute minimizers in the quasiconformal setting. The goal here is to find a candidate PDE that would play

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

39

the role similar to that of the infinity-Laplacian in the AMLE theory. Following the approach of Sheffield and Smart in [63] we focus on the C 2 case. Although this is an unnatural smoothness hypothesis for quasiconformal mappings, it does provide some insights into the general problem. The first step in this approach consists in studying extremal mappings for the corresponding Lp problem. If p > 1, ˝  Rn and the diffeomorphism u 2 C 2 .˝; Rn / is a critical point of the functional Z Fp .u; ˝/ WD

jdujnp dx .detdu/p

(15)

then the mapping u satisfies the system of Euler-Lagrange equations .Lp u/ D np@j i

   np2 2 1;T du S.g/ ; for i D 1; : : : ; n trace.g/ ij

If we let p ! 1 then formally one obtains the Aronsson PDE, .L1 u/i D S.g/ij @j

p trace.g/ D 0 for i D 1; : : : ; n:

(16)

p This PDE tells us that the dilation of the mapping u (i.e. trace.g/ ) is constant along curves tangent to the sub-bundle generated by the rows of S.g/. Problem 9. What is the lowest regularity for the mapping u for which the PDE .L1 u/i D S.g/ij @j

p trace.g/ D 0 for i D 1; : : : ; n

is meaningful? Remark 14. It is tempting to define solutions of (16) as quasiconformal mappings such that their dilation trace.g/ is constant along all curves tangent to the sub-bundle generated by the rows of S.g/. Observe that for this definition to be meaningful at the very least one would need regularity for u such that constant linear combinations of the rows of S.g/ generate integral flows (for instance S.g/ 2 BV ) and the quantity trace.g/ must be continuous (so it can be evaluated along such integral curves). It is important to note that classical solutions of the extremal quasiconformal problems, e.g. Teichm¨uller mappings, solve (16) in the regions where they are C 2 smooth. Proposition 4. (1) Any Teichm¨uller map of the form u WD ı v ı  1 with ;  conformal and v affine is a solution of L1 u D 0. (2) the quasiconformal mappings u.x/ D jxj˛1 x for ˛ > 0 solve L1 u D 0 away from the origin. (3) Let 0 < ˛ < 2 and .r; ; z/ be cylindrical coordinates for x D .x1 ; : : : ; xn / where x1 D r cos , x2 D r sin and xj D zj , 3  j  n. The quasiconformal mapping

40

L. Capogna

( u.r; ; z/ D

.r;  =˛; z/ .r;  C

˛  2˛ ; z/

0 ˛ ˛ < < 2

(17)

solves L1 u D 0 away from the set r D 0. The following theorem establishes a link between the formal PDE (16) and the L1 -variational problem. Theorem 33 ([16]). If u W ˝ ! Rn C 2 is a quasiconformal solution of L1 u D 0 in ˝, then p p (i) For every D  ˝ one has supDN p trace.g/ D sup@Dp trace.g/. (ii) For every D  ˝ one has infDN trace.g/ D inf@D trace.g/. (iii) There exists C D C.n/ > 0 such that for every C 2 domain D  ˝ and wpW DN ! Rn C 2 quasiconformal such that u D w on @ D one has p supD trace.g.u//  C supD trace.g.w//j. (iv) If n D 2 the dilation jgj is constant in ˝ and if u is affine in a neighborhood of @˝ then u must be an affine transformation throughout ˝. Sketch of the proof. Show that any interior maximum points for jgj propagate along curves tangent to the span of the rows of S.g/ until they reach the boundary. This is achieved by using the fact that S.g/ is either vanishing or has at least rank higher than two. This is used to construct a non self-intersecting curve of this kind and showing that (i) its total length must be finite; (ii) the curve cannot vanish in ˝. Points (i) and (ii) imply then that the curve must reach the boundary. t u Theorem 34 ([16]). If u W ˝ ! Rn C 2 is a quasiconformal absolute extremal, i.e. for every D p˝ and w W DN ! Rn p C 2 quasiconformal such that u D w on @ D one has supD trace.g.w//  supD trace.g.w//, then L1 u D 0 in ˝. Sketch of the proof. Arguing by contradiction we assume there is a ball B  ˝ s.t. L1 u ¤ 0 in B. We construct a better competitor for the variational problem: i.e. a C 2 quasiconformal diffeomorphism V W BN ! Rn with same boundary values as u on @B and supB trace g.V / < supB trace g.u/. This is done by perturbing u with a finite number of “bumps” that reduce the dilation near the boundary. t u Remark 15. A similar result was proved much earlier by Barron et al. in their important work [12] with a different, less constructive proof. The advantage of the approach in [16] is that it provides a competitor which is also quasiconformal. Remark 16. Recently in [46], Katzourakis applied the refined derivation technique we described earlier to the quasiconformal setting and obtained the formal extended system: duak Jki dubl Jlj @kl ub C jduj2 ŒN.duJ / ab Jij @ij ub D 0 where J D g 1 S.g/ and g D duT du. The equation is composed of two linearly independent parts. The first, in the case of diffeomorphisms between domains of Rn

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

41

coincides with the system we have described earlier. The second component is new but it is not yet clear how it relates to the variational problem. The paper [46] also provides a necessary and sufficient condition for C 2 mappings to satisfy this system.

8.1 A Gradient Flow Approach Let ˝  Rn is a bounded, C 2;˛ smooth, open set. Currently we do not know how to prove existence of solutions of (16) or how to attack the extremal problems for a fixed homotopy class of quasiconformal mappings. A possible strategy for a proof consists in finding solutions of a gradient flow up .x; t/ for the functional Fp .u; ˝/ defined in (15). The long term existence and suitable estimates (independent of p as p ! 1) for such flow then would yield the existence of the asymptotic mapping wp .x/ D limt !1 up .x; t/ which would be a candidate for the Lp minimization problem within the homotopy class of the initial data. The solution to the L1 problem then could be achieved by establishing estimates on wp independent of p and letting p ! 1. For a fixed diffeomorphism u0 W ˝ ! Rn , we want to study diffeomorphism solutions u.x; t/ of the initial value problem ( @t u D Lp u in ˝  .0; T /: (18) at ˝  ft D 0g u D u0 where we recall that   np2 .Lp u/i D np@j jgj 2 du1;T S.g/ ; for i; j D 1; : : : ; n ij

If there is a T > 0 such that a solution u 2 C 2 .˝  .0; T // exists with det du > 0 in ˝  .0; T /, then   Z d 1 2 Fp .u; ˝/ D  jLp uj dx  0; dt j˝j ˝ i.e., the p-distortion is nonincreasing along the flow. Hence we obtain Proposition 5. If u 2 C 2 .˝  Œ0; T /; Rn / \ C 1 .˝N  Œ0; T /; Rn / is a solution of p (18) with det du > 0 in ˝N  Œ0; T /, then for all 0  t < T , kKup kLp .˝/ D R T p kKu kLp .˝/  0 kLp u. ; t/kL2 .˝/ dt and consequently kKu kLp .˝ft g/  kKu0 kLp .˝/ :

(19)

It is immediate to show that the functional Fp .u; ˝/ is invariant by conformal deformation. Therefore, if we let s 7! Fs W Rn ! Rn be a one-parameter semi-group of conformal transformations, then solutions to the PDE system

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L. Capogna

ˇ ˇ d @t u D Lp u C Fs .u/ˇˇ ds sD0 would also satisfy (19). Recall that the flow Fs is conformal if 1 d D C d DT  trace .d D/In D 0 2 n ˇ ˇ ˇ ˇ d d Fs /ˇˇ ı F01 D . ds Fs /ˇˇ and S denotes the Ahlfors operator. where D D . ds sD0 sD0 If n D 2 then this amounts to @zN D D 0: If n  3 there is more rigidity and conformality requires that S.d D/ D

D.x/ D a C Bx C 2.c x/x  jxj2 c for any vectors a; c and matrix B with S.B/ D 0 (see [59]). We observe that in light of conformal invariance, if u.x; t/ is a solution of (18) in ˝  .0; T / and v.x; t/ D ıu. x; ı 2 t/ for some ; ı > 0, then v.x; t/ is still a solution with initial data v0 .x/ D ıu0 . x/ in an appropriately scaled domain. Applying inversions in a suitable way will also yield new solutions from u.x; t/. Usual elliptic/parabolic PDE techniques do not apply. The main difficulty consists in the fact that the functional is not convex but only quasi-convex (in the sense of Morrey). In order to study the gradient flow it helps to rewrite the system in non-divergence form.3 k .Lp u/i D Aik j ` .du/uj ` :

with Aik j ` .q/

" # qij qk` jqjnp2 ji `k 2 `i jk `k ji np.qk` q C qij q /  n.np  2/ D p  jqj .q q C pq q /  nıki ıj ` : .det q/p jqj2

This form of the PDE has a remnant of ellipticity in the form of the so-called Legendre-Hadamard property: There exists constants C1 ; C2 > 0 depending respectively only on n and on p and on n such that for a.e. q 2 Rnn and for all ;  2 Rn C1 .n; p/pjj2 jj2

3

jqjnp2 j `  Aik j ` .q/i  k  . det q/p   jqjnp2 jqjn.pC2/2  C2 .n/p 2 jj2 jj2 C . det q/p .det q/pC2

To do this however one has to assume existence of two derivatives for the solution.

L1 -Extremal Mappings in AMLE and Teichm¨uller Theory

43

Using the latter, Raich and the author established in [16] certain Schauder type estimates for the gradient flow (i.e. a gain of two derivatives with respect to the regularity of the right hand side and the coefficients of the PDE). The Schauder estimates in turn allow to rephrase the system (18) as a fixed point problem for a contraction map, leading to the short time existence and uniqueness result Definition 18. Let ˝  Rn be a smooth bounded domain and for T > 0 let Q D ˝  .0; T /. The parabolic boundary is defined by @par Q D .˝  ft D 0g/p[ .@˝  .0; T //. The parabolic distance is d..x; t/; .y; s// WD max.jx  yj; jt  sj/. For ˛ 2 .0; 1/ we define the parabolic H¨older class C 0;˛ .Q/ WD fu 2 C.Q; R/j kukC ˛ .Q/ WD Œu˛ C kuk0 < 1g, where Œu˛ WD

ju.x; t/  u.y; s/j ˛ .x;t /;.y;s/2Q and .x;t /¤.y;s/ d..x; t/; .y; s// sup

and juj0 D supQ juj: For K 2 N we let C K;˛ .Q/ D fu W Q ! Rj @xi1 @xiK u 2 C 0;˛ .Q/g. Proposition 6. Let u0 W ˝ ! Rn be a C 2;˛ diffeomorphism for some 0 < ˛ < 1 N Assume that Lp u0 D 0 for all x 2 @˝. with det du0  " > 0 in ˝. There exist constants C; T > 0 depending on p; n; ˝; "; ku0 kC 1;˛ .˝/ N , and a sequence h 2;˛ of diffeomorphisms fu g in C .Q/ with Q D ˝  .0; T / so that (a) det uh  2" for all .x; t/ 2 Q, (b) kuh kC 2;˛ .Q/ C k@t uh kC 0;˛ .Q/  C ku0 kC 2;˛ .˝/ , ( h1 /@j @l uh;k D 0 in Q @t uh;i  Aik jl .du (c) on @par Q: uh D u0 Theorem 35. If u.x; 0/ 2 C 2;˛ Cboundary conditions then there exists a unique C12;˛ .˝  .0; T /; Rn / solutions for small T D T .p; n; u0 ; ˝/ > 0. Although the previous result establishes short time existence, the dependence of the interval of existence from p remains an obstacle to the study of the asymptotic limit p ! 1. In order to carry out the program we outlined earlier, one would need a global existence result, as well as estimates independent of p as p ! 1. Currently there is very little literature about gradient flows of quasi-convex functionals but a an important paper of Evans et al. [21] lays out a strategy to obtain global estimates: Following [21], Raich and the author in [16] let ˇ D det du1 then show that ˇ solves the scalar PDE @t ˇ D Œaij .du/ˇij with   dujk duik p np aij D p ıij  n jgj : jduj2

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Although the lack of a sign in the symbol prevents us from using the maximum principle and establishing immediate global bounds, this PDE is a starting point for the study of global estimates. Acknowledgements We wish to thank C. Gutierrez and E. Lanconelli for the scientific organization of the C. I. M.E. course and for inviting the author to present these lectures. We are also grateful to P. Zecca and to all the staff of C. I.M.E. for their logistic support and hospitality. The author is partially supported by the US National Science Foundation through grants DMS-1101478 and DMS-0800522

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Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs Pengfei Guan

Abstract The notes consider two special fully nonlinear partial differential equations arising from geometric problems, one is of elliptic type and another is of parabolic type. The elliptic equation is associated to the problem of prescribing curvature measures, while an inverse mean curvature type of parabolic equation is introduced to prove the isoperimetric type inequalities for quermassintegrals of kconvex starshaped domains.

The material in the notes is compiled from the lectures given in the CIME Summer School in Cetraro, 2012. It treats some nonlinear elliptic and parabolic partial differential equations arising from geometric problems of hypersurfaces in RnC1 . A curvature type of elliptic equation is used to solve the problem of prescribing curvature measures, which is a Minkowski type problem. An inverse mean curvature type of parabolic equation is employed for the proof of isoperimetric type inequalities for quermassintegrals of k-convex starshaped domains. Both types of equations are fully nonlinear, they belong to the category of general geometric fully nonlinear PDE. The emphasis of the notes is the a priori estimates, which is the key in the theory of fully nonlinear PDE. These estimates are intend to be self-contained here, with minimal assumptions on basic knowledge in PDE and geometry, namely the standard maximum principles for linear elliptic and parabolic equations, the elementary formulas of Gauss, Codazzi and Weingarten for hypersurfaces in RnC1 , and the curvature commutator identities. Two theorems we would use without proof for higher regularity are: the Evans-Krylov Theorem [11, 31] for uniformly fully nonlinear elliptic equations and the Krylov Theorem [31] for uniformly parabolic

P. Guan () Department of Mathematics, McGill University, Montreal, QC H3A 2K6, Canada e-mail: [email protected] L. Capogna et al., Fully Nonlinear PDEs in Real and Complex Geometry and Optics, Lecture Notes in Mathematics 2087, DOI 10.1007/978-3-319-00942-1 2, © Springer International Publishing Switzerland 2014

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P. Guan

fully nonlinear PDE, since the proofs of these deep results would take considerable space. The topics dealt in this notes are special samples of geometric nonlinear PDE. It is our hope they can serve as an introduction to the general theory of geometric analysis. The notes are organized as follows. The curvature measures are introduced through the Steiner formula in differential geometric setting in Sect. 1, where the Steiner formula and the Minkowski identity are proved. As the geometric objects and the associated differential equations are involved the elementary symmetric functions, some important properties of these functions are collected in Sect. 2 with proofs, except the theory of hyperbolic polynomials of Garding which is put in the Appendix. Section 3 deals with the problem of prescribing curvature measures. A k-curvature fully nonlinear elliptic equation is set up there together with the a priori estimates of the solutions of the equation. Section 4 is devoted to the proof of the isoperimetric inequalities for quermassintegrals of k-convex star shaped domains, via parabolic approach. Again, the main part is the a priori estimates for the solutions of the corresponding parabolic equation. The literature comments appear at the end of the notes.

1 The Steiner Formula and Curvature Measures Suppose  is a domain in RnC1 , for each x 2 RnC1 , denote p.; x/ to be the set of the nearest points in  to x. Given any Borel set ˇ 2 B.RnC1 /, 8s > 0, consider As .; ˇ/ WD fx 2 RnC1 j0 < d.; x/  s and p.; x/ 2 ˇg which is the set of all points x 2 RnC1 for which the distance d.; x/  s and for which the nearest point p.; x/ belongs to ˇ. If @ is smooth and ˇ is open, for s > 0 small, one may write As .; ˇ/ D fX C t.X /

jX 2 ˇ \ M; 0  t  s; g

where .X / is the outer normal of M at X . We assume the boundary of , M D @, is C 2 (or smoother). Let .X / D .1 .X /; ; n .X // be the principal curvatures of X 2 M . To calculate the volume of As .; ˇ/, pick any local orthonormal frame of M , so that the second fundamental form .Wij .X // of M at X is diagonal. As .X C t.X //i D .1 C tWii /Xi , and .X / is orthogonal to Xi , the volume element at X C t.X / is simply dV D .

n Y

.1 C tWii //dM dt D

i D1

n X i D0

i ..X //t i dM dt;

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

49

where i ./ is the i -th elementary symmetric function of  (see Definition (6)), and where dg is the volume element with respect to the induced metric g of M in RnC1 . Therefore, Z sZ V .As .; ˇ// D 0

n X

n Z X i ..X //t dM dt D . i

ˇ\M i D0

i D0

i ..X //dM / ˇ\M

s i C1 : i C1

Set Cm ./ D nm ./dM ;

m D 0; 1; ; n:

(1)

We have proved the Steiner formula, V .As .; ˇ// D

n X

s nC1m Cm .; ˇ/; nC1m mD0

(2)

for ˇ 2 B.RnC1 / and s > 0. In the context of classical convex geometry, the coefficients C0 .; /; ; Cn .; / in (2) are called curvature measures of the convex body . Formula (1) indicates that Cm .; / is well defined if @ is C 2 without convexity assumption. In general, Cm ./ is a signed measure. The positivity of Cm ./ for 0  m  k is related to the notion of k-convexity (Definition 3.1). The global quantities Z Vnm ./ D Cn;k

m ./dM ;

m D 0; 1; ; n;

(3)

M k .1; ;1/ where Cn;k D k1 , are called the quermassintegrals of  in convex geometry, .1; ;1/ if  is convex. Again, we note that these quantities are well defined for general C 2 domain  without convexity condition. It is clear that the curvature measures capture the geometry of M .

1. What are the relations between quermassintegrals? 2. How much information can we extract from the curvature measures? These are the main questions we want to deal with in this notes. The first question has satisfactory answer when  is convex, which corresponds to the classical Alexandrov-Fenchel inequalities. Generalization of these inequalities to non-convex domains has gained much interest recently, but remains largely unsettled. We will focus on a class of non-convex star-shaped domains, where a clean result can be established. The second question can be answered in terms of the Minkowski type problem, the problem of prescribing curvature measures. It turns out there is an affirmative answer if we restrict ourselves to the class of non-convex star-shaped domains.

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There is a different expression for Vnm ./ involving the support function u.X / D hX; .X /i. The Minkowski identity states that 8k  1, Z

Z uk ./dM D Cn;k

k1 ./dM ;

M

(4)

M

By the Divergent theorem, VnC1 ./ D

1 nC1

Z udM : M

From (4), we may define Z V.nC1/k ./ D

uk ./dM ;

(5)

M

for k D 0; ; n. VnC1 ./ is multiple of the volume of  by a dimensional constant, Vn ./ is a multiple of the surface area of @ by another dimensional constant. In convex geometry, u is called the support function of . The Minkowski identity (4) can be verified using the fact that k has divergent free structure (Lemma 2.1). Again, pick a local orthonormal frame on M , let h D .Wij / be the second fundamental from and let g 1 h D .hij / be the Weingarten tensor. We compute .

jX j2 /ij D Xi Xj C Xij D ıij  hX; .X /iWij D ıij  uWij : 2 ij

Contracting with k D

@k .g 1 h/ @hij

Z ij

M

k .

and integrating over M

jX j2 /ij D 2

Z . M

X

ij

ij

k ıij  uk Wij /:

i

As ij

k ıij D .n  k C 1/k1 ;

ij

k Wij D kk ;

and by (8), we get Z 0 D .n  k C 1/ M

k1 .g 1 h/  k

Z

uk .g 1 h/: M

This is exactly the identity (4). The Minkowski addition of two sets 1 ; 2  RnC1 is defined as 1 C 2 D fz D x C yjx 2 1 ; y 2 2 g:

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

51

The Minkowski addition is one of the basic operation in convex geometry. For general domain , when 0  s small, one may define s D fz D x C yjx 2 ; y 2 Bs g; where Bs is the ball centered at the origin with radius s. s D fX C t.X /

jX 2 ; 0  t  s:g

If M D @ is smooth, the boundary @s D Ms is also smooth and can be written as Ms D fX C s.X / jX 2 M:g Moreover, the normal of Ms at Xs D X C s.X / is the same as .X / for each X 2 M . The support function of s is us .X s / D u.X / C s. For any local orthonormal frame e1 ; ; en on M such that h D .Wij / is diagonal at the point, one may calculate the induced metric gs on M s gs D

n X

.1 C hii /2 ei ˝ ei ;

i D1

and the area element of M s dMs D det.I C sg 1 h/dM : By the Minkowski identity, the volume of s can be computed as V .s / D D

D

1 nC1 1 nC1 1 nC1

Z

us det.I C sg 1 h/dM M

Z X n .u C s/s i i .g 1 h/dM M i D0

Z X n .us i i .g 1 h/ C s i C1 /i .g 1 h/dM M i D0

1 X n C 1 i C1 s D n C 1 i D0 i C 1 n

D

nC1 X i D0

Z

i cnC1 t nC1i Vi ./;

1

i .g h/dM M

1 C nC1

Z udM M

52

P. Guan

2 Some Properties of Elementary Symmetric Functions The elementary symmetric functions appear naturally in the geometric quantities in the previous section. In order to carry on analysis, we need to understand properties of the elementary symmetric functions. For 1  k  n, and D . 1 ; : : : ; n / 2 Rn , the k-th elementary symmetric function is defined as X

i1 : : : ik ; (6) k . / D where the sum is taken over all strictly increasing sequences i1 ; : : : ; ik of the indices from the set f1; : : : ; ng. The definition can be extended to symmetric matrices. Denote .W / D . 1 .W /; : : : ; n .W // to be the eigenvalues of the symmetric matrix W , set k .W / D k . .W //: It is convenient to set 0 .W / D 1;

k .W / D 0;

for k > n.

It follows directly from the definition that, for any n  n symmetric matrix W , and 8t 2 R, n .I C tW / D det.I C tW / D

n X

i .W /t i :

(7)

i D0

Conversely, (7) can also be used to define k .W /, 8k D 0; ; n. An important property of k is the divergent free structure. Suppose M is a general Riemannian manifold of dimension n, W is a symmetric tensor on M . We call W is Codazzi if DW D 0. This property is equivalent to say that, for any local orthonormal frame .e1 ; ; en / on M , write W D .wij /, then wij;l D rel wij is symmetric with respect to i; j; l. Some classical examples are 1. Second fundamental form h of any hypersurface in space form N.c/ with constant sectional curvature c, this follows from the Codazzi equation; 2 2. W D r v C cv, 8v 2 C 3 .N.c//. Throughout the rest of the notes, we will use Einstein summation convention, unless it is otherwise indicated. Below is the statement of divergent free structure of k . Lemma 2.1. Suppose e1 ; ; en is a local orthonormal frame on M , W D .wij / is a Codazzi tensor on M , then for each i , n X @k . /j .W / D 0: @w ij j D1

(8)

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

53

Proof. We first verify (8) for k D n. Denote C il to be the cofactor of W , i.e., @n D C il ; @wil

C il wlj D det.W /ıji :

Differentiate above identity in em direction and contract with C jm , C jm Cmil wlj C C il wlj;m C jm D ıji .det.W //m C jm : If det.W / ¤ 0 at the point, we get Cmim D C pq C im wpq;m  C il C jm wlj;m D C pq C im wpq;m  C il C jm wjm;l D 0: If det.W / D 0 at the point, we may approximate W by Codazzi tensor WQ D W Ctg where g is the metric tensor on M such that det.WQ / ¤ 0 for t small. Equation (8) is verified for the case k D n. Observe that, for t 2 R, n .WQ / D

n X

t m nm .W /:

mD0

Apply (8) for the case k D n, n X mD0

tm

X @nm . .W //j D 0: @wij j

Since it is true for all t 2 R, we must have 8m, X @nm . .W //j D 0: @wij j t u The following gives explicit algebraic formulas for k .W /. Proposition 2.2. If W D .Wij / is an n  n symmetric matrix, let F .W / D k .W / for 1  k  n. Then the following relations hold. k .W / D

F ˛ˇ WD

1 kŠ

n X i1 ;:::;ik D1 j1 ;:::;jk D1

@F .W / @W˛ˇ

ı.i1 ; : : : ; ik I j1 ; : : : ; jk /Wi1 j1 Wik jk ;

54

P. Guan

D

F ij;rs WD D

1 .k  1/Š

n X

ı.˛; i1 ; : : : ; ik1 I ˇ; j1 ; : : : ; jk1 /Wi1 j1 Wik1 jk1

i1 ;:::;ik1 D1 j1 ;:::;jk1 D1

@2 F .W / @Wij @Wrs 1 .k 2/Š i

n X

ı.i; r; i1 ; : : : ; ik2 I j; s; j1 ; : : : ; jk2 /Wi1 j1 Wik2 jk2 ;

1 ;:::;ik2 D1 j1 ;:::;jk2 D1

where the Kronecker symbol ı.I I J / for indices I D .i1 ; : : : ; im / and J D .j1 ; : : : ; jm / is defined as

ı.I I J / D

X

8 ˆ ˆ1;
0; : : : ; k . / > 0g:

A n  n symmetric matrix W is called belong to k is .W / 2 k . Let W 1 ; ; W n be n  n symmetric matrices, define n .W 1 ; : : : ; W n / to be the coefficient in front of the factor t1 tn of the polynomial det.t1 W 1 C C tn W n /. It is called the mixed determinant of W 1 ; ; W n . In general, for 1  k  n, we n 1 k define k .W ; : : : ; W / D k n .W 1 ; : : : ; W k ; I; ; I /, where the identity matrix I appears .n  k/ times. k .W 1 ; : : : ; W k / is called the complete polarization of the symmetric function k . The following Garding inequality plays important role in geometric PDE. Lemma 2.4. k is a convex cone. 8W i 2 k ; i D 1; : : : ; k, k2 .W 1 ; W 2 ; W 3 ; ; W k /  k .W 1 ; W 1 ; W 3 ; ; W k /k .W 2 ; W 2 ; W 3 ; ; W k /; (11) equality hold if and only if W 1 and W 2 are proportional. And 1

1

k .W 1 ; ; W k /  kk .W 1 ; ; W 1 / kk .W k ; ; W k /;

(12)

the equality holds if and only if W i ; W j are pairwise proportional.

Lemma 2.4 is a special case of Garding’s theory of hyperbolic polynomials, which can be found in Appendix. The convexity of k follows from Proposition 5.2, (11) and (12) follow from Corollary 5.4 and Proposition 5.6 in Appendix. Inequality (11) yields the Newton-MacLaurin inequality. Lemma 2.5. For W 2 k , .n  k C 1/.k C 1/k1 .W /kC1 .W /  k.n  k/k2 .W /;

(13)

and kC1

kC1 .W /  cn;k k k .W /;

(14)

kC1

where cn;k D c > 0.

kC1 .I / k  k

.I /. The equality holds if and only if W D cI for some

Proof. If kC1 .W /  0, as W 2 k , (13) is trivial. We may assume kC1 .W / > 0, so W 2 kC1 . Replace k by k C 1 in (11), and set W 1 D I , W 2 D D W kC1 D W 2 k , (13) follows from (11). The similar argument yields (14) using (12). t u We remark that the Newton-MacLaurin inequality is valid for general symmetric matrix W (e.g., [28]). The following lemma establish connection of k with the ellipticity of Hessian and curvature equations.

56

P. Guan

@F Lemma 2.6. Let F D k , then the matrix . @W / is positive definite for W 2 k . ij where Wij are the entries of W . If W 2 k , then .W ji / 2 k1 ; 8k D 0; 1; ; n, i D 1; 2; ; n, where .W ji / is the matrix qP with i -th column and i -th row deleted. 2 Furthermore, if W 2 k and kW k D i;j wij  R for some R > 0, then there is cn;k > 0 depending only on n; k, such that

k .W / 1 k1

I .

R.1 C cn;k k1 .I //

@F /  Rk1 k1 .I /I: @Wij

(15)

Proof. Fix W 2 k , for any positive definite matrix A D .aij /, by Lemma 2.4, 0 < k .W; ; W; A/ D

X @F .W /aij : @wij ij

@F @F This implies the positivity of . @W /. By Proposition 2.2 and the positivity of . @W /, ij ij for each l  k, W 2 k , and for any i 2 f1; ; ng,

0
0, then there is cn;k > 0 depending only on n; k, such that

F .W / 1 k1

R.1 C cn;k k1 .I //

I .

@F /  .n  k C 1/I: @wij

Moreover, the function F is concave in k1 .

(16)

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

57

Proof. To simplify notation, define Qm D

m : m1

For any l < k, kl Y k D QlCj : l j D1

(17)

As QlCj > 0 for j D 1; ; k  l, for the first statement in lemma, we only need @Q .W / to check the positivity of . mC1 / for W D .wij / 2 k and for m D l; ; k  1. @wij By product rule, m .W / @QmC1 .W / D @wij

@mC1 .W /  mC1 .W @wij m2 .W /

By Proposition 2.2, the positivity of .

@j .W / @wij /

m .W / / @@w ij

is invariant under orthonormal

transformations, we only need to check the positivity of i 2 f1; ; ng and m D l; ; k  1. Again, m . / @QmC1 . / D @ i

@mC1 . / @ i

:

@QmC1 . / @ i

m . /  mC1 . / @@

i

m2 . /

D

m . /m . ji /  mC1 . /m1 . ji / m2 . /

D

m . ji /m . ji /  mC1 . ji /m1 . ji / m2 . /



m2 . ji / n .n  m/.m C 1/ m2 . /

for 2 k ,

(18)

> 0; the Newton-MacLaurine inequality (13) is used in the last step as . ji / 2 k1 for each i . In particular, if m D k  1 and W 2 k , for each i , 0< 

@Qk . / X @Qk . /  @ i @ i i X k1 . ji / i

k1 . /

D n  k C 1:

58

P. Guan

This provides the upper bound in (16). By (14) 1 k .W /  cn;k1 kk .W /: k1 .W /

For each i D 1; ; n, k1 . ji / k . / k1 . ji / D : k1 . / k1 . / k . / Now the lower bound in (16) follows from (18) and (15). Notice that if f1 > 0 and f2 > 0 are two concave function, for any 1  ˛  0,  f D f1˛ f21˛ is also concave. Hence, we only need to check the concavity of mC1 m  in mC1 . In fact, we show mC1 m in m . m D 0 is trivial. For m D 1, there is a useful explicit formula. 8 ; ˙  2 1 , we have algebraic identity P . i .i 1 . /  i 1 .///2 : 2Q2 . /  Q2 . C /  Q2 .  / D 1 . /1 . C /1 .  / This yields, @2 Q2 . D @2 

P

i .i 1 . /  i 1 ./// 13 . /

2

This gives the concavity of 21 on 1 . For m > 1, we use induction. For 2 m , for each i 2 f1; ; ng fixed, by (10) and Corollary 2.6,

i C Qm . ji / D

mC1 . / > 0: m . ji /

Apply the last identity in Proposition 2.2, .m C 1/Qm . / D D

X m1 . ji / / . i  2i m . / i X . i  2i i

D

X . i  i

m1 . ji / / m . ji / C i m1 . ji //

2i /:

i C Qm . ji /

For any  2 Rn with jj D 1, set ˙ D ˙ . Take  > 0 small enough such that

˙ 2 m , using the above identity for ; ˙ , one compute

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

59

.m C 1/.2QmC1. /  QmC1 . C /  QmC1 .  // D

X i

 C

. i C i /2 . i  i /2 C Qm . C ji / C i C i Qm .  ji / C i  i

.2 i /2  Qm . C ji / C Qm .  ji / C 2 i X i

D

X i

2

.2 i /2 2 2i   Qm . C ji / C Qm .  ji / C 2 i

i C Qm . ji /

.. i C i /Qm .  /  . i  i /Qm . C //2 .Qm . C / C i C i /.Qm .  / C i  i /.Qm . C /CQm .  /C i /

X i

2i

Qm . C ji / C Qm .  ji /  2Qm . / .Qm . C ji / C Qm .  ji / C 2 i /. i C Qm . ji //

Thus, 

@2 QmC1 2QmC1. /  QmC1 . C /  QmC1 .  / D lim 2 !0 @  2 X Qm . C ji / C Qm .  ji /  2Qm . /  lim 2

2i 2 !0  .Q .

m C ji / C Qm .  ji / C 2 i /. i C Qm . ji / i D

2

X

2i . @@Q2m /. ji /

i

.m C 1/.Qm. ji / C i /2

As . ji / 2 m1 , by induction hypothesis,

:

@2 Q m . ji / @ 2

 0.

t u

The following lemma will play key role for the problem of prescribing curvature measures. 1 Lemma 2.8. Let ˛ D k1 , if W 2 k is a symmetric tensor on a Riemannian manifold M . For any local orthornormal frame fe1 ; ; en g, denote Wij;s D res Wij . Then    .k /s .1 /s .k /s .1 /s .˛  1/ : (19) .k /ij;lm Wij;s Wlm;s  k   .˛ C 1/ k 1 k 1

 Proof. By the concavity of

0

k 1

1  k1

.W /, we have

  1  k k1 @2 Wij;s Wlm;s : @Wij @Wlm 1

(20)

60

P. Guan

Denote ˛ D

1 k1 .

Direct computations yield,  ˛ @2 k 0 Wij;s Wlm;s @Wij @Wlm 1  ˛  D˛

k 1 ij

.k /ij;lm k lm

1/  2˛.k k/ . C 1

C

.˛1/.k /ij .k /lm k2

.˛C1/.1 /ij .1 /lm 12

(21)



Wij;s Wlm;s

Equivalently, .k /ij;lm Wij;s Wlm;s k

 ij lm ij lm 1/   .˛1/.k2/ .k /  2˛.k k/ . 1 k  .˛C1/.1 /ij .1 /lm Wij;s Wlm;s C 2 1

   .kk/s 

.1 /s 1

(22)

  .k /s .1 /s .˛  1/ k  .˛ C 1/ 1 t u

Note in Lemma 2.8, one may replace k by any positive function F with the property that . F1 /˛ is concave for some ˛ > 0. The following is a corollary of Lemma 2.8. Corollary 2.9. If

.1 /s 1

D

.k /s k

 r for some r,

.k /ij;lm WijIs WijIs  max 2r.k /s 

k r 2 k ; 0 : k1

(23)

3 Prescribing Curvature Measures Assume   RnC1 is a bounded star-shaped domain with respect to the origin. We may parametrize M D @ over Sn by positive radial function  Due to the parametrization, the prescribe curvature measure problem for this class of domains can be reduced to a curvature type nonlinear partial differential equation of  on Sn . We want to establish the existence theorems of prescribing general .n  k/-th curvature measure problem with k > 0 on bounded C 2 star-shaped domains. When k D n, the prescribing curvature measure C0 is the Alexandrov problem corresponding to a Monge-Amp`ere type equation on Sn , which won’t be treated here.

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

61

In order to make the problem in proper PDE setting, we need to impose some geometric condition on @. Definition 3.1. A domain  is called k-convex if its principal curvature vector .x/ D .1 ; ; n / 2 k at every point x 2 @. For each star-shaped domain  with M D @, express M as a radial graph of Sn , RM W Sn ! M z 7! .z/z: From (1) the .n  k/-th curvature measure on each Borel set ˇ in Sn can be defined as Z Ck .M; ˇ/ WD

k ./dg : RM .ˇ/

The precise statement of the problem for prescribing .nk/-th curvature measure is: given a positive function f 2 C 2 .Sn /,Rfind a closed hypersurface M as a radial graph over Sn , such that Cnk .M; ˇ/ D ˇ f d for every Borel set ˇ in Sn , where d is the standard volume element on Sn . 2 n For the C p graph M on S , denote the induced metric to be g and the density function is det g. Then Z

Z

Cnk .M; ˇ/ D

k dg D RM .ˇ/

k

p det gd Sn :

(24)

ˇ

We now write down the local expressions of the induced metric, support function u, second fundamental form and Weingarten curvatures in terms of positive function  and its derivatives r; r 2 . Let fe1 ; ; en g be a local orthonormal frame on Sn , and denote eij the standard spherical metric with respect to this frame (which is the identity matrix). We use r as the gradient operator with respect to standard metric on Sn . To simplify notation, for any function v on Sn , we will write r ei v D vi as covariant derivative with respect to ei on Sn in this subsection, if there is no confusion. From the radial parametrization X.x/ D .x/x, Xi D i x C ei ; Xij D ij x C i ej C j ei C .ei /j D ij x C i ej C j ei  eij x: The following identities can be read off from the above.

62

P. Guan

 D pxr 2 2  Cjj 2

uD p

2 Cjrj2

gij D 2 ıij C i j g ij D

1 .ı ij 2



(25)

i j / 2 Cjrj2

q hij D . 2 C jrj2 /1 .r i r j  C 2i j C 2 eij /

hij D

2

p

1

2 Cjrj2

.e ik 

i k /.r k r j  2 Cjrj2

C 2k j C 2 ekj /:

From (25), q p n1 det g D  2 C jrj2 : The prescribing .n  k/-th curvature measure problem can be deduced to the following curvature equation on Sn : k .1 ; ; n / D k .hij / D

q

f

;

(26)

n1 2 C jrj2

where f > 0 is the given function on Sn . A solution of (26) is called admissible if .X / 2 k at each point X 2 M . We note that any positive C 2 function  on Sn satisfying (26) is automatically an admissible solution. Since the principal curvatures at a maximum point of  are positive, solution is admissible at this point. As k and Sn are connected, and .X / varies continuously, the fact of k ..X // > 0 implies solution is admissible at each point of M . The following is the statement of solvability of the problem of the prescribing curvature measures. Theorem 3.2. Let n  2 and 1  k  n  1. Suppose f 2 C 2 .Sn / and f > 0. Then there exists a unique k-convex star-shaped hypersurface M 2 C 3;˛ , 8˛ 2 .0; 1/ such that it satisfies (26). Moreover, there is a constant C depending only on k; n; kf kC 1;1 ; k1=f kC 0 ; and ˛ such that, kkC 3;˛  C:

(27)

The rest of the section is devoted to the proof of Theorem 3.2. The main task will be the a priori estimates for solutions of (26). We will use the radial parametrization on Sn for the estimates up to C 1 . Then we will work directly on M for the curvature estimates, which is equivalent to C 2 estimates. It will be convenient to introduce a new variable  D log . Set q ! WD

1 C jr j2 :

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

63

The unit outward normal and support function can be expressed as  1 e ! .1; 1 ; ; n / and u D ! respectively. Moreover,

D

gij g ij hij hij

D e 2 .ıij C i j /;  D e 2 .e ij  !i 2j /  D e! .ij C i j C eij /  D e ! .e ik  !i 2k /.kj C k j C ekj /:

(28)

Notice that the Weingarten tensor in (28) is in general not symmetric with respect local lo orthonormal frames .e1 ; ; en / on Sn , even though it is symmetric with respect to local orthonormal frames on M . We observe that the symmetric matrix  .e ij  !i 2j / has an obvious square root S . That is, S D .Sij / D .eij 

i j /; !.! C 1/

.e ij 

i j / D S 2: !2

(29)

S can be used to symmetrize the Weingarten tensor. The eigenvalues of .hij / is the  same as eigenvalues of e ! B, with B defined as B D W .bij / D S.lm C l m C elm /S P P i j l;m l lm m l .i lj C j il /l  D .ij C ıij C /: !.! C 1/ ! 2 .1 C !/2

(30)

Curvature equation (26) can be rewritten as e .nk/ k .B/ D f: ! k1

(31)

2

As B is a function in r ; r only, it is independent of  . Set 2 FQ .r ; r / D k .B/: ij

Denote k .B/ D

@k @bij ,

(32)

we compute @FQ ij .FQ ij / D . / D S.k .B//S: @ij Q

@F Since S in (29) is positive definite, we have . @ / > 0. ij

(33)

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P. Guan

3.1 Uniqueness and C 1 -Estimates Lemma 3.3. Let 1  k < n. Let L denote the linearized operator at a solution  of (26), if v satisfies L.v/ D 0 on Sn , then v 0 on Sn . Moreover, suppose , Q are two solutions of (26) and .i / 2 k , for i D 1; 2. Then 1 2 . Proof. (31) can be put in the form of e .nk/ Q 2 F .r ; r / D f: ! k1

(34)

The linearized operator at  is L.v/ D

X e .nk/ Q ij F v C bl vl  .n  k/fv; ij ! k1 l

for some function bl ; l D 1; ; n. The first statement in lemma follows immediately from the maximum principle. q Suppose  D log  and Q D log Q are two solutions of (31), denote !Q D 1 C jr Q j2 and BQ to be the corresponding tensor B in (30) with  replaced by Q . For t 2 Œ0; 1, set q  D t C .1  t/Q ; t

!t D

1 C jr t j2 ;

Q B t D tB C .1  t/B:

Set v D   Q , as B t 2 k , e .nk/ e .nk/Q Q F .B/  k1 F .B/ k1 ! !Q Z 1 t d e .nk/ . k1 F .B t //dt D !t 0 dt Z 1 .nk/ t Z 1 t e .nk/ e .n  k/. k1 F .B t //dt C . k1 F ij .B t //dt.bij  bQij / C mod.rv/: D !t !t 0 0

0D

2 Write S D .Sji /, and observe that S only involves r; r  (and so is SQ ), by the Mean Value Theorem, 2 B  BQ D S.r v/S C mod.rv/;

and Z 0D. 0

1

t

.n  k/.

e .nk/ Q t F .B //dt/v  !tk1

Z

1

. 0

e .nk/ !tk1

t

ˇ

F ij .B t //dt/Si˛ Sj v˛ˇ C mod.rv/:

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

65

R 1 .nk/ t R1 .nk/ t Since . 0 . e ! k1 F ij .B t //dt/S i ˛ S ˇj / > 0, 0 .n  k/. e ! k1 FQ .B t //dt > 0, v t t satisfies the following elliptic equation, aij .x/vij .x/ C b k .x/vk .x/ C c.x/v.x/ D 0;

8x 2 Sn ;

with c.x/ < 0 for all x 2 Sn . The maximum principle yields v 0. That is  D . Q t u It is useful to write down some differential identities for general C 1 symmetric function F . F .W / is symmetric if it is invariant under orthonormal transformation. 2 @F @FQ With B is defined in (30), set FQ .r ; r / D F .B/. Define F ij D @b , FQ ij D @ . ij ij It follows from (30) that .FQ ij / D S.F ij /S:

(35)

Lemma 3.4. For any C 1 symmetric function F .B/, set  D 2

jr j2 , 2

then there exist

cm depending on .r ; r; F /, such that FQ ij ij D

X

cm m 

m

X

l .F .B//l C F ij .ıij jr j2  j i C ıij ii2 /:

(36)

l

Proof. By (30), ij D

X .l lij C li lj / l

D

X

l .lij C ıli j  j ıil / C li lj

l

D

X

l .ijl C ıij l  j ıil / C li lj

l

D

X l

l .bijl C .





P  i  j m  m m  i j C  j i  /l / !.! C 1/ ! 2 .1 C !/2

Cıij jr j2  j i C ıij ii2 D

X l

P i lj C j li i j m m ml  l .bijl C . // !.! C 1/ ! 2 .1 C !/2

Cıij jr j2  j i C ıij ii2 C cijm m ; where we used the fact that tensor Aij WD ij C eij is Codazzi for any function  2 C 3 .Sn /. We rewrite above identity as

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P. Guan

P X i l lj C l j li i j m;l l m ml ij D . /cijm m  !.! C 1/ ! 2 .1 C !/2 l X Cıij jr j2  j i C ıij ii2  l bijl ; l

or equivalently 2

2

S r S  .cijm m / D jr j2 I  .i j / C .r  /2  .

X

l bijl /:

l

Set cm D

P ij

FQ ij ij 

F ij cijm , contracting above identity with F ij , it follows from (35),

X

cm m D 

m

X

F ij .B/l bijl C F ij .ıij jr j2  j i C ıij ii2 /

l

D

X

l .F .B//l C F ij .ıij jr j2  j i C ıij ii2 /:

l

t u Proposition 3.5. If M satisfies (26), then 1 1  max n f  nk  min n f  nk S S  min jX j  max jX j  : Sn Sn Cnk Cnk

Moreover, there exits a constant C depending only on n, k, minSn f , jf jC 1 such that jrj  C: max n S

Proof. .ij / is semi-negative definite at maximum point of  and r D 0. By (31), f D

e .nk/ k .B/ D e .nk/ k .B/  e .nk/ : ! k1

This yields an upper bound of  . A lower bound of  follows similarly, as .ij / is semi-positive definite at any minimum point of . To obtain an upper bound for jrj is now equivalent to obtain an upper bound of 2  D jr2 j . Suppose p 2 Sn is a maximum point of . At p, rjr j2 D 0;

r! D 0;

B D .ij C ıij /:

(37)

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

67

It follows from (36) with F .B/ D k .B/, at p, 0

X

F ij ij

ij

D

X

l .k .B//l C



ij

k .ıij jr j2  i j C ıij i2i /

ij

l

X

X

l .e .nk/ ! k1 f /l

l

 D .n  k/jr j2 f  r rf e .kn/ ! k1  c.jr j2  C jr j/e .kn/ ! k1 ;

(38)

where c  ı; C  1ı are two positive constants with ı depending only on n; k; inf f; jrf j. The gradient estimate follows from (38). u t

3.2 C 2 -Estimates and the Existence We precede to prove C 2 a priori estimates, this is equivalent to obtain curvature estimate for M due to C 1 estimates we have already obtained. For this purpose, it is convenient to work directly on induced metric g on M  RnC1 . For X 2 M , choose local orthonormal frame fe1 ; ; en g on M , and  D enC1 is the unit outer normal of the hypersurface, such that fe1 ; ; enC1 g of RnC1 is a local orthonormal frame in RnC1 . We use lower indices to denote covariant derivatives with respect to the induced metric. The second fundamental form is the symmetric .2; 0/-tensor given by the matrix j fhij g, and we denote the Weingarten tensor fhi g D fg jl hli g, hij D h@i X; @j i:

(39)

We have the following identities, Xij ./i hijk Rijkl

D hij  .Gauss formula/ j D hi Xj .Weigarten equation/ D hikj .Codazzi formula/ D hik hjl  hil hjk .Gauss equation/;

(40)

where Rijkl is the .4; 0/-Riemannian curvature tensor. We also have hijkl D hijlk C hmj Rimlk C him Rjmlk D hklij C .hmj hil  hml hij /hmk C .hmj hkl  hml hkj /hmi :

(41)

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Since fe1 ; ; en g is an orthonormal frame on M , gij D ıij , hij D hij . The principal curvatures .1 ; ; n / are the eigenvalues of the second fundamental form with respect to the metric which satisfy det.hij  gij / D 0: The curvature equation (26) on Sn can also be equivalently expressed as a curvature equation on M , k .1 ; ; n /.X / D

u.X /  X ; f jX jnC1 jX j

8X 2 M:

(42)

Proposition 3.6. For 1 < k < n, let F k D ˆu and denote H 1 , then at a maximum point of Hu , F ij

H u ij

  D 1u Œˆss u C 2ˆs us   Hu ˆl hX; Xl i  .k  1/ Hu ˆ C.k  1/jAj2  1u F ijIml hijIs hmlIs ;

(43)

where A denotes the second fundamental form. Proof. By definition, u D hX; i. Compute the first and second order covariant derivatives, we have us D hsl hX; Xl i uij D hijIl hX; Xl i C hij  .h2 /ij u

(44)

Also since .hij / is Codazzi, by Ricci identity and Gauss equation, hijIkl D hklIij C .hlk him  hlm hik /hmj C .hlj him  hlm hij /hmk F ij hijIst D Fst  F ijIml hmlIs hijIt : At any maximum point P 2 M n of 

F ij Hu ij

H u

 DF D

ij

Hij u



1 ij u F Hij

uj u

,

H u i

H

.P / D 0. At P , H

u i   ij F uij :  1 H u u

(45)

ui u

u j



 H uij u

u

 (46)

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69

Apply formulas (44) and (45), 1 ij F Hij u

D 1u F ij hssIij

 D 1u F ij hijIss C .hij hsm  hjm hsi /hms C .hjs hsm  hjm hss /hmi D 1u F ij hijIss C kˆjAj2  1u F ij .h2 /ij H D 1u Fss  1u F ijIml hijIs hmlIs C kˆjAj2 

H u

F ij .h2 /ij

D 1u Œˆss u C 2ˆs us C ˆuss   1u F ijIml hijIs hmlIs C kˆjAj2   Hu F ij .h2 /ij 

D 1u Œˆss u C 2ˆs us  C ˆu Hl hX; Xl i C H  jAj2 u   1u F ijIml hijIs hmlIs C kˆjAj2  Hu F ij .h2 /ij  D 1u Œˆss u C 2ˆs us  C ˆu Hl hX; Xl i C Hu ˆ   1u F ijIml hijIs hmlIs C .k  1/jAj2  Hu F ij .h2 /ij :

(47)

We also compute    ij  ij 1 H 2 F uij D  u u F hijIl hX; Xl i C hij  .h /ij u    D  1u Hu Fl hX; Xl i  k Hu C Hu F ij .h2 /ij     D  ˆu Hu ul hX; Xl i  Hu ˆl hX; Xl i  kˆ Hu C Hu F ij .h2 /ij ; (48) where .h2 /ij D hik hkj . Adding up (47) and (48), and using the critical point condition, we obtain  1u Hu

F ij

H u ij

  D 1u Œˆss u C 2ˆs us  C  Hu l hX; Xl i  Hu ˆl hX; Xl i H .k  1/ u ˆ  1u F ijIml hijIs hmlIs C .k  1/ˆjAj2 D

1 u Œˆss u C 2ˆs us   1u F ijIml hijIs hmlIs

H

  u ˆl hX; Xl i  .k  1/ Hu ˆ C .k  1/ˆjAj2 ;

(49)

t u

(43) is verified. 2

C estimates can be established with the help of Proposition 3.6 and Corollary 2.9. Lemma 3.7. If M satisfies (42) for some 1  k  n, then there exists a constant C depending only on n, k, minS n f , jf jC 1 , and jf jC 2 , such that max 1  C; M

jr 2 j  C:

(50)

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Proof. We have already obtained the C 0 and C 1 estimates for . For the case of k D 1, (42) is a mean curvature type equation which is of divergent form of quasilinear PDE. C 2 estimates follows from the classical quasilinear elliptic PDE theory. We work on 2  k  n  1 cases. When k > 1, the estimation of the curvature bound is equivalent to the estimation of mean curvature H (which yields C 2 bound on ). To see this, suppose mean curvature H  C is bounded from above. Since  2 k  2 , .ji / 2 1 . Hence, for each i , C  H D 1 ./ D i C 1 .ji /  i : This give an upper bound of curvature. A lower bound follows from the fact 1 ./ > 0 and i  C for each i . As u is bounded from below and above, we only need to get an upper bound of Hu . Suppose P 2 M where Hu achieves its maximum, it follows from (43)  0  F ij Hu ij   D 1u Œˆss u C 2s us   Hu ˆl hX; Xl i  .k  1/ Hu ˆ  1u F ijIml hijIs hmlIs C .k  1/ˆjAj2 :

(51)

X Recall ˆ.X / D jX j.nC1/ f . jX / and with C 0 , C 1 estimates of  D jX j , we j have the following estimates.

jˆi j.P /  C.n; k; minS n f; jf jC 1 /  jˆi i j.P /  C.n; k; minS n f; jf jC 1 ; jf jC 2 / 1 C jAj.P / On the other hand, jui j D jhij j j  c3 jAj. By (42), 1  1 D : u k At a maximum point P of the test function

1 , u

one has

.1 /s .k /s s D  : 1 k  In Corollary 2.9, set r D

s  .P /,

then

k r 2 u F ijIml hijIs hmlIs  2r.u/s  k1  C1 .n; k; minS n f; jf jC 1 /jAj C C2 .n; k; minS n f; jf jC 1 /:

With the above estimates, (51) can be simplified as jAj2 .P / C c4 jAj.P / C c5  0;

(52)

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

71

where c4 and c5 are constants depending only on n, k, minS n , jf jC 1 , and jf jC 2 . Hence at P , jAj.P /  C . In turn 1 .X /  u.X /

1 .P /  C; u.P /

for any X 2 M . t u

This implies (50). We prove Theorem 3.2 using the method of continuity.

Proof. For any positive function f 2 C 2 .Sn /, for 0  t  1 and 1  k < n  1, set 1

ft .x/ D Œ1  t C tf  k .x/k : Consider the following family of equations for 0  t  1: 1

1

kk .1 ; ; n /.x/ D .ft .x/1n .2 C jrj2 /1=2 / k ;

on Sn ;

(53)

where n  2. We want to find admissible solutions in the class of star-shaped hypersurfaces. Set I D ft 2 Œ0; 1 W such that .53/ is solvable:g 1

I is nonempty because  D ŒCnk  n2 is a solution for t D 0. By Lemmas 3.5, 3.7, 2.6 and 2.7, equation (53) is unform elliptic and concave, apply the Evans-Krylov theorem and the Schauder theorem, we have kkC 3;˛ .Sn /  C; where C depends only on only on n, k, minS n f , maxS n f , jf jC 1 , jf jC 2 and ˛. The a priori estimates guarantee that I is closed. The openness comes from Lemma 3.3 and the inverse function theorem. This proves the existence part of the theorem. The uniqueness part of the theorem follows from Lemma 3.3. t u

4 Isoperimetric Inequality for Quermassintegrals on Starshaped Domains In this section, we use a geometric flow to establish isoperimetric inequalities for quermassintegrals of k-convex starshaped domains in RnC1 . Theorem 4.1. Suppose 1  n  1, and suppose  is a k-convex starshaped domain in RnC1 , then the following inequality holds, 1

1

.V.nC1/k .// nC1k  Cn;k .Vnk .// nk ;

(54)

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where 1

Cn;k D

.V.nC1/k .B// nC1k 1

;

.Vnk .B// nk

B is the standard ball in RnC1 . The equality holds if and only if  is a ball. We consider the following normalized evolution equation on hypersurface M n in RnC1 . @t X D .

1  ru/; F ./

(55)

where F . ; t/ and r.t/ are to be determined, u D< X;  > is the supporting function of the hypersurface. We derive the evolution equations of various geometric quantities for the following general flow. @t X D f :

(56)

Proposition 4.2. Under flow (56), the following evolution equations hold. @t gij D 2f hij @t  D rf @t hij D ri rj f C f .h2 /ij @t hij D r i rj f  f .h2 /ij  P ij @t k D  ij k .g 1 h/fij  f 1 .g 1 h/k .g 1 h/  .k C 1/kC1 .g 1 h/ (57) Proof. Pick any local coordinate chart .x1 ; ; xn / of M , denote Xi D 1; ; n, as hXi ; i D 0; 8i , by Weingarten equation (40), .gij /t D hXi ; Xj it D hXi;t ; Xj i C hXi ; Xj;t i D hXt;i ; Xj i C hXi ; Xt;j i D h.f /i ; Xj i C hXi ; .f /j i D f h./i ; Xj i C f hXi ; ./j i X X Dfh hli Xl ; Xj i C f hXi ; hlj Xl i l

Df

X l

D 2f hij

hli glj

Cf

X l

l

hlj gli

@X @xi

,i D

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

73

Since  is a unit vector field, t has only tangential component. We only need to compute ht ; Xi i. As h; Xi i 0, ht ; Xi i D h; Xi;t i D h; .f /i i D h; .f /i i D fi : This verifies the second identity in the proposition. For the third identity, again using the fact  is a unit vector field, by the second identity we just proved and the Gauss formula in (40), hij;t D hXij ; it D hXij;t ; i  hXij ; t i D h.f /ij ; i C hhij ; rf i D fij  f hij ; i D fij  f h.hli Xl /j ; i D fij  f h.hli /j Xl i  f hhli Xlj ; i D fij C f hhli hlj ; i D fij C f hli hlj : ij

The fourth identity follows from the first and third, and the fact gt D g il g mj glm;t . The final identity in the proposition follows from the fourth identity and Proposition 2.2. t u Corollary 4.3. Under flow (55), where F is homogeneous of degree 1, then we have the following evolution equations. 1  ru/hij F 1 r.  ru/ F 1 1 ri rj .  ru/ C .  ru/.h2 /ij F F 1 1 r i rj .  ru/  .  ru/.h2 /ij F F P 1 1  ij  ij rj ri .  ru/  .  ru/k1Ii 2i F F 1 1 FP ij r i rj .  ru/  .  ru/FP ij .h2 /ij F F

@t gij D 2. @t  D @t hij D @t hij D @t k D @t F D

(58)

Furthermore, the following heat type evolution equation for Weingarten map hij is valid.

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P. Guan

Proposition 4.4. @t hij D

1 P kl k 1 1 F r rl hij C 2 FP .h2 /hij C 2 FR .rh; rh/ 2 F F F 2  F23 r i F rj F  .h2 /ij C rr i hlj < rl X; X > Crhij : F

(59)

t u

Proof. It follows from previous corollary, (41) and (44).

4.1 Monotonicity Properties We want to choose F and r in flow (55) such that the corresponding global geometric quantities are monotone along the flow. The Minkowski identity (4) plays key role here. From identities in Corollary 4.3, for 1  l  n  1, Z

Z

1 @t l C l g ij @t gij dg 2 MZ X 1 D .  ru/ l1Ii 2i  l 1 dg M F i Z 1 D .l C 1/ .  ru/lC1 dg  M Z F Z 1 lC1 dg  r D .l C 1/ ulC1 dg M Z  ZM F  1 D .l C 1/ l dg ; lC1 dg  rCn;l M F M

l dg D

@t M



(60)

.I /

is the constant in the Minkowski equality. where Cn;l D lC1 l .I / For the special case l D n and for any f , by Proposition 4.2, along flow (56), Z

Z

1 @t n C n g ij @t gij dg 2 MZ X D f n1Ii 2i  n 1 dg M Z i  D .l C 1/ f n 1  n 1 dg

n dg D

@t M

D0

(61)

M

That is, V0 ./ is a topological invariant. This gives topological obstruction for the problem of prescribing curvature measure C0 .

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

From (60), if one wants to fix define r as

R M

r.t/ D

k dg , one may choose F D R Mt

kC1 k1 dg k

Cn;k

R

M

k dg

k k1

75

in (55) and

:

To be precise, we consider the normalized flow   k1 @t X D  ru ; k

(62)

(63)

The first step is to get an estimate on r.t/. Lemma 4.5. r.t/ is invariant under rescaling, and r.t/  .

k1 /.I / D Cn;k1 ; k

(64)

equality holds if and only if Mt is the standard sphere. Proof. The inequality follows directly from the Newton-MacLaurin inequality. If the equality holds, this means the Newton-MacLaurin inequality holds at every point t u of Mt . So Mt is umbilical at every point, it is a sphere. The following monotonicity property is crucial. Proposition 4.6. For any k-convex domain , under flow equation (63), we have Z k dg is a constant; 1. ZM 2. k1 dg is monotonically non-decreasing. M

Proof. By the choice of r and (60), Z @t k dg D 0:

(65)

M

This proves the first part of the statement. From (60),  Z Z Z 1 k dg  rCn;k1 k1 dg D k k1 dg @t M  M Z M F 1 k Dk  rCn;k1 k1 dg F k1 R kC1 k1  ZM  dg M R k 1 Dk Cn;k1 k1 dg Cn;k M k dg  ZM  kC1 .I /k1 .I / Cn;k1 1 k1 dg D 0; k Cn;k k2 .I / M where we used the Newton-MacLaurine inequality in the last step.

(66)

t u

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P. Guan

We want to establish the following longtime existence and convergence of flow (63). Theorem 4.7. If 0 is k-convex starshaped domain with smooth boundary M0 , flow (63) exists all time t > 0, it converges to a standard sphere centered at the origin. By a proper rescaling, we will assume Vk .0 / D Vk .B/ where B is the standard ball in RnC1 . The rest of the section is devoted to the proof of Theorem 4.7.

4.2 The Harnack Estimate If M n is starshaped, it can be parametrized as X D .x/x, where x 2 S n . All the geometric information of the hypersurface except the parametrization are encoded in the function .x/. Write  D jX.t/j D .x.t/; t/, where X evolves according to Xt D f :  satisfies d D t C x xt : dt By (25), x  r Dq : 2 C jrj2 We have, x  r fq D f D Xt D .x/t D .t C x xt /x C xt : 2 C jrj2 Note that xt ? x, equalize the tangential components of Sn in (67), f r xt D  q :  2 C jrj2 Therefore, f jrj2 x xt D r xt D  q :  2 C jrj2

(67)

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

77

Put the above identity to (67), equalize the normal component of Sn in (67), q 2 C jrj2 f f D t D x xt C q :  2 C jrj2 In particular, if X satisfies (55),  satisfies q @t  D

2 C jrj2 1  r:  F

(68)

Equation (68) is equivalent to (55) up to diffeomorphism, if we can prove that the starshapedness is preserved along the flow. For the gradient estimate, we prefer to work on (68). As in the previous section dealing to the problem of prescribing curvature measure, let  ln , and we choose a local orthonormal frame fei gniD1 on S n . By the homogeneity of F , @t  D

!2  r; F .B/

(69)

where q ! D 1 C jr j2 ;

P B D .ij C ıij C

P i j l;m l lm m C j il /l /;  !.! C 1/ ! 2 .1 C !/2

l .i lj

as defined in (30).

Proposition 4.8. Let  D

jr j2 , 2

assume (69) preserves .t/ 2 k ,

@t  D Llj r l r j  C Wk r k  

! 2 X @F .ıij jr j2  j i C ıij ii2 /: (70) F 2 .B/ ij @bij

where Wk is a one-parameter family of vector fields depending on time, and Lij is an elliptic operator defined as follows, Lij

!2 FQ ij ; F 2 .B/

(71)

where FQ ij defined as in (35). In consequence, r is bounded from above independent of time t. Proof.  2 k is equivalent to B 2 k , hence F .B/ > 0. Rewrite the last equation in (69) as F .B/ D

!2 : t C r

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Proposition follows from Lemma 3.4 with a straightforward computation using identity (36). t u The following Harnack type gradient estimate is a directly consequence. Corollary 4.9. Let  be a positive solution to (68) on S n  Œ0; T /. Then there exists a constant C which depends on . ; 0/ but independent of t, such that at each time t 2 Œ0; T /, max . ; t/  C min . ; t/ n n S

S

(72)

Proof. We prove the corollary for each fixed time t0 2 Œ0; T /. Assume . ; t0 / achieves maximum at xC and minimum at x , and let  W Œs1 ; s2  ! M n be a path joining x and xC . We have .xC ; t0 / D log .x ; t0 /

Z

s2

d Œlog ..s/; t0 /ds ds Zs1s2 r d  ds D Zs1s2  ˇ ds ˇ ˇd ˇ ˇds  r ˇˇ ˇ ds s1Z ˇ ˇ s2 ˇ ˇ ˇ d  ˇds:  CQ ˇ ds ˇ s1

(73)

By taking Zto beˇ theˇshortest geodesic with constant speed 1 which joins x and s2 ˇ ˇ ˇ d  ˇds D d.x ; xC /  . xC , we obtain t u ˇ ds ˇ s1

Lemma 4.10. Suppose that  > 0 satisfies (68), then at any time t0  0, if x0 2 Sn is a minimum point of .x; t0 /, then .x0 ; t0 /t  0, strict inequality holds unless M.t0 / is a round unit sphere at the origin. Proof. The minimum point of .x; t0 / is the same as minimum point of .x; t0 /. By (69), t .x0 ; t0 / D

! 2 .x0 ; t0 /  r.t0 /: F .B.x0 ; t0 //

As x0 is a minimum point, r.x0 ; t0 / D 0, so at .x0 ; t0 /, ! D 1 and 2

B D .r  C I /  I:

(74)

F .B.x0 ; t0 //  F .I /:

(75)

Hence,

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

79

That is, 1 ! 2 .x0 ; t0 /  : F .B.x0 ; t0 // F .I /

(76)

1 unless M.t0 / is a round sphere (by normalization, it F .I / is a sphere of radius 1). We have

By Lemma 4.5, r.t0 /
0; F .B.x0 ; t0 // F .I /

unless M.t0 / is a round sphere of radius 1. We claim if t .x0 ; t0 / D 0, this round sphere must centered at the origin. Suppose its center z is not the origin, we may assume z D .0; ; 0; s/ for some 1 < s < 0. Now .x; t0 / D

1 log.1 C s 2 C 2sxnC1 /: 2

The minimum point is x0 D .0; ; 0; 1/, it is easy to compute that 2

r .x0 ; t0 / D

s I: .1 C s/2

The strictly inequalities will occur in (74)–(76). Thus, t .x0 ; t0 / D

! 2 .x0 ; t0 / 1  r.t0 / >  r.t0 / D 0: F .B.x0 ; t0 // F .I / t u

contradiction.

The following C 0 estimate is a direct consequence of Corollary 4.9 and Lemma 4.10. Corollary 4.11. Let  be a positive solution to (68) on S n  Œ0; T /. Then there exists a uniform positive constant C which does not depend on time t, such that for 8t 2 Œ0; T /, 0
0 for some constant c independent of t. Proof. By Lemma 4.10, .x0 ; t0 /t > 0 at any minimum point x0 of .x; t0 /, unless M.t0 / is a round unit sphere centered at 0. That is, minx2Sn .x; t/ is strictly increasing at t0 unless M.t0 / is a round sphere centered at 0. In any case, min .x; t/  minn .x; 0/:

x2Sn

x2S

An upper bound of  follows from the Harnack inequality (72).

(78)

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P. Guan

The last statement in lemma follow from the identity 2

uD q 2

C

: jrj2 t u

Since u is bound from below by a positive constant independent of t, flow (68) preserves the starshapedness. We want to show that k is also preserved along the flow. From the property of k , we only need to show k > 0 is preserved. This is equivalent to show F > 0 is preserved.  C.

1 F

Proposition 4.12. There is C > 0, such that

Proof. We consider function G D t C r. We may rewrite (69) as GD

!2 2 DW F .r; r  /; F .B/

(79)

ij

@F where .F / D . @ / > 0. Differentiate (79) in t variable, and notice that r is ij independent of x,

Gt D

X

ij

F .t /ij C

ij

D

X ij

X @F l

ij

F Gij C

@l

X @F l

@l

.t /l

Gl :

G is bounded from above by the maximum principle. Since r is bounded, bounded. The boundedness of F1 follows from C 0 and C 1 estimates.

1 F .B/

is t u

4.3 C 2 Estimates 1 , ' satisfies the following evolution equation. u Proposition 4.13. Let  be a positive solution to (68) on S n  Œ0; T /. We have Denote '

@t ' D

1 P ij i ' 2 F r rj '  2 FP .h2 /  2 FP .r'; r'/ C r' C r' 1 rl ' < X; rl X > : F2 F F ' (80)

Proof. We first write down the evolution equation of u using (55), (58) and (44). We work on local orthonormal frames on M.t/.

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

81

ut D hXt ; i C hX; t i X 1 1  ru  hX; Xl i.  ru/l D F F l

X F ij hij;l 1  ru C D hX; Xl i. C rul / F F2 l

D

X F ij .uij  hij C .h2 /ij u/ 1  ru C Cr hX; Xl iul 2 F F l

D

X 1 1 F ij uij F ij .h2 /ij u  Cr hX; Xl iul  ru C C 2 2 F F F F l

Proposition follows from above identity by inserting u D '1 .

t u

Proposition 4.14. Let  be a positive solution to (68) on S  Œ0; T /. We have n

@t .'hij / D

1 P kl k 2' ' F r rl .'hij //  3 r i F rj F C 2 FR kl;mn r i hkl rj hm n 2 F F F  F 22 ' FP kl r k 'rl .'hij / C rrl .'hij / < rl X; X >  2i  .h / 2' F j  rhij :

Proof. Proof follows from (59) and Proposition 4.13.

(81)

t u

Proposition 4.15. Let  be a positive solution to (68) on S n  Œ0; T / and let .t/ Q D maxn .1 .x/; ; n .x//. Then for t > 0,

x2Mt

max ' .t/ Q  max ' Q .0/; n n Mt

M0

(82)

with the equality holds if and only if M0 is a sphere centered at the origin. Since 1 ./ > 0, we have uniform curvature bounds. Proof. Let x0 be a point such that h11 .x0 ; t0 / D .t0 / for some direction e1 . By (81), and concavity of F ,   1 2 .h1 / 1 1  rh1 : .'h1 .x0 ; t0 //t  2' F ./ Q  i for all i . By the monotonicity, homogeneity of F and by At x0 , h11 D .t/ Lemma 4.5, 1 h11   r: F ./ F .I / We obtained at x0 , .'h11 .x0 ; t0 //t  0.

(83)

82

P. Guan

We claim for any t0 , .'h11 .x0 ; t//t > 0 unless M.t0 / is the unit sphere centered at 0. Now suppose .'h11 .x0 ; t//t D 0, all inequalities in (83) must be equalities. In particular, r.t/ D

1 : F .I /

By Lemma 4.5 and normalization, M.t0 / must be a sphere of radius 1. So 1 .x; t0 / D ; n .x; t0 / D 1; 8x 2 Sn and we may use the standard spherical paramerization for M.t0 /. Suppose its center is z ¤ 0, we may assume z D .0; ; 0; s/ for some 1 < s < 0. Now u.x; t0 / D 1 C sxnC1 ;

'.x; t0 / D

1 : 1 C sxnC1

The minimum point is x0 D .0; ; 0; 1/, it follows from (81), @t .'hij / D

  2 i .h /j 1 P kl k 1 P kl k i i  rh F F r rl ' < 0; r r .'h /  2' l j j D F2 F F2 t u

contradiction. We now prove Theorem 4.7.

Proof. By C 2 estimates and Proposition 4.12,  2 k is preserved along flow (68). By Lemma 2.7, the equation is uniform parabolic. We may apply the Krylov Theorem [31] and the standard parabolic theory to conclude the longtime existence and regularity for the flow. To get exponential convergence, we use the uniform ellipticity of F . There is c0 > 0 independent of t, .

@F .B/ /.x; t/  c0 I; @bij

8.x; t/:

Thus, as n  2, X @F .B/ i

@bii

 c0 C M .

@F .B/ /; @bij

where M .W / denoting the largest eigenvalue of W . By C 2 estimates, there is ˇ > 0 independent of t such that ! 2 X @F .B/ .ıij jr j2  i j /  ˇjr j2 : F 2 ij @bij

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

83

By Proposition 4.8, @t Set Q D e ˇt

 jr j2 2

 jr j2

 Llj r l r j

2

C Wk r k

 jr j2 2

 ˇjr j2 :

(84)

jr j2 , Q satisfies differential inequality 2 @t Q  Llj r l r j Q C Wk r k Q:

(85)

Therefore, Q is bounded from above independent of t. From there, we conclude jr j2 ! 0 exponentially as t ! 1. By our normalization,  ! 1 and r ! 0 exponentially as t ! 1. m For the exponential convergence of r , apply integration by parts, Z

m

Sn

Z

jr j2 dSn  C.

jr

mC1

Sn

Z

1

j2 dSn / 2 .

jr

m1

Sn

1

j2 dSn / 2 :

mC1

kL1 .Sn /  cm for some cm independent of t. An By the a priori estimates, kr induction argument yields that, for each m 2 NC , there is Cm > 0; ˇm > 0, such that m

kr kL2 .Sn /  Cm e ˇm t : m

The Sobolev Lemma implies r  ! 0 exponentially and t ! 1, for each m 2 NC . t u We prove Theorem 4.1. In fact, the following is true. Theorem 4.16. Suppose  is a C 2 starshaped domain in RnC1 . Assume 1  k  n  1, that .x/ 2  k D f 2 Rn jl . /  0; 8l D 1; ; k:g; then the following inequality holds, 1

1

.V.nC1/k .// nC1k  Cn;k .Vnk .// nk ; where 1

Cn;k D

.V.nC1/k .B// nC1k 1

;

.Vnk .B// nk

B is the standard ball in RnC1 . The equality holds if and only if  is a ball.

(86)

84

P. Guan

Proof. Case 1.  is k-convex. Inequality (86) follows directly from the above Proposition 4.6 and Theorem 4.7. We examine the equality case. Recall (66), Z 

Z @t M

R

 R 1 k1 ./dg D k Cn;k1 k1 dg Cn;k M k ./dg  ZM  kC1 .I /k1 .I / Cn;k1 1 k1 dg D 0: k Cn;k k2 .I / M M

kC1 ./k1 ./ dg k ./

(87)

At any time t0  0, inequality is strict in (87) unless kC1 ./k1 ./ k ./ D 2 ; kC1 .I /k1 .I /k ./ k .I /

a.e. in M.t0 /.

That is the equality is the case in (13), this implies M.t0 / is umbilical almost everywhere. As M.t0 / is C 2 , it is umbilical everywhere. M.t0 / is a round sphere for each t  t0 . In particular, if equality is held in (86), then M is a sphere. Case 2. General case. We may approximate  by k-convex starshaped domains. The inequality follows from the approximation. We now treat the equality case. We first note that both R R  d and  d g g are positive, since there exists at least one elliptic point M k M k1 on an embedded compact hypersurface in Euclidean space and also the k-convexity condition. Suppose  is a weakly k-convex starshaped domain with equality in (86) attained. Let MC D fx 2 M jk ..x// > 0g. MC is open and nonempty since M is compact and embedded in RnC1 . We claim that MC is closed. This would imply M D MC , so  is k-convex, by Case 1, we may conclude  is a standard ball. We now prove that MC is closed. Pick any  2 C02 .MC / compactly supported in MC . Let Ms be the hypersurface determined by position function Xs D X C s, where X is the support function of M and  is the unit outernormal of M at X . Let s be the domain enclosed by Ms . It is easy to show Ms is k-convex starshaped when s is small enough. Define 1

Ik ./ D

nC1k ./ V.nC1/k 1

:

(88)

nk Vnk ./

Therefore Ik .s /  Ik ./  0 for s small, i.e. d Ik .s /jsD0 D 0: ds Simple calculation yields d ds

Z

Z l .s /dgs jsD0 D .l C 1/ Ms

lC1 ./dg : M

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

85

Therefore, d Ik .s /jsD0 D A ds

Z

for some constant A > 0 with c1 D

.kC1 ./  c1 k .//dg D 0; M k.nk/ .kC1/.nkC1/

 2 C02 .MC /. Thus,

1 I.B/nkC1 .

kC1 ..x// D c1 k ..x//;

R

1

M

> 0 and for all

k / nk

8x 2 MC :

(89)

It follows from the Newton-MacLaurine inequality, there is a dimensional constant CQ k;n such that kC1 ..x//  CQ k;n k

1C1=k

..x//;

8x 2 MC :

In view of (89), there is a positive constant c2 , such that k ..x//  c2 > 0;

8x 2 MC ;

(90)

where c2 D . CQc1 /k is a positive constant depending only on n, k, and . (90) k;n implies MC is closed. t u

5 Appendix We present Garding’s theory of hyperbolic polynomials here. Definition 5.1. Let P be a homogeneous polynomial of degree m in a finite vector space V . For 2 V , P is called hyperbolic at if P . / ¤ 0 and the equation P .xC t / D 0 (as a polynomial of t 2 C) has only real roots for every x 2 V . We say P is complete if P .x C ty/ D P .x/ for all x; t implies y D 0. Proposition 5.2. Suppose P is hyperbolic at , then the component  of in fx 2 V I P .x/ ¤ 0g is a convex cone, the zeros of P .x C ty/ as a polynomial in t are real 8x; y 2 V . The polynomial PP .x/ . / is real, and it is positive when x 2 . Furthermore, 1

. PP .x/ / m is concave and homogeneous of degree 1 in , equal to 0 on the boundary . / of . Proof. We normalize P . / D 1, then there exist tj 2 R; j D 1; ; m, such that P .x C t / D .t  t1 /  : : :  .t  tm /:

86

P. Guan

In particular, P .x/ D .t1 /  : : :  .tm / 2 R. Set  D fx 2 V I P .x C t / ¤ 0; t  0g:  is open and 2  as P . C t / D .1 C t/m P . / only has the zero t D 1. Notice that  is also closed in fx 2 V I P .x/ ¤ 0g. If x 2 N , then P .x C t / ¤ 0, when t > 0. Hence,  D fx 2 N ; P .x/ ¤ 0g: If x 2  , then x C t 2  when t > 0. This implies that  is connected, Therefore x C  2  for all > 0;  > 0. That is,  is star-shaped with respect to and  D . For y 2  and ı > 0 fixed, Ey;ı D fx 2 V I P .x C i ı C isy/ ¤ 0; Re.s/  0g is open. If s ¤ 0, P .i ı C isy/ D .i s/m P . ı s C y/ D 0, the hyperbolicity implies s < 0. That is, 0 2 Ey;ı . If x 2 EN y;ı and Res > 0, then Hurwitz’ theorem implies P .x C i ı C isy/ ¤ 0. This is still true when Re.s/ D 0 since x C isy is real. Therefore, Ey;ı is both open and closed, and Ey;ı D V . Thus, P .x C i.ı C y// ¤ 0; 8x 2 Rn ; y 2 ; ı > 0: For  is open, the above remains true for ı D 0. Equation P .x C ty/ D 0 has only 1y C iy/ D 0. real roots, for if t D t1 C i t2 is a root with t2 ¤ 0 we would get P . xCt t2 This means that y can play the role of ,  is star-shaped with respect to every point in . The convexity of  follows. We also have P .y/ > 0 for all y 2 . We now prove the concavity statement in the proposition. As P .x C ty/ has only real roots for y 2 , there are tj 2 R, j D 1; : : : ; m, P .x C ty/ D P .y/.t  t1 /  : : :  .t  tm /: In turn, P .sx C y/ D P .y/.1  st1 /  : : : .1  stm /: If sx C y 2 , we must have 1  stj > 0 for every j . If f .s/ D logP .sx C y/, then 0

f .s/ D 

X

tj ; 1  stj

f ” .s/ D 

X

tj2 .1  stj /2

:

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

87

Therefore, by Cauchy-Schwarz inequality, f .s/

2 

m e

f .s/ m

d 2 .e m / 0 D f .s/2 C mf ” .s/ ds 2 X X tj tj2 /2  m  0: D. 1  stj .1  stj /2 t u

If P is a homogeneous polynomial ofPdegree m. For x l D .x1l ; : : : ; xnl / 2 V , @ l D 1; : : : ; m, we denote < x l ; @x >D n1 xjl @x@j as a vector field. We define the complete polarization of P as 1 @ @ < x1 ; > : : : < xm; > P .x/: PQ .x 1 ; : : : ; x m / D mŠ @x @x It is a multilinear and symmetric in x 1 ; : : : ; x m 2 V , independent of x, and that 1 dm P .tx/ D P .x/; 8x 2 V: PQ .x; : : : ; x/ D mŠ dt m And P .t1 x 1 C : : : C tm x m / D mŠt1 : : : tm PQ .x 1 ; : : : ; x m / C : : : where the dots denote terms not containing all the factors tj . Lemma 5.3. If P is hyperbolic at and m > 1, then for any y D .y1 ; : : : ; yn / 2 , Q.x/ D

n X 1

yj

@ P .x/ @xj

is also hyperbolic at . In general, if x 1 ; : : : ; x l 2  for some l < m, then QQ l .x/ D PQ .x 1 ; : : : ; x l ; x; : : : ; x/ is hyperbolic at . The proof is immediate by Rolle’s theorem. Using polarization and Lemma 5.3, we list some of important examples of hyperbolic polynomials. Corollary 5.4. The following polynomials are hyperbolic. 1. The polynomial P D .x1 /2  .x2 /2  : : :  .xn /2 is hyperbolic at .1; 0; : : : ; 0/. 2. The polynomial P D x1 : : : xn is complete hyperbolic at any with P . / ¤ 0. The positive cone  of P at .1; : : : ; 1/ is  D fx D .x1 ; : : : ; xn /I xj > 0;

8j g:

88

P. Guan

3. In general the elementary symmetric function k .x/ is complete hyperbolic at .1; : : : ; 1/, the corresponding positive cone k is k D fl .x/ > 0; 8l  kg: 4. Let S denote set of all real n  n symmetric matrices. Then k .W /; W 2 S is complete hyperbolic at the identity matrix, the corresponding positive cone is k D fl .W / > 0; 8l  kg: 5. For W 1 ; : : : ; W l 2 k , l < k, then Ql .W / D PQ .W 1 ; : : : ; W l ; W; : : : ; W / is complete hyperbolic in k . Lemma 5.5. Suppose P is a second order complete hyperbolic polynomial. Suppose both roots of f .s/ D P .sy C w/ vanishing for some y 2  and w 2 V . Then, all the roots of g.s/ D P .sz C w/ are vanishing for any z 2 . Proof. Since P .y C tw/ D P .y/ ¤ 0 for all t, we must have y C tw 2 . By the convexity of , we have z C tw 2  for all t. So, P .z C tw/ ¤ 0. For any z 2  and all t, P .z/.1 C t 1 /.1 C t 2 / D P .z C tw/ ¤ 0;

1 ; 2 are the roots of P .sz C w/. Since t is arbitrary, this gives 1 D 2 D 0.

t u

Lemma 2.4 is a special case of the following proposition. Proposition 5.6. Suppose P a homogenous polynomial of degree m, suppose it is hyperbolic at and P . / > 0, then 8x 1 ; : : : ; x m 2 , P 2 .x 1 ; x 2 ; x 3 ; ; x m /  P .x 1 ; x 1 ; x 3 ; ; x m /P .x 2 ; x 2 ; x 3 ; ; x m / 1

1

P .x 1 ; : : : ; x m /  P .x 1 / m : : : P .x m / m :

(91)

If P is complete, the equality holds if and only if all x j are pairwise proportional. This is also equivalent that for x; y 2  not proportional, the function h.t/ D 1 P .x C ty/ m is strictly concave in t > 0. If P is complete, then QQ l .X / D PQ .x 1 ; : : : ; x l ; x; : : : ; x/ is complete if m  l  2 and x 1 ; : : : ; x l 2 . In particular, PQ .x 1 ; : : : ; x m / > 0 if x 1 2 N and x j 2  when m  2. 1

Proof. Since P m .X / is concave in , it follows that for any x; y 2 , h.t/ D 1 P .x C ty/ m is concave in t > 0. So, h” .t/  0. A direct computation yields h” .0/ D .m  1/.PQ .y; y; x; : : : ; x/P .X /  PQ .y; x; : : : ; x/2 /P .x/ m 2 : 1

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

89

We get the inequality PQ .y; y; x; : : : ; x/P .X /  PQ .y; x; : : : ; x/2 : In turn, it implies PQ .y; x; : : : ; x/m  P .y/P .x/m1 : We now apply induction argument. Take y D x 1 and assuming that (91) is already proved for hyperbolic polynomials of degree m  1. Let Q.x/ D PQ .y; x; : : : ; x/, we get PQ .x 1 ; : : : ; x m /  .Q.x 2 / : : : Q.x m // .m1/ 1

1

 .P .x 1 /P .x 2 /m1 : : : P .x 1 /P .x m /m1 / m.m1/ ; which proves (91). To prove the last statement in the proposition, it suffices to show that if m  3, Q (defined above) is complete. suppose Q.x/ D Q.x C tz/ for all x; t. In particular, Q.y Ctz/ D Q.y/. That means that Q.ty Cz/ D Q.ty/, so P .ty Cz/P .ty/ D a is independent of t. Since the zeros of P .ty/ C a D t m P .y/ C a must all be real, it follows that a D 0. This P .y C sz/ D P .y/ ¤ 0 for all s, so it follows that y C sz 2 . Hence, .sx C y C sz/ 2 ; 8x 2 ; s > 0: .s C 1/ Letting s ! 1, we conclude that x C z 2 N for all x 2 . This implies x C z 2 . We can replace z by tz for any t, so x C tz 2  for all t and x 2 . Thus P .z C sx/ can not have any zeros ¤ 0, so P .z C sx/ D s m P .x/. That is P .x C tz/ D P .x/ for all t and all x 2 . Since P is analytic, that means P .x C tz/ D P .x/ for all t and all x 2 V . By the completeness assumption on P , z D 0. Finally, we discuss the equality case in (91). By the above, we may assume m D 2. If the equality holds, we have P .y/P .x/ D PQ .y; x/2 . This implies the roots of the second order polynomial p.t/ D P .x Cty/ are equal, i.e., t1 D t2 D  ¤ 0. In turn, for all t, P .y C .t C /1 .x  y// D .t C /2 P .ty C x/ D P .y/: That is both roots of the polynomial f .s/ D P .sy C .x  y// are vanishing. From Lemma 5.5, we have P .z C t.x  y// D P .z/ for all z 2  and all t. Since  is open and P is analytic, P .z C t.x  y// D P .z/ for all z and all t. By the completeness of P , x  y D 0. That is, x and y are proportional. t u

90

P. Guan

6 Notes 1. The definition of curvature measures in this notes follows from Federer [12], where he used Steiner’s formula to define them for sets of positive reach. Alexandrov [3] initiated the problem of prescribing curvature measure C0 , which he called the integral curvature. The problem of prescribing 0-th curvature measure is often referred as the Alexandrov problem in literature. It was Alexandrov who formulated the problem through radial parametrization. The existence and uniqueness of solutions were obtained by A.D. Alexandrov [3]. It can be deduced to a Monge-Amp´ere type equation on Sn . For n D 2 the regularity of solutions of the Alexandrov problem in the elliptic case was proved by Pogorelov [37] and for higher dimensional cases, it was solved by Oliker [35]. The general regularity results (degenerate case) of the problem were obtained in [20]. The problem of prescribing general k-th curvature measures was settled for starshaped hypersurfaces recently in [27], though C 0 and C 1 estimates were obtained in [19] some time ago. The proof of Lemma 3.4 presented here is due to Junfang Li (Li, private notes (2012)), which can apply to more general curvature equations. Another proof of gradient estimate for (26) appeared in [25], there the question of when solution to (26) is discussed. 2. The presentation of theory of hyperbolic polynomials in Appendix basically follows the original paper of Garding [14]. Caffarelli-Nirenberg-Spruck [5] developed the study of k-Hessian equation in the category of k , followed by [6] for k-curvature equation. The proof of Lemma 2.7 is from [30], which in turn is inspired by Marcus and Lopes [32]. Lemma 2.8 was proved in [27]. Using  in C 2 estimates for k-curvature equation on star-shaped hypersurfaces was u introduced in [6]. The complication for (26) is that the right hand side depends 1

on r, the standard concavity of kk is not sufficient in this case. C 2 estimate is still open for k-curvature equation on star-shaped k-convex hypersurfaces with general right hand side k ./ D f .r.x/; .x/; x/;

x 2 Sn :

In a recent work [26] established C 2 estimates for admissible solutions of above equation in the case k D 2 and for convex solutions for general k. 3. The classical isoperimetric inequalities for quermassintegrals of convex bodies are the consequence of the Alexandrov-Fenchel inequality [1, 2] in convex geometry. Trudinger was the first to consider such inequalities for k-convex domains in [40]. Theorem 4.1 was proved in [17]. The proof in [17] used un-normalized inverse mean curvature type flow for starshaped hypersurfaces studied by Gerhardt [15] and Urbas [41], where they established longtime existence and exponential convergence for a class of more general type of inverse mean curvature flow. In Sect. 3, we use normalized flow (63), which was initially devised in (Guan and Li, private notes) when they did not realize that the work of [15,41] would imply the monotonicity of the isoperimetric ratio Ik in (88). Flow

Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs

91

(63) considered here has an advantage that one can see how to design a flow to fit the monotonicity. Similar design was used previously in conformal geometry 1 in [22,23]. Junfang Li pointed out that, one may also pick r.t/ F .I / in (63), as 0 in a recent paper [18]. With this choice of r, the proof of C estimates for flow (63) can be simplified. The monotonicity in Proposition 4.6 is reversed as Z

Z k dg is monotonically non-increasing; M

k1 dg is a constant. M

It is an open question if (54) is valid without the starshapedness condition. In the case k D 1, Huisken [29] verified the inequality replacing the star shapedness by the assumption that @ is outward-minimizing. Again, in the case k D 1, (54) was proved for general 1-convex domains in [7] for some constant c which is a not sharp. Under additional condition that  is k C 1-convex (without starshapedness assumption), inequality (54) is proved in [8] with some no-sharp constant c. 4. The normalized inverse mean curvature flow Xt D .

u 1  / H n

(92)

preserves the surface area and increases the enclosed volume. This implies the isoperimetric inequality for mean convex star-shaped domain. The statement can be checked as below. Z

Z d dt

u 1  /Hd n M Z H D n1 .n  uH /d

dg D M

.

D 0:

(93)

M

The evolution of the volume V .t/ is Z d V dt

D

ZM

D M

 0:

u 1  /d H n nC1 1 d  V H n

.

(94)

where the last inequality comes from an inequality proved by Ros in [39], see formula (5) on page 449. 5. The prescribing measure problem is a counter part of the Christoffel-Minkowski problem, which is the problem of prescribing area measures for convex bodies. The Minkowski problem was considered by Minkowski in [33] in 1897. The differential geometric setting of the problem was solved in early 1950s by

92

P. Guan

Nirenberg [34] and Pogorelov [36] for n D 2. The solution of the Minkowski problem in higher dimension came much later in 1970s by Cheng-Yau [9] and Pogorelov [38]. The Minkowski problem is a special case (k D n) of the problem of prescribing general k-th (1  k  n) area measures in convex geometry. At the other end (k D 1), it is the Christoffel problem. This case has been settled completely by Firey [13]. In general, the problem of prescribing k-th is termed the Christoffel-Minkowski problem. It is equivalent to solve the following equation k .uij C uıij / D '

on Sn ;

(95)

with convexity requirement .uij C uıij / > 0. The intermediate Christoffel-Minkowski problem (1 < k < n) is still open, except for some special cases. There are also some sufficient conditions, we refer to [38] and [21]. The necessary and sufficient condition for the existence of admissible solutions of (95) is known (e.g., [24]). The main difficulty lies in the question of convexity for the admissible solutions (which in general are not convex) of (95). 6. The Minkowski problem can also be considered as a problem of prescribing the Gauss curvature on outernormals of convex hypersurfaces. The similar question was raised for other Weingarten curvature functions k .1 ; ; n / for fixed 1  k  n in [4] and [10]. The corresponding equation is n .uij C uıij / D f nk

on Sn :

(96)

When 1  k < n, very little is known for this problem. No uniqueness result is known except the case n D 2 (e.g., see [4]). If the prescribed curvature function is invariant under an automorphic group G without fixed points, the problem is solvable [16]. Acknowledgements Large part of the material in this lecture notes are based on joint works with Junfang Li [17,27]. I would like to thank him for many helpful discussions and valuable comments regarding the exposition of the notes. Research of the first author was supported in part by an NSERC Discovery Grant.

References 1. A.D. Alexandrov, Zur Theorie der gemischten Volumina von konvexen korpern, II. Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen (in Russian). Mat. Sbornik N.S. 2, 1205–1238 (1937) 2. A.D. Alexandrov, Zur Theorie der gemischten Volumina von konvexen korpern, III. Die Erweiterung zweeier Lehrsatze Minkowskis uber die konvexen polyeder auf beliebige konvexe Flachen (in Russian). Mat. Sbornik N.S. 3, 27–46 (1938)

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3. A.D. Alexandrov, Existence and uniqueness of a convex surface with a given integral curvature. Doklady Akademii Nauk Kasah SSSR 36, 131–134 (1942) 4. A.D. Alexandrov, Uniqueness theorems for surfaces in the large I. Vestnik Leningrad Univ. 11, 5–17 (1956); Am. Soc. Trans. Ser. 2 21, 341–354 (1962) 5. L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, III: functions of the eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985) 6. L.A. Caffarelli, L. Nirenberg, J. Spruck, Nonlinear Second Order Elliptic Equations IV: Starshaped Compact Weigarten Hypersurfaces, ed. by Y. Ohya, K. Kasahara, N. Shimakura. Current Topics in Partial Differential Equations (Kinokunize, Tokyo, 1985), pp. 1–26 7. P. Castillon, Submanifolds, isoperimetric inequalities and optimal transportation. J. Funct. Anal. 259, 79–103 (2010) 8. A. Chang, Y. Wang, The Aleksandrov-Fenchel Inequalities for Quermassintegrals on k C 1Convex Domains. Preprint (2011) 9. S.Y. Cheng, S.T. Yau, On the regularity of the solution of the n-dimensinal Minkowski problem. Comm. Pure Appl. Math. 24, 495–516 (1976) 10. S.S. Chern, Integral formulas for hypersurfaces in Euclidean space and their applications to uniqueness theorems. J. Math. Mech. 8, 947–955 (1959) 11. L.C. Evans, Classical solutions of fully nonlinear, convex, second order elliptic equations. Comm. Pure Appl. Math. 35, 333–363 (1982) 12. H. Federer, Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959) 13. W.J. Firey, The determination of convex bodies from their mean radius of curvature functions. Mathematik 14, 1–14 (1967) 14. L. Garding, An inequality for hyperbolic polynomials. J. Math. Mech 8, 957–965 (1959) 15. C. Gerhardt, Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32, 299–314 (1990) 16. B. Guan, P. Guan, Hypersurfaces with prescribed curvatures. Ann. Math. 156, 655–674 (2002) 17. P. Guan, J. Li, The quermassintegral inequalities for k-convex starshaped domains. Adv. Math. 221, 1725–1732 (2009) 18. P. Guan, J. Li, A Mean Curvature Type Flow in Space Forms. Preprint (2012) 19. P. Guan, Y.Y. Li, Unpublished Research Notes (1995) 20. P. Guan, Y.Y. Li, C 1;1 estimates for solutions of a problem of Alexandrov. Comm. Pure Appl. Math. 50, 189–811 (1997) 21. P. Guan, X. Ma, The Christoffel-Minkowski problem I: convexity of solutions of a Hessian equation. Inventiones Mathematicae 151, 553–577 (2003) 22. P. Guan, G. Wang, A fully nonlinear conformal flow on locally conformally flat manifolds. Journal fur die reine und angewandte Mathematik 557, 219–238 (2003) 23. P. Guan, G. Wang, Geometric inequalities on locally conformally flat manifolds. Duke Math. J. 124, 177–212 (2004) 24. P. Guan, X. Ma, F. Zhou, The Christoffel-Minkowski problem III: existence and convexity of admissible solutions. Comm. Pure Appl. Math. 59, 1352–1376 (2006) 25. P. Guan, C.S. Lin, X. Ma, The existence of convex body with prescribed curvature measures. Int. Math. Res. Not. 2009, 1947–1975 (2009) 26. P. Guan, C. Ren, Z. Wang, Global Curvature Estimates for Convex Solutions of k -Curvature Equations. Preprint (2012) 27. P. Guan, J. Li, Y.Y. Li, Hypersurfaces of prescribed curvature measures. Duke Math. J. 161, 1927–1942 (2012) 28. G. Hardy, J. Littlewood, G. Polya, Inequalities (Cambridge University Press, London, 1952) 29. G. Huisken, in preparation 30. G. Huisken, C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta. Math. 183, 45–70 (1999) 31. N.V. Krylov, Boundely inhomogeneous elliptic and parabolic equations in domains. Izvestin Akad. Nauk. SSSR 47, 75–108 (1983) 32. M. Marcus, L. Lopes, Inequalities for symmetric functions and Hermitian matrices. Can. J. Math. 9, 305–312 (1957)

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Refraction Problems in Geometric Optics Cristian E. Guti´errez

Abstract We present a description of the far field and the near field problems for refraction when the source of energy is located at one point. The far field problem is solved using mass transportation and also a variant of the Minkowski method. Maxwell equations are developed and the boundary conditions studied to obtain Fresnel formulas. These are used to present a model for refraction that takes into consideration the energy used in internal reflection.

1 Introduction These are notes expanding the material of a mini course I taught on MongeAmp`ere type equations and geometric optics in June 2012 at Cetraro. The purpose of these lectures was to explain basic facts about the Monge-Amp`ere equation and the application of these ideas to pose and solve problems in geometric optics concerning refraction with prescribed input and output energies. These problems are basically of two types: the far field and the near field. In the far field case the goal is to send radiation into a set of directions and in the second is to send radiation to a specific target set. The far field case can be treated with optimal transportation methods, Sect. 3, but the near field does not. Instead in this case, a method based on the Minkowski method in convex geometry can be used. In fact, with this method one can treat both far and near fields problems. The method is illustrated in Sect. 4 in the far field case. Refraction and reflection occur simultaneously, in other words, if an incident ray is refracted there is always a percentage of the incident ray that is internally reflected. The fractions of energy refracted and reflected are given by the Fresnel formulas. Beginning with Maxwell’s equations this is developed and

C.E. Guti´errez () Department of Mathematics, Temple University, Philadelphia, PA 19122, USA e-mail: [email protected] L. Capogna et al., Fully Nonlinear PDEs in Real and Complex Geometry and Optics, Lecture Notes in Mathematics 2087, DOI 10.1007/978-3-319-00942-1 3, © Springer International Publishing Switzerland 2014

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explained in Sect. 5. The application to the refraction problem with loss of energy is described in Sect. 5.13. These notes include the some of my work with Qingbo Huang and Henok Mawi, and the relevant references are [7–9]. The equations describing the solutions to these problems are Monge-Amp`ere type equations whose derivation is included in [7], and [9]. The notes are organized as follows. Section 2 contains the Snell law of refraction in vector form. This is simple but essential for the development of the results. In particular, we introduce in this section the notion of surfaces having the uniform refraction property used later to give the definition of refractors. Section 3 contains basic facts about optimal mass transport and its application to solve the refractor problem in the far field case. Section 4 introduces a method to solve the far field refractor problem that has its roots in the Minkowski method. Section 5 contains a detailed development of Maxwell’s equations, the so called boundary conditions explaining the propagation across two different materials, and its application to deduce the Fresnel formulas. These formulas are written in a convenient form that is used finally to propose, in Sect. 5.13, a model for the refractor problem with loss of energy in internal reflection. It is a pleasure to thank the CIME for their support in the organization of this series of lectures and in particular, to Elvira Mascolo and Pietro Zecca for their great support and help that made this series possible.

2 Snell’s Law of Refraction 2.1 In Vector Form Suppose  is a surface in R3 that separates two media I and II that are homogeneous and isotropic. Let v1 and v2 be the velocities of propagation of light in the media I and II respectively. The index of refraction of the medium I is by definition n1 D c=v1 , where c is the velocity of propagation of light in the vacuum, and similarly n2 D c=v2 . If a ray of light1 having direction x 2 S 2 and traveling through the medium I hits  at the point P , then this ray is refracted in the direction m 2 S 2 through the medium II according with the Snell law in vector form: n1 .x  / D n2 .m  /;

(1)

where  is the unit normal to the surface to  at P going towards the medium II. The derivation of this formula is in Sect. 2.2. This has several consequences:

1

Since the refraction angle depends on the frequency of the radiation, we assume radiation is monochromatic.

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(a) the vectors x; m;  are all on the same plane (called plane of incidence); (b) the well known Snell law in scalar form n1 sin 1 D n2 sin 2 ; where 1 is the angle between x and  (the angle of incidence), 2 the angle between m and  (the angle of refraction), From (1), .n1 x  n2 m/   D 0, which means that the vector n1 x  n2 m is parallel to the normal vector . If we set  D n2 =n1 , then x   m D ;

(2)

for some 2 R. Takingpdot products we get D cos 1  cos 2 , cos 1 D x  > 0, and cos 2 D m  D 1   2 Œ1  .x /2 , so

Dx 

p 1   2 .1  .x /2 :

(3)

Refraction behaves differently for  < 1 and for  > 1. 2.1.1  < 1 This means v1 < v2 , and so waves propagate in medium II faster than in medium I, or equivalently, medium I is denser than medium II. In this case the refracted rays tend to bent away from the normal, that is the case for example, when medium I is glass and medium II is air. Indeed, from the scalar Snell law, sin 1 D  sin 2 < sin 2 and so 1 < 2 . For this reason, the maximum angle of refraction 2 is =2 which, from the Snell law in scalar form, is achieved when sin 1 D n2 =n1 D . So there cannot be refraction when the incidence angle 1 is beyond this critical value, that is, we must have 0  1  c D arcsin .2 Once again from the Snell law in scalar form, 2  1 D arcsin. 1 sin 1 /  1

(4)

and it is easy to verify that this quantity is strictly increasing for 1 2 Œ0; c ,   c . We then have x m D cos. 2  1 /  and therefore 0  2  1  2 cos.=2  c / D , and therefore obtain the following physical constraint for refraction:

2

If 1 > c , then the phenomenon of total internal reflection occurs, see Fig. 1c.

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a

b

c

Fig. 1 Snell’s law  < 1, e.g., glass to air. (a) x   m parallel to . (b) Critical angle. (c) Internal reflection

if  D n2 =n1 < 1 and a ray of direction x through medium I is refracted into medium II in the direction m, then m x  :

(5)

Notice also that in this case > 0 in (3). Conversely, given x; m 2 S 2 with x m   and  < 1, it follows from (4) that there exists a hyperplane refracting any ray through medium I with direction x into a ray of direction m in medium II. 2.1.2  > 1 In this case, waves propagate in medium I faster than in medium II, and the refracted rays tend to bent towards the normal. By the Snell law, the maximum angle of refraction denoted by c is achieved when 1 D =2, and c D arcsin.1=/. Once again from the Snell law in scalar form 1  2 D arcsin. sin 2 /  2

(6)

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 which is strictly increasing for 2 2 Œ0; c , and 0  1  2   c . We therefore 2 obtain the following physical constraint for the case  > 1: if a ray with direction x traveling through medium I is refracted into a ray in medium II with direction m, then m x  1=.

(7)

Notice also that in this case < 0 in (3). On the other hand, by (6), if x; m 2 S 2 with x m  1= and  > 1, then there exists a hyperplane refracting any ray of direction x through medium I into a ray with direction m in medium II. We summarize the above discussion on the physical constraints of refraction in the following lemma. Lemma 2.1. Let n1 and n2 be the indices of refraction of two media I and II, respectively, and  D n2 =n1 . Then a light ray in medium I with direction x 2 S 2 is refracted by some surface into a light ray with direction m 2 S 2 in medium II if and only if m x  , when  < 1; and if and only if m x  1=, when  > 1. 2.1.3  D 1 This corresponds to reflection. It means x  m D :

(8)

Taking dot products with x and then with m yields 1 m x D x  and x m 1 D

m , then x  D m . Also taking dot product with  in (8) then yields

D 2 x . Therefore m D x  2.x /:

2.2 Derivation of the Snell Law At time t, .x; y; z; t/ D 0 denotes a surface that separates the part of the space that is at rest with the part of the space that is disturbed by the electric and magnetic fields. For each t fixed the surface defined by .x; y; z; t/ D 0 is called a wave front. The light rays are the orthogonal trajectories to the wave fronts. We assume that t ¤ 0 and so we can solve .x; y; z; t/ D 0 in t obtaining that .x; y; z/ D ct; where c is the speed of light in vacuum. Therefore, it t runs, then we get that the wave fronts are the level sets of the function .x; y; z/.

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Let us assume that the wave fronts travel in an homogenous and isotropic medium I with refractive index n1 .D c=v1 /, v1 is the speed of propagation in medium I . This wave front is transmitted to another homogeneous and isotropic medium II having refractive index n2 . Let † be the surface in 3-d separating the media I and II, and suppose it is given by the equations x D f .; /; y D g.; / and z D h.; /. Let 1 .x; y; z/ D ct be the wave front in medium I and 2 .x; y; z/ D ct be the wave front, to be determined, in medium II. On the surface † the two wave fronts agree, that is, 1 .f .; /; g.; /; h.; // D 2 .f .; /; g.; /; h.; // : Differentiating this equation with respect to  and  yields  

@1 @2  @x @x @1 @2  @x @x



 f C



 f C

@1 @2  @y @y @1 @2  @y @y



 g C



 g C

@1 @2  @z @z @1 @2  @z @z

 h D 0  h D 0:

This means that the vector D1  D2 is perpendicular to both vectors .f ; g ; h / and .f ; g ; h /, and therefore it is normal to the surface †. Let  be the outer normal at the surface †. Then we have .D1  D2 / D ;

(9)

for some scalar . A light ray 1 .t/ in medium I has constant speed v1 and a light ray 2 .t/ in II constant speed v2 . Say 1 .t/ is the light ray in medium I and 2 .t/ the light ray in medium II. So we have 1 .1 .t// D ct and 2 .2 .t// D ct. Differentiating with respect to t yields Di .i .t// 10 .t/ D c, i D 1; 2. Let i be the angle between the vectors Di .i .t//; 10 .t/. Since the rays are orthogonal trajectories we get that i D 0 for i D 1; 2. If i is parametrized so that ji0 .t/j D vi , we then obtain that jDi .i .t//j D

If we let x D

c D ni : vi

D2 .2 .t// D1 .1 .t// and m D , we then obtain from (9) that jD1 .1 .t//j jD2 .2 .t//j n1 x  n2 m D 

which is equivalent to (1).

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2.3 Surfaces with the Uniform Refracting Property: Far Field Case Let m 2 S 2 be fixed, and we ask the following: if rays of light emanate from the origin inside medium I, what is the surface , interface of the media I and II, that refracts all these rays into rays parallel to m? Suppose  is parameterized by the polar representation .x/x where  > 0 and x 2 S 2 . Consider a curve on  given by r.t/ D .x.t// x.t/ for x.t/ 2 S 2 . According to (2), the tangent vector r 0 .t/ to  satisfies r 0 .t/ .x.t/   m/ D 0. That is, .Œ.x.t//0 x.t/ C .x.t//x 0 .t// .x.t/   m/ D 0, which yields ..x.t//.1  m x.t///0 D 0. Therefore .x/ D

b 1m x

(10)

for x 2 S 2 and for some b 2 R. To understand the surface given by (10), we distinguish two cases  < 1 and  > 1. 2.3.1 Case  < 1 For b > 0, we will see that the surface  given by (10) is an ellipsoid of revolution about the axis of direction m. Suppose for simplicity that m D en , the nth-coordinate vector. If y D .y 0 ; yn / 2 Rn is a point on , then y D .x/x with x D y=jyj. From (10), jyj   yn D b, that is, jy 0 j2 C yn2 D . yn C b/2 which yields jy 0 j2 C .1   2 /yn2  2byn D b 2 . This surface  can be written in the form



jy 0 j2 b

p 1  2

 yn  2 C

b 1  2  2 b 1  2

2 D1

(11)

which is an ellipsoid of revolution about the yn axis with foci .0; 0/ and .0; 2b=.1 y en    2 //. Since jyj D yn C b and the physical constraint for refraction (5), jyj b is equivalent to yn  . That is, for refraction to occur y must be in the upper 1  2 part of the ellipsoid (11); we denote this semi-ellipsoid by E.en ; b/, see Fig. 2. To verify that E.en ; b/ has the uniform refracting property, that is, it refracts any ray emanating from the origin in  the directionen , we check that (2) holds at eachpoint.  y y y  en  1 > 0, and  en en  Indeed, if y 2 E.en ; b/, then jyj jyj jyj y  en is an outward normal to E.en ; b/ at y. 0, and so jyj

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Fig. 2 Only half of the ellipsoid refracts in the direction m D en

Rotating the coordinates, it is easy to see that the surface given by (10) with  < 1 and b > 0 is an ellipsoid of revolution about the axis of direction m with foci 2b m. Moreover, the semi-ellipsoid E.m; b/ given by 0 and 1  2 E.m; b/ D f.x/x W .x/ D

b ; x 2 S n1 ; x m  g; 1m x

(12)

has the uniform refracting property, any ray emanating from the origin O is refracted in the direction m. 2.3.2 Case  > 1 Due to the physical constraint of refraction (7), we must have b < 0 in (10). Define for b > 0

Refraction Problems in Geometric Optics

H.m; b/ D f.x/x W .x/ D

103

b ; x 2 S n1 ; x m  1=g: m x1

(13)

We claim that H.m; b/ is the sheet with opening in direction m of a hyperboloid of revolution of two sheets about the axis of direction m. To prove the claim, set for simplicity m D en . If y D .y 0 ; yn / 2 H.en ; b/, then y D .x/x with x D y=jyj. From (13),  yn " jyj D b, and therefore jy 0 j2 C#yn2 D . yn  b/2 which yields 2  2  b b 0 2 2 D b 2 . Thus, any point y on jy j  .  1/ yn  2   1 2  1 H.en ; b/ satisfies the equation  2 b yn  2 jy 0 j2  1 (14) 2    2 D 1 b b p 2  1 2  1 which represents a hyperboloid of revolution of two sheets about the yn axis with foci .0; 0/ and .0; 2b=. 2  1//. Moreover, the upper sheet of this hyperboloid of revolution is given by v u b u b jy 0 j2 u1 C  C 2 yn D 2 2 p t  1  1 b=  2  1 b .  en x  1 Similarly, the lower sheet satisfies yn  b < 0 and has polar equation .x/ D b . For a general m, by a rotation, we obtain that H.m; b/ is the sheet  en x C 1 with opening in direction m of a hyperboloid of revolution of two sheets about the 2b m. axis of direction m with foci .0; 0/ and 2  1 Notice that the focus .0; 0/ is outside the region enclosed by H.m; b/ and the 2b y focus 2 m is inside that region. The vector m  is an inward normal to  1 jyj H.m; b/ at y, because by (13)     y 2b 2 2 b 2b m  m  y   2  m y C jyj 2 2 jyj  1  1  1 and satisfies yn  b > 0, and hence has polar equation .x/ D

2b b.  1/ b D > 0:  C1 C1     y y y m    1 and m  > 0. Therefore, H.m; b/ Clearly, m  jyj jyj jyj satisfies the uniform refraction property. D

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We remark that one has to use H.en ; b/ to uniformly refract in the direction en , and due to the physical constraint (7), the lower sheet of the hyperboloid of (14) cannot refract in the direction en . From the above discussion, we have proved the following. Lemma 2.2. Let n1 and n2 be the indexes of refraction of two media I and II, respectively, and  D n2 =n1 . Assume that the origin O is inside medium I, and E.m; b/; H.m; b/ are defined by (12) and (13), respectively. We have: (i) If  < 1 and E.m; b/ is the interface of media I and II, then E.m; b/ refracts all rays emitted from O into rays in medium II with direction m. (ii) If  > 1 and H.m; b/ separates media I and II, then H.m; b/ refracts all rays emitted from O into rays in medium II with direction m. 2.3.3 Case  D 1 When  D 1 we see this is a paraboloid. Indeed, let m D en , then a point X D .x/x is on the surface (10) if jX j D b  xn . The distance from X to the plane xn D b is b  xn , and the distance from X to 0 is jX j. So this is a paraboloid with focus at 0, directrix plane xn D b and axis in the direction en .

2.4 Uniform Refraction: Near Field Case The question we ask is: given a point O inside medium I and a point P inside medium II, find an interface surface S between media I and II that refracts all rays emanating from the point O into the point P . Suppose O is the origin, and let X.t/ be a curve on S. By the Snell law of refraction the tangent vector X 0 .t/ satisfies X 0 .t/



P  X.t/ X.t/  jX.t/j jP  X.t/j

 D 0:

That is, jX.t/j0 C jP  X.t/j0 D 0: Therefore S is the Cartesian oval jX j C jX  P j D b:

(15)

Since f .X / D jX j C jX  P j is a convex function, the oval is a convex set. We need to find and analyze the polar equation of the oval. Write X D .x/x with x 2 S n1 . Then writing j.x/x  P j D b  .x/, squaring this quantity and solving the quadratic equation yields

Refraction Problems in Geometric Optics

.x/ D

.b   2 x P / ˙

p

105

.b   2 x P /2  .1   2 /.b 2   2 jP j2 / : 1  2

(16)

Set .t/ D .b   2 t/2  .1   2 /.b 2   2 jP j2 /:

(17)

2.5 Case 0 <  < 1 We have .x P / >  2 .x P  b/2 ;

if jx P j < jP j:

(18)

If b  jP j, then O and P are inside or on the oval, and so the oval cannot refract rays to P . If the oval is non empty, then jP j  b. In case jP j D b, the oval reduces to the point O. The only interesting case is then jP j < b < jP j. From the equation of the oval we get that .x/  b. So we now should decide which values ˙ to take in the definition of .x/. Let C and  be the corresponding ’s. We claim that C .x/ > b and  .x/  b. Indeed, p .b   2 x P / C .x P / C .x/ D 1  2 

.b   2 x P / C jb  x P j 1  2

DbC

 2 .b  x P / C jb  x P j 1  2

 b: The equality C .x/ D b holds only if jx P j D jP j and b D x P . So C .x/ > b if jP j < b < jP j. Similarly, p .b   2 x P /  .x P /  .x/ D 1  2 

.b   2 x P /  jb  x P j 1  2

DbC

 2 .b  x P /  jb  x P j 1  2

 b: So the claim is proved. Therefore the polar equation of the oval is then given by

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a

b 0

P

P 0

Fig. 3 Cartesian ovals  < 1, e.g., glass to air. (a) jXj C 2=3jX  P j D 1:4  1:9, P D .2; 0/. (b) jXj C 2=3jX  P j D 1:7, P D .2; 0/

p .b   2 x P /  .x P / h.x; P; b/ D  .x/ D : (19) 1  2   Ph.x;P; b/x  From the physical constraint for refraction we must have x jPh.x;P; b/xj , and from the equation of the oval we then get that to have refraction we need x P  b:

(20)

Concluding with the case  < 1, given P 2 Rn and jP j < b < jP j, keeping in mind (19) and (20), a refracting oval is the set (Fig. 3) O.P; b/ D fh.x; P; b/ x W x 2 S n1 ; x P  bg where h.x; P; b/ D

.b   2 x P / 

p .b   2 x P /2  .1   2 /.b 2   2 jP j2 / : 1  2

Remark 2.3. If jP j ! 1, then the oval converges to an ellipsoid which is the surface having the uniform refraction property in the far field case, see Sect. 2.3.1. In fact, if m D P =jP j and b D jP j C C with C positive constant we have

h.x; P; b/ D D

b 2   2 jP j2 p b   2 x P C .x P / .jP j   2 x mjP j C C / C !

as jP j ! 1.

C.2jP j C C / p .jP j   2 x mjP j C C /2  .1   2 /C.2jP j C C /

C 2C p D 1  x m .   2 x m/2

.   2 x m/ C

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2.6 Case  > 1 In this case we must have jP j  b, and in case b D jP j the oval reduces to the point P . Also b < jP j, since otherwise the points 0; P are inside the oval or 0 is on the oval, and therefore there cannot be refraction if b  jP j. So to have refraction we must have jP j < b < jP j and so the point P is inside the oval and 0 is outside the oval. Rewriting  in (16) we get that p . 2 x P  b/2  . 2  1/. 2 jP j2  b 2 / ˙ .x/ D 2  1 p b C . 2  1/. 2 jP j2  b 2 / for .x P /  0 which amounts x P  . Notice 2 2 that ˙ .x/ < p0 for  x P  b < 0. We have that  .x/  C .x/  . 2 jP j  b/ C .jP j/ jP j C b D < b. To have refraction, by the physical 2  1 C1 P  x˙ .x/ constraint we need to have x  1=, which is equivalent to jP  x˙ .x/j  2 x P  b  . 2  1/˙ .x/. Therefore, the physical constraint is satisfied only by  . Therefore if  > 1, refraction only occurs when jP j < b < jP j, and the refracting piece of the oval is then given by . 2 x P  b/ ˙

( O.P; b/ D h.x; P; b/x W x P 

bC

) p . 2  1/. 2 jP j2  b 2 / 2

(21)

with h.x; P; b/D .x/D

. 2 x P  b/ 

p . 2 x P  b/2  . 2  1/. 2 jP j2  b 2 / ; 2  1 (22)

see (Fig. 4). Remark 2.4. If jP j ! 1, then the oval O.P; b/ converges to the semi hyperboloid appearing in the far field refraction problem when  > 1, see Sect. 2.3.2. Indeed, let P 2 S n1 and b D jP j  a with a > 0 a constant. For x 2 .P; b/ we mD jP j have bC

p p . 2  1/. 2 jP j2  b 2 / jP j  a C . 2  1/. 2 jP j2  .jP j  a/2 / 1 D !  2 jP j  2 jP j 

as jP j ! 1. On the other hand, if x m > 1=, we get

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C.E. Guti´errez

a

b 0

P

P

0

Fig. 4 Cartesian ovals  > 1, e.g., air to glass. (a) jXj C 3=2jX  P j D 2:9  2:4, P D .2; 0/. (b) jXj C 3=2jX  P j D 2:7, P D .2; 0/

h.x; P; b/ D

. 2 jP jx m  jP j C a/ C !

p

a.2jP j  a/ . 2 jP jx m  jP j C a/2  . 2  1/a.2jP j  a/

a2 a p D ; x m  1 . 2 x m  /2

2x m   C

as jP j ! 1.

3 Optimal Mass Transportation Let D, D  be two domains on S n1 or domains in a manifold (D might be contained in one manifold and D  in another) with j@Dj D 0. Let N be a multi-valued mapping from D onto D  such that N .x/ is singlevalued a.e. on D. We also assume that jfx 2 DN W N .x/ D ;gj D 0. For F  D  , we set T .F / D N 1 .F / D fx 2 D W N .x/ \ F ¤ ;g: We say N is measurable if T .F / is Lebesgue measurable for any Borel set F  D  . For example, if N D @u with u convex, then N is measurable. Because for u convex we have m 2 @u.x0 / iff x0 2 @u .m/, where u is the Legendre transformation of u. Therefore .@u/1 .F / D @u .F /, and since @u .F / is Lebesgue measurable for each Borel set F , we get that @u is measurable. Suppose g 2 L1 .D/ is nonnegative and  on D  is a finite Radon measure satisfying the conservation condition Z g.x/ dx D .D  / > 0:

(1)

D

R Notice that the set function .F / D T .F / g.x/ dx is Borel measure because N is single valued a.e., and measurable. Indeed, if S is the set of measure zero such that N .x/ is single valued for all x 2 DN n S , and Fj is a sequence of disjoint Borel sets,

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109

then jT .Fi / \ T .Fj /j D 0 for i ¤ j , then  is -additive. Therefore  is a finite Borel measure and so is regular. We say N is measure preserving from g.x/dx to  if for any Borel F  D  Z g.x/ dx D .F /:

(2)

T .F /

Lemma 3.1. N is a measure preserving mapping from g.x/dx to  if and only if for any v 2 C.D  / Z

Z v.N .x//g.x/ dx D D

D

v.m/ d .m/:

(3)

We remark that v.N .x// is well defined R for x 2 D n S where N .x/ is single-valued on D n S and jS j D 0, and D v.N .x//g.x/ dx is understood as R DnS v.N .x//g.x/ dx. Proof. Let N be a measure preserving mapping. To show (3), it suffices to prove it for v D F , the characteristic function of a Borel set F , because for each v continuous there exists a sequence of simple functions converging uniformly to v. It is easy to verify that T .F / .x/ D F .N .x// for x 2 D n S . Therefore by (2) Z

Z D

F .m/ d  D

Z g dx D T .F /\.DnS /

F .N .x//g.x/ dx: DnS

To prove the converse, assume that (3) holds. We will show that for any Borel set E  D Z g dx  .E/: (4) T .E/

Indeed, let us first assume that E D G is open, then given a compact set K  G, choose v 2 C.D  / such that 0  v  1, v D 1 on K, and v D 0 outside G. By (3), one gets Z

Z g.x/ dx  T .K/

v.N .x//g.x/ dx  .G/; D

for each compact K  G. Since  is regular, (4) follows for E open. For a general Borel set E  D  , since  is also regular, given  > 0 there exists G open E  G with .G n E/ < . Then Z

Z g.x/ dx  T .E/

and so (4) follows.

g.x/ dx  .G/ D .E/ C .G n E/ < .E/ C  T .G/

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We next prove that equality holds in (4). First notice that fx 2 DN W N .x/ ¤ ;g \ .T .F //c  T .F c /; for any set F  DN  . Then applying (4) to D  n F with F Borel set yields Z

Z N fx2DWN .x/¤;g\.T .F //c

g.x/ dx  .D  n F / D .D  /  .F /:

g.x/ dx  T .F c /

Since jfx 2 DN W N .x/ D ;gj D 0, we have Z

Z N fx2DWN .x/¤;g\.T .F //c

Z

g.x/ dx D

g.x/ dx D

Z DN

.T .F //c

g.x/ dx 

g.x/ dx: T .F /

So from the conservation condition (1), we obtain the reverse inequality in (4).

t u

Lip.DD  /, the space of Lipschitz

Consider the general cost function c.x; m/ 2 functions on D  D  , and the set of admissible functions

K D f.u; v/ W u 2 C.D/; v 2 C.D  /; u.x/Cv.m/  c.x; m/; 8x 2 D; 8m 2 D  g: Define the dual functional I for .u; v/ 2 C.D/  C.D  / Z

Z u.x/g.x/ dx C

I.u; v/ D D

D

v.m/ d ;

and define the c- and c  -transforms uc .m/ D inf Œc.x; m/  u.x/ ; x2D

m 2 D I

vc .x/ D inf Œc.x; m/  v.m/ ; m2D 

x 2 D:

Definition 3.2. A function  2 C.D/ is c-concave if for x0 2 D, there exist m0 2 D  and b 2 R such that .x/  c.x; m0 /  b on D with equality at x D x0 . Obviously vc is c-concave for any v 2 C.D  /. We collect the following properties: 1. For any u 2 C.D/ and v 2 C.D  /, vc 2 Lip.D/ and uc 2 Lip.D  / with Lipschitz constants bounded uniformly by the Lipschitz constant of c. Indeed, (x0 the point where the minimum is attained) uc .m1 /  uc .m2 /  uc .m1 /  .c.x0 ; m2 /  u.x0 //  c.x0 ; m1 /  u.x0 /  c.x0 ; m2 / C u.x0 /  Kjm1  m2 j:

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111

2. If .u; v/ 2 K, then v.m/  uc .m/ and u.x/  vc .x/. Also .vc ; v/; .u; uc / 2 K. 3.  is c-concave iff  D . c /c . Indeed, if .x/  c.x; m0 /  b on D and the equality holds at x D x0 , then b D  c .m0 /. So .x0 / D c.x0 ; m0 /   c .m0 / which yields .x0 /  . c /c .x0 /. On the other hand, from the definitions of c and c  transforms we always have that . c /c   for any . Definition 3.3. Given a function .x/, the c-normal mapping of  is defined by Nc; .x/ D fm 2 D  W .x/ C  c .m/ D c.x; m/g;

for x 2 D;

1 .m/ D fx 2 D W m 2 Nc; .x/g. and Tc; .m/ D Nc;

We assume that the cost function c.x; m/ satisfies the following: For any c-concave function , Nc; .x/ is single-valued a.e. on D

(5)

and Nc; is Lebesgue measurable. Notice that if c.x; m/ D x m, then Nc; .x/ D @ .x/, where @  is the superdifferential of  @ .x/ D fm 2 Rn W .y/  .x/ C m .y  x/ 8 y 2 g; and we have @ .x/ D @./.x/. Lemma 3.4. Suppose that c.x; m/ satisfies the assumption (5). Then (i) If  is c-concave and Nc; is measure preserving from g.x/dx to , then .;  c / is a maximizer of I.u; v/ in K. (ii) If .x/ is c-concave and .;  c / maximizes I.u; v/ in K, then Nc; is measure preserving from g.x/dx to . Proof. First prove (i). Given .u; v/ 2 K, obviously u.x/ C v.Nc; .x//  c.x; Nc; .x// D .x/ C  c .Nc; .x//;

a.e. x on D:

Integrating the above inequality with respect to g dx yields Z

Z

Z

ug dx C D

Z

v.Nc; .x//g.x/ dx  D

g dx C D

 c .Nc; .x//g.x/ dx: D

By Lemma 3.1, it yields I.u; v/  I.;  c / and from (2) above the conclusion follows. To prove (ii), let D  c , and for v 2 C.D  /, let .m/ D .m/ C v.m/ where 0 <  0 with 0 small, and let  D . /c . We shall prove that lim

!0

I. ;

 I.; / D

/

Z

Z v.Nc; .x// g dx C D

D

v.m/ d :

(6)

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C.E. Guti´errez

Since . ; / 2 K, we have I. ; /  I.; / for 0 <  0 , and hence the existence of the limit (6) implies it must be zero. Therefore the measure preserving property of Nc; follows from Lemma 3.1. To prove (6) we write I. ;

 I.; / D

/

Z D

   g dx C

Z D

v.m/ d :

By Lebesgue dominated convergence theorem, to show (6), it is enough to show  .x/  .x/  .x/  .x/ is uniformly bounded, and ! v.Nc; .x// for that all x 2 D n S , where Nc; .x/ is single-valued on D n S and jS j D 0. Let us first N we have by continuity that  .x/ D prove the uniform boundedness. Fix x 2 D, c.x; m /  .m / for some m 2 D  . Since  is c-concave there exists m1 2 D  and b 2 R such that .y/  c.y; m1 /  b for all y 2 DN with equality when y D x. This implies that b D  c .m1 / and so .x/ D c.x; m1 /  .m1 /. Hence  .x/  .x/ D c.x; m /  

c .x/ 

.m /  v.m /  .x/

v.m /  .x/ D . c /c .x/  v.m /  .x/   v.m /;

by (3) above. We also have  .x/  .x/ D  .x/  c.x; m1 / C   .x/  .

/c .x/ 

.m1 / D  .x/  c.x; m1 / C

.m1 / 

v.m1 /

v.m1 / D  v.m1 /:

Then we get  v.m /   .x/  .x/   v.m1 /: Moreover, if x 2 DN n S , then m1 D Nc; .x/ since D  c . To finish the proof, we show that m converges to m1 as ! 0. Otherwise, there exists a sequence m k such that m k ! m1 ¤ m1 . So .x/ D lim !0  .x/ D c.x; m1 /  .m1 /, which yields m1 2 Nc; .x/. We then get m1 D m1 , a contradiction. The proof is complete. t u Lemma 3.5. There exists a c-concave  such that I.;  c / D supfI.u; v/ W .u; v/ 2 Kg: Proof. Let I0 D supfI.u; v/ W .u; v/ 2 Kg; and let .uk ; vk / 2 K be a sequence such that I.uk ; vk / ! I0 . Set uN k D .vk /c and vN k D .Nuk /c . From property (2) above, .Nuk ; vN k / 2 K, uk  uN k , vk  vN k , and so I.Nuk ; vN k / ! I0 . Let ck D minD uN k and define

Refraction Problems in Geometric Optics ]

uk D uN k  ck ; ]

113 ]

vk D vNk C ck :

]

Obviously .uk ; vk / 2 K and by the mass conservation condition on gdx and ] ] ] , equation (2), I.Nuk ; vNk / D I.uk ; vk /. Since uN k are uniformly Lipschitz, uk are ] ] uniformly bounded. In addition, vk D .Nuk /c C ck D .uk /c and consequently ] ] ] vk are also uniformly bounded. By Arzel´a-Ascoli’s theorem, .uk ; vk / contains a  subsequence converging uniformly to .; / on D  D . We then obtain that .; / 2 K and I0 D supfI.u; v/ W .u; v/ 2 Kg D I.; /. Notice that this shows in particular that the supremum of I over K is finite. From property (2) above, . c ; . c /c / is the sought maximizer of I.u; v/, and c is c-concave. t u c Lemma 3.6. Suppose that c.x; m/ satisfies the assumption Z (5). Let .;  / with c.x; s.x//g.x/ dx is  c-concave be a maximizer of I.u; v/ in K. Then inf s2S

D

attained at s D Nc; , where S is the class of measure preserving mappings from g.x/dx to . Moreover Z c.x; s.x//g.x/ dx D supfI.u; v/ W .u; v/ 2 Kg:

inf

s2S

Proof. Let

(7)

D

D  c . For s 2 S, we have

Z

Z c.x; s.x//g.x/ dx  D

..x/ C

.s.x/// g.x/ dx

D

Z

Z

.x/g.x/ dx C

D D

Z D

Z .x/g.x/ dx C

D

Z D

.s.x//g.x/ dx



.x/ C

D

D

.m/ d  D I.; /

.Nc; .x// g.x/ dx; from Lemma 3.4(ii)

D

Z D

c.x; Nc; .x//g.x/ dx: D

t u Obviously, for any c-concave function , Nc; has the following converging property (C): if mk 2 Nc; .xk /, xk ! x0 and mk ! m0 , then m0 2 Nc; .x0 /. R Lemma 3.7. Assume that c.x; m/ satisfies the assumption (5) Z and that G g dx > 0 for any open G  D. Then the minimizing mapping of inf c.x; s.x//g.x/ dx is s2S

D

unique in the class of measure preserving mappings from g.x/dx to  with the converging property (C).

114

C.E. Guti´errez

Proof. From Lemmas 3.5 and 3.6, let Nc; be a minimizing mapping associated with a maximizer .;  c / of I.u; v/ with  c-concave. Suppose that N0 is another minimizing mapping with the converging property (C). Clearly Z .c.x; N0 .x//  .x/   c .N0 .x/// g.x/ dx D

Z

Z

D inf

s2S

c.x; s.x//g.x/ dx 

.x/g.x/ dx C

D



Z D

c

D

 .m/ d 

D 0;

and since .x/ C  c .N0 .x//  c.x; N0 .x//, it follows that .x/ C  c .N0 .x// D c.x; N0 .x// on the set fx 2 D W g.x/ > 0g which is dense in D. Hence from (5) and the converging property (C), we get N0 .x/ D Nc; .x/ a.e. on D. t u We remark fromR the above proof that if g.x/ > 0 on D, then the minimizing mapping of infs2S D c.x; s.x//g.x/ dx is unique in the class of measure preserving mappings from g.x/dx to .

3.1 Application to the Refractor Problem  < 1 Let n1 and n2 be the indexes of refraction of two homogeneous and isotropic media I and II, respectively. Suppose that from a point O inside medium I light emanates with intensity f .x/ for x 2  . We want to construct a refracting surface R parameterized as R D f.x/x W x 2 g, separating media I and II, and such that all rays refracted by R into medium II have directions in  and the prescribed illumination intensity received in the direction m 2  is f  .m/. We first introduce the notions of refractor mapping and measure, and weak solution. In the next section we then convert the refractor problem into an optimal 1 mass transport problem from  to  with the cost function log and 1x m establish existence and uniqueness of weak solutions. Let ,  be two domains on S n1 , the illumination intensity of the emitting beam is given by nonnegative f .x/ 2 L1 ./, and the prescribed illumination intensity of the refracted beam is given by a nonnegative Radon measure  on  . Throughout this section, we assume that j@j D 0 and the physical constraint inf

x2;m2

x m  :

(1)

We further suppose that the total energy conservation Z f .x/ dx D . / > 0; 

and for any open set G  

(2)

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115

Z f .x/ dx > 0;

(3)

G

where dx denotes the surface measure on S n1 .

3.2 Refractor Measure and Weak Solutions We begin with the notions of refractor and supporting semi-ellipsoid. Definition 3.8. A surface R parameterized by .x/x with  2 C./ is a refractor from  to  for the case  < 1 (often simply called as refractor in this section) if for any x0 2  there exists a semi-ellipsoid E.m; b/ with m 2  such that b b for all x 2 . Such E.m; b/ is and .x/  .x0 / D 1   m x0 1m x called a supporting semi-ellipsoid of R at the point .x0 /x0 . From the definition, any refractor is globally Lipschitz on . Definition 3.9. Given a refractor R D f.x/x W x 2 g, the refractor mapping of R is the multi-valued map defined by for x0 2  NR .x0 / D fm 2  W E.m; b/ supports R at .x0 /x0 for some b > 0g: Given m0 2  , the tracing mapping of R is defined by 1 TR .m0 / D NR .m0 / D fx 2  W m0 2 NR .x/g:

Definition 3.10. Given a refractor R D f.x/x W x 2 g, the Legendre transform of R is defined by 1 ; m 2  g: x2 .x/.1   x m/

R D f .m/m W  .m/ D inf

We now give some basic properties of Legendre transforms. Lemma 3.11. Let R be a refractor from  to  . Then (i) R is a refractor from  to . (ii) R D .R / D R. (iii) If x0 2  and m0 2  , then x0 2 NR .m0 / iff m0 2 NR .x0 /. Proof. Given m0 2  , .x/.1  x m0 / must attain the maximum over  at some x0 2 . Then  .m0 / D 1=Œ.x0 /.1  x0 m0 /. We always have 1 1  ; .x0 /.1  x0 m/ x2 .x/.1  m x/

 .m/ D inf

8m 2  :

(4)

116

C.E. Guti´errez

Hence E.x0 ; 1=.x0 // is a supporting semi-ellipsoid to R at  .m0 /m0 . Thus, (i) is proved. To prove (ii), from the definitions of Legendre transform and refractor mapping we have .x0 /  .m0 / D

1 1  m0 x0

for m0 2 NR .x0 /:

(5)

1=.x0 / . By 1  x0 m0 (4),  .m/.1  kx0 m/ attains the maximum 1=.x0 / at m0 . Thus,

For x0 2 , there exists m0 2 NR .x0 / and so from (5)  .m0 / D

 .x0 / D inf

m2

 .m/.1

1 1 D :  kx0 m/ .x0 /1

To prove (iii), we get from the proof of (ii) that if m0 2 NR .x0 /, then the semiellipsoid E.x0 ; 1=.x0 // supports R at  .m0 /m0 and so x0 2 NR .m0 /. On the other hand, if x0 2 NR .m0 /, we get that m0 2 NR .x0 /, and since R D R, m0 2 NR .x0 /. t u The next two lemmas discuss the refractor measure. Lemma 3.12. C D fF   W TR .F / is Lebesgue measurableg is a -algebra containing all Borel sets in  . Proof. Obviously, TR .;/ D ; and TR . / D . Since TR .[1 i D1 Fi / D  T .F /, C is closed under countable unions. Clearly for F   [1 R i i D1 TR .F c / D fx 2  W NR .x/ \ F c ¤ ;g D fx 2  W NR .x/ \ F D ;g [ fx 2  W NR .x/ \ F c ¤ ;; NR .x/ \ F ¤ ;g D ŒTR .F /c [ ŒTR .F c / \ TR .F /:

(6)

If x 2 TR .F c /\TR .F /\, then R parameterized by  has two distinct supporting semi-ellipsoids E.m1 ; b1 / and E.m2 ; b2 / at .x/x. We show that .x/x is a singular point of R. Otherwise, if R has the tangent hyperplane ˘ at .x/x, then ˘ must coincide both with the tangent hyperplane of E.m1 ; b1 / and that of E.m2 ; b2 / at .x/x. It follows from the Snell law that m1 D m2 . Therefore, the area measure of TR .F c / \ TR .F / is 0. So C is closed under complements, and we have proved that C is a -algebra. To prove that C contains all Borel subsets, it suffices to show that TR .K/ is compact if K   is compact. Let xi 2 TR .K/ for i  1. There exists mi 2 NR .xi / \ K. Let E.mi ; bi / be the supporting semi-ellipsoid to R at .xi /xi . We have .x/.1  mi x/  bi

for x 2 ;

(7)

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117

where the equality in (7) occurs at x D xi . Assume that a1  .x/  a2 on  for some constants a2  a1 > 0. By (7) and (1), a1 .1  /  bi  a2 .1   2 /. Assume through subsequence that xi ! x0 , mi ! m0 2 K, bi ! b0 , as i ! 1. By taking limit in (7), one obtains that the semi-ellipsoid E.m0 ; b0 / supports R at .x0 /x0 and x0 2 TR .m0 /. This proves TR .K/ is compact. To show that C is closed by complements, it is enough to notice the formula that    TR  n F D TR . / n TR .F / [ TR . n F / \ TR .F / : t u Lemma 3.13. Given a nonnegative f 2 L1 ./, the set function Z MR;f .F / D f dx TR .F /

is a finite Borel measure defined on C and is called the refractor measure associated with R and f . Proof. Let fFi g1 i D1 be a sequence of pairwise disjoint sets in C. Let H1 D TR .F1 /, and Hk D TR .Fk / n [k1 i D1 TR .Fi /, for k  2. Since Hi \ Hj D ; for i ¤ j and 1 [1 H D [ T .F k /, it is easy to get kD1 k kD1 R MR;f .[1 kD1 Fk / D

Z f dx D

1 Z X

f dx:

kD1 Hk

[1 kD1 Hk

// is a subset of the singular Observe that TR .Fk / n Hk D TR .Fk / \ .[k1 i D1 TR .Fi R set of R and has area measure 0 for k  2. Therefore, Hk f dx D MR;f .Fk / and the -additivity of MR;f follows. t u The notion of weak solutions is introduced through the conservation of energy. Definition 3.14. A refractor R is a weak solution of the refractor problem for the case  < 1 with emitting illumination intensity f .x/ on  and prescribed refracted illumination intensity  on  if for any Borel set F   Z MR;f .F / D f dx D .F /: (8) TR .F /

3.3 Solution of the Refractor Problem We introduce the cost c.x; m/ D log

1 1x m

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C.E. Guti´errez

for x 2  and m 2  where we assume    . From Definitions 3.2 and 3.8, R D f.x/x W x 2 g is a refractor iff log  is c-concave. Using Definitions 3.3 and 3.9 we get that Nc; .x/ D NR .x/;

R D f.x/x W x 2 g;

.z/ D e .z/:

Furthermore, log  D .log /c , log  D .log  /c by Remark (3) after Definition 3.2, and NR .x0 / D Nc;log  .x0 / by (5). By the Snell law and Lemma 3.12, c.x; m/ satisfies (5). From the definitions, R is a weak solution of the refractor problem iff log  is c-concave and Nc;log  is a measure preserving mapping from f .x/dx to . By Lemma 3.5, there exists a c-concave .x/ such that .;  c / maximizes Z

Z uf dx C

I.u; v/ D 



v d.m/

in K D f.u; v/ 2 C./  C. / W u.x/ C v.m/  c.x; m/; for x 2 ; m 2  g. Then by Lemma 3.4, Nc; .x/ is a measure preserving mapping from f dx to . Therefore, R D fe .x/x W x 2 g is a weak solution of the refractor problem. It remains to prove the uniqueness of solutions up to dilations. Let Ri D fi .x/x W x 2 g, i D 1; 2, be two weak solutions of the refractor problem. Obviously, Nc;log i have the converging property (C) stated before Lemma 3.7. It follows from Lemmas 3.4, 3.6 and 3.7 that Nc;log 1 .x/ D Nc;log 2 .x/ a.e. on . x  NRi .x/ That is, NR1 .x/ D NR2 .x/ a.e. on . From the Snell law i .x/ D jx  NRi .x/j is the unit normal to Ri towards medium II at i .x/x where Ri is differentiable. So 1 .x/ D 2 .x/ a.e. and consequently 1 .x/ D C 2 .x/ for some C > 0.

4 Solution to the Refractor Problem for  < 1 with the Minkowski Method In this section we solve the refractor problem using a method having its roots in the Minkowski method from convex analysis, [13, Sect. 7.1]. In addition, to the definitions and lemmas from Sect. 3.2, we have the following. Lemma 4.1. We have for a refractor R that (i) ŒTR .F /c  TR .F c / for all F   , with equality except for a set of measure zero. (ii) The set C D fF   W TR .F / i s Lebesgue measurableg is a -algebra containing all Borel sets in  . Lemma 4.2. Let Rj D fj .x/x W x 2 g, j  1 be refractors from  to  : Suppose that 0 < a1  j  a2 and j !  uniformly on : Then:

Refraction Problems in Geometric Optics

119

(i) R WD f.x/x W x 2 g is a refractor from  to  : (ii) For any compact set K   lim sup TRj .K/  TR .K/: j !1

(iii) For any open set G   ; TR .G/  lim inf TRj .G/ [ S; j !1

where S is the singular set of R: Proof. (i) Obviously  2 C./ and  > 0: Fix xo 2 : Then there exist mj 2  and bj > 0 such that E.mj ; bj / supports Rj at .xo /xo and thus bj 1  mj xo

j .xo / D

and

j .x/ 

and

a1 

bj 1  mj x

for all x 2 : Consequently bj  a2 1  mj xo

bj 1  mj x

for all j and therefore a1 .1  /  bj  a2 for all j: If need be by passing to a subsequence we obtain mo and bo such that mj ! mo 2  and bj ! bo : We claim E.mo ; bo / supports R at .xo /xo : Indeed .xo / D lim j .xo / D lim j

j

bj bo D 1  mj xo 1  mo xo

and .x/ D lim j .x/  lim j

j

bj bo D 1  mj x 1  mo x

for all x 2 : Thus R is a refractor. (ii) Let xo 2 lim sup TRj .K/: Without loss of generality assume that xo 2 TRj .K/ for all j  1: Then there exist mj 2 NRj .xo / \ K and bj such that j .xo / D

bj 1  mj xo

and

j .x/ 

bj 1  mj x

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for all x 2 : As in the proof of (i) we may assume that mj ! mo 2 K and bj ! bo and conclude that E.mo ; bo / supports R at .xo /xo ; proving that xo 2 TR .mo /: Hence xo 2 TR .K/: (iii) Let G be an open subset of  : By (ii) lim sup TRj .G c /  TR .G c / as G c is compact. Also lim supŒTRj .G/c  lim supfŒTRj .G/c [ ŒTRj .G/ \ TRj .G c /g j !1

j !1

(1)

and by Lemma 4.1 the right hand side of (1) is equal to lim supj !1 TRj .G c /: By (ii) we will then have lim supŒTRj .G/c  TR .G c / D fŒTR .G/c [ ŒTR .G/ \ TR .G c /g: j !1

Taking complements we obtain flim supŒTRj .G/c gc  ŒTR .G/ \ ŒTR .G/ \ TR .G c /c : j !1

Consequently lim inf TRj .G/  ŒTR .G/ \ ŒTR .G/ \ TR .G c /c j !1

and thus ŒŒTR .G/ \ ŒTR .G/ \ TR .G c /c  [ S  lim inf TRj .G/ [ S: j !1

But TR .G/ \ TR .G c /  S: Thus TR .G/  TR .G/ [ S  lim inf TRj .G/ [ S j !1

as required. t u Remark 4.3 (Invariance by dilations). Suppose that R is a refractor weak solution N in the sense of Definition 3.14 with intensities f;  and defined by .x/x for x 2 . N Then for each ˛ > 0, the refractor ˛R defined by ˛ .x/x for x 2  is a weak solution in the sense of Definition 3.14 with the same intensities. In fact, E.m; b/ is a supporting ellipsoid to R at the point y if and only if E.m; ˛ b/ is a supporting N . ellipsoid to ˛R at the point y. This means that TR .m/ D T˛R .m/ for each m 2 

4.1 Existence of Solutions in the Discrete Case Theorem 4.4. Let f 2 L1 ./ with inf f > 0, g1 ; ; gN positive numbers, m1 ; ; mN 2  distinct points, N  2, with x mj   for all x 2  and 1  P j  N . Let  D N j D1 gj ımj , and assume the conservation of energy condition

Refraction Problems in Geometric Optics

Z

121

f .x/ dx D . /:

(2)



Then there exists a refractor R such that N D [N TR .mj /, (a)  j D1 R (b) TR .mj / f .x/ dx D gj for 1  j  N . To prove the theorem, we prove first a sequence of lemmas. Lemma 4.5. Let ( W D

)

Z b D .1; b2 ; ; bN / W bj > 0; MR.b/;f .mj / D

TR.b/ .mj /

f .x/ dx  gj ; j D 2; ; N

;

(3)

where .x/ D R.b/.x/ D min

1j N

bi : 1   x mj

(4)

Then, with the assumptions of Theorem 4.4, we have (a) W ¤ ; (b) if b D .1; b2 ; ; bN / 2 W , then bj >

1 for j D 2; ; N . 1C

Proof. (a) If for some j ¤ 1, the semi-ellipsoid E.mj ; b/ supports R.b/ at some b b x 2 , then .z/  for all z 2 , and .x/ D . Since 1   z mj 1   x mj x mj  , we have b b 1 1 ;  D .x/   1  2 1   x mj 1   x m1 1 and so b  1 C . Therefore, if bi > 1 C  for 2  i  N , then E.mi ; bi / cannot be a supporting ellipsoid to R.b/ at any x 2 . On the other hand, if x 2 TR .mj /, then mj 2 NR .x/ and if x is not a singular point of R.b/ there is a unique ellipsoid E.mj ; b/ supporting R at x. But from the definition of R there is an ellipsoid E.mk ; bk / that supports R at x, and so E.mk ; bk / D E.mj ; b/, i.e., k D j and b D bj . Consequently the set TR .mj / is contained in the set of singular points and therefore has measure zero. So MR.b/;f .mj / D 0 < gj for j D 2; ; N and so any point b D .1; b2 ; ; bN / 2 W as long as bi > 1 C  for i D 2; ; N . (b) First notice that if E.mj ; bj / and E.mk ; bk / support R.b/ at x0 , then x0 is a singular point. And therefore, jTR .mj / \ TR .mk /j D 0 for k ¤ j . Claim 1. If b 2 W , then g1  MR.b/;f .m1 /.

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Indeed, N X

MR.b/;f .mi / D

iD1

N Z X iD1

Z f .x/ dx D TR.b/ .mi /

Z [N i D1 TR.b/ .mi /

f .x/ dx D . / D

f .x/ dx D 

N X

gi ;

iD1

from (2). Hence g1  MR.b/;f .m1 / C

N X  gi  MR.b/;f .mi / D 0: i D2

If b 2 W , then gi  MR.b/;f .mi /  0 for i D 2; ; N , and Claim 1 follows.  c ¤ ;. Claim 2. For each b 2 W , TR.b/ .m1 / \ [N i D2 TR.b/ .mi / Otherwise, TR.b/ .m1 /  [N T .m / which means that each point in i i D2 R.b/ TR.b/ .m1 / is singular, and therefore jTR.b/ .m1 /j D 0. This contradicts Claim 1, since g1 > 0.  c Therefore, if b 2 W , then we can pick x0 2 TR.b/ .m1 / \ [N i D2 TR.b/ .mi / and so .x0 / D

1 bi < ; 1   x0 m1 1   x0 mi

i D 2; ; N

so bi >

1   x0 mi 1   x0 mi 1 1 :   D 1   x0 m1 1  2 1  2 1C t u j

j

Lemma 4.6. If bj D .b1 ; ; bN / ! b0 D .b10 ; ; bN0 / as j ! 1, then j D N as j ! 1. R.bj / ! 0 D R.b0 / uniformly in  b`0 . 1   y m`

N there exists 1  `  N such that 0 .y/ D Proof. Given y 2 , Hence j

j .y/  0 .y/ 

j

j

b` b`0 jb`  b`0 j jb  b`0 j    ` ! 0; 1   y m` 1   y m` 1   y m` 1

as j ! 1.

t u

Lemma 4.7. Let ı > 0 and the region Rı D f.1; b2 ; ; bN / W bj  ı; 2  j  N g. The functions GR.b/ .mi / WD MR.b/ .mi / are continuous for b 2 Rı for i D 1; 2; ; N . j

j

Proof. Let bj D .1; b2 ; ; bN / 2 Rı with bj ! b0 as j ! 1. By Lemma 4.6, j b` N Given x 2 , N we have j .x/ D for j ! 0 uniformly in . 1   x m`

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123

minf1; ıg . On the other hand, j .x/ D 1C j b` 1 1 . Therefore min   1`N 1   x m` 1   x m1 1 some 1  `  N and so j .x/ 

minf1; ıg 1  j .x/  1C 1 N and for all j . for all x 2  N  be a neighborhood of mi such that m` … G Let us fix 1  i  N . Let G   for ` ¤ i . If x0 2 TR.bj / .G/ and x0 is not a singular point, then there exists a unique b b and j .z/  for all m 2 G and b > 0 such that j .x0 / D 1   x0 m 1z m N From the definition of R.bj / and since x0 is not singular, m D m` . Since x 2 . m 2 G, we get m D mi . Therefore TR.bj / .G/  TR.bj / .mi / [ S; where S is the set of singular points. By Lemma 4.2 TR.b0 / .G/  lim inf TR.bj / .G/ [ S; j !1

and we therefore obtain TR.b0 / .G/  lim inf TR.bj / .mi / [ S: j !1

Thus Z

Z

Z

f .x/ dx 

f .x/ dx 

TR.b0 / .mi /

TR.b0 / .G/

Z

 lim inf j !1

f .x/ dx lim infj !1 TR.bj / .mi /

f .x/ dx

by Fatou.

TR.bj / .mi /

We next prove that Z

Z f .x/ dx 

lim sup j !1

TR.bj / .mi /

f .x/ dx: TR.b0 / .mi /

By Lemma 4.2 lim sup TR.bj / .K/  TR.b0 / .K/ j !1

for each K compact. Hence

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C.E. Guti´errez

Z

Z f .x/ dx  lim supj !1 TR.bj / .mi /

f .x/ dx: TR.b0 / .mi /

By reverse Fatou we have Z

Z f .x/ dx 

lim sup j !1

TR.bj / .mi /

f .x/ dx lim supj !1 TR.bj / .mi /

t u

and therefore the lemma is proved. Proof of Theorem 4.4. Fix bQ D .1; bQ2 ; ; bQN / 2 W and let WQ D fb D .1; b2 ; ; bN / 2 W W bj  bQj ; j D 2; ; N g:

WQ is compact. Let d W WQ ! R be given by d.b/ D 1 C b2 C C bN ; d attains its minimum value in WQ at a point b  D .1; b2 ; ; bN / (notice that the minimum is strictly positive by Lemma 4.5(b)). We prove that R.b  / is the refractor R that solves the problem. By conservation of energy it is enough to show that T  .mj / f .x/ dx D gj for j D 2; ; N . Since b  2 W , we have R.b / R TR.b  / .mj / f .x/ dx  gj for j D 2; ; N . Suppose by contradiction that this inequality is strict for some j , suppose for example that Z TR.b  / .m2 /

f .x/ dx < g2 :

(5)

Let 0 < < 1 and b  D .1; b2 ; b3 ; ; bN /. We claim that TR.b  / .mi / n set of measure zero  TR.b  / .mi /

(6)

for i D 3; 4; ; N and all 0 < < 1. Indeed, if x0 2 TR.b  / .mi / is not a a singular point of R.b  /, then there is a unique ellipsoid that supports 1   x mi .b  /i R.b  / at x0 for some a > 0. Since R.b  /.x/ D min1i N , there exists 1   x mi .b  /j .b  /j 1  j  N such that R.b  /.x0 / D . That is, the ellipsoid 1   x mj 1   x mj  / .b a j

supports R.b  / at x0 . Therefore, D implying j D i and 1   x mi 1   x mj bi  R.b  /.x/  R.b  /.x/ for all x, it so a D .b  /i D bi . Since 1   x mi bi follows that is a supporting ellipsoid to R.b  / at x0 . This proves the 1   x mi claim.

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125

This implies that Z

Z f .x/ dx  TR.b  / .mi /

TR.b  / .mi /



f .x/ dx  gi

for i D 3; 4; ; N and all 0 < < 1. Finally from Lemma 4.7, inequality (5) holds for all sufficiently close to one, and therefore the point b  2 WQ for all close to one. This is a contradiction because d.b  / < d.b  /. t u Remark 4.8. Notice that the solution in Theorem 4.4 has the form given by formula (4), where b1 D 1 and .1; b2 ; ; bN / 2 W . So from Lemma 4.5(b), we have bi > 1=.1 C / for i D 2; ; N . This implies that infN .x/ D ˛ > 0.

4.2 Solution in the General Case Lemma 4.9. Let R D f.x/x W x 2 g be a refractor from  to  such that infx2 .x/ D 1: Then there is a constant C , depending only on , such that sup .x/  C: x2

Proof. Suppose infx2 .x/ is attained at xo 2 ; and let E.m; b/ be a supporting semi-ellipsoid to R at .x0 /x0 : Then 1 D .x0 / D

b b 8x 2 : and .x/  1   m x0 1m x

From the first equation we get that b  1 C , and using this in the inequality we obtain .x/ 

1C for all x 2  1  2 t u

which proves the lemma.

Theorem 4.10. Let f 2 L1 ./ with inf f > 0; and let  be a Radon measure on  ; such that Z f .x/ dx D . / 

Then there exists a weak solution R of the refractor problem for the case  < 1; with emitting illumination intensity f and prescribed refracted illumination intensity :

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C.E. Guti´errez

Proof. Fix l 2 N; l  2: Partition  into a finite number of disjoint Borel subsets 1 !1l ; : : : ; !kl l such that diam.!il /  : Choose points mli 2 !il and define a measure l on  l D

kl X

.!il /ıml : i

i D1

Then kl X

l . / D

Z .!il / D . / D

i D1



f .x/ dx:

If h 2 C. /; then Z

Z 

h dl 



h d D

!il

i D1

D

!il

D

kl Z X i D1

!il

h.x/ dl 

h.x/ d !il

!

Z

Z kl X i D1

!

Z

Z kl X

h.mli /d



h.x/d !il

.h.mli /  h.x//d:

1 ; we obtain that l Z h dl ! h d as l ! 1

Since h 2 C. / and diam.!il / < Z 



and hence l converges weakly to : By Theorem 4.4, let Rl D fl .x/x W x 2 g be the solution corresponding to l , that is, MRl ;f .!/ D l .!/ for every Borel subset ! of  : Notice that from Remark 4.8, infN l .x/ D ˛l > 0. l .x/ solves the same In view of Remark 4.3, the refractor defined by the function ˛l l .x/ problem and infN D 1. So normalizing l , we may assume that inf l .x/ D ˛l 1: Then by Lemma (4.9) there exists a uniform bound C D C./ such that

Refraction Problems in Geometric Optics

127

sup l .x/  C

for all l  1:

x2

Also if xo ; x1 2  and E.mo ; bo / is a supporting semi ellipsoid to Rl at l .xo /xo then for x1 2  we have l .x1 /  l .xo /  

bo bo  b0 mo .x1  x0 /  D 1   mo x1 1   mo xo .1  mo x1 /.1  mo xo /   b0 jmo j jx1  x0 j C jx1  xo j: .1  mo x1 /.1  mo xo / 1

By changing the roles of xo and x1 we conclude that jl .x1 /  l .xo /j  C

 jx1  xo j for all l  1: 1

Thus fl W l  1g is an equicontinuous family which is bounded uniformly. Then by Arzel`a-Ascoli Theorem, if need be by taking subsequence, we have that l !  uniformly on : By Lemma 4.2(i), R D f.x/x W x 2 g is a refractor. We claim that MRl ;f converges weakly to MR;f . Indeed, if F is any closed subset of  then by Lemma 4.2(ii) and reverse Fatou we have Z lim sup MRl ;f .F / 

Z f .x/ dx  lim supl!1 TRl .F /

l!1

TR .F /

f .x/ dx D MR;f .F /:

Moreover for any open set G   we claim that Z MRl ;f .G/ D

TR .G/

f .x/ dx  lim inf MRl ;f .G/:

Indeed, from Lemma 4.2(iii) we have Z Z MR;f .G/ D f .x/ dx  Z D

TR .G/

`!1

(7)

f .x/ dx lim inf`!1 TR` .G/

lim inf TR` .G/ .x/ f .x/dx

N  `!1 

Z

 lim inf `!1

N 

TR` .G/ .x/ f .x/ dx D lim inf MRl ;f .G/; `!1

by Fatou’s lemma. Consequently MRl ;f ! MR;f weakly. Since MRl ;f .!/ D l .!/ for each Borel set !, it follows by uniqueness of the weak limit that MR;f D . t u

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4.3 Uniqueness Discrete Case With the method used in this section we show uniqueness when the target measure is discrete. Notice that uniqueness (up to dilations) was already proved with the mass transport approach in Sect. 3.3. But the method we describe here is applicable to the near field problem which is not a mass transport problem. Recall that .b1 ; ; bN / are positive numbers, m1 ; ; mN are different points in the sphere S n1 , and   S n1 with infx2;1j N N x mj  . We let .x/ D min

1i N

bi ; 1  x mi

N and let R D R.b/ D f.x/x W x 2 g. Lemma 4.11. Suppose that the set TR .mj / has positive measure. If x0 2 TR .mj /, then the semi-ellipsoid E.mj ; bj / supports R at the point x0 . Proof. We have mj 2 NR .x0 /, that is, there exists a supporting semi-ellipsoid E.mj ; b/ to R at the point x0 for some b > 0. We claim that b D bj , and bj therefore supports R at x0 . Since E.mj ; b/ supports R, we have 1  mj x b b N with equality at x D x0 . Hence .x/  for all x 2   1  x mj 1  x0 mj bj , and so b  bj . If b D bj we are done. If b < bj , then 1  x0 mj .x/ 

bj b < ; 1  x mj 1  x mj

N 8x 2 ;

bk , and therefore R cannot refract in the direction mj 1  mk x (except on a set of directions with measure zero) and so TR .mj / has measure zero, a contradiction. t u so .x/ D mink¤j

Lemma 4.12. Let Rb ; Rb  be two solutions from Theorem 4.4, with b D .b1 ; ; bN /, and b  D .b1 ; ; bN /. Assume that f > 0 a.e. in . If b1  b1 , then bi  bi for all 1  i  N . In particular, if b1 D b1 , then bi D bi for all 1  i  N. Proof. Let J D fj W bj < bj g and I D fi W bi  bi g. Suppose by contradiction bj that J ¤ ;. We have I ¤ ; since 1 2 I . Fix j 2 J , we have < 1  z mj bj bi bi N since bj < b  . And also for all z 2   for all j 1  z mj 1  z mi 1  z mi N Fix j 2 J and let x 2 TR  .mj /. Since Rb  is a solution to the i 2 I and all z 2 . b

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129

discrete problem and gi > 0 for all 1  i  N , we have that TRb .mj / has positive bj supports Rb  at the point measure. So from Lemma 4.11, the ellipsoid 1  mj z bi , and x. Since the function defining Rb  is given by  .z/ D min1i N 1  mi z  bj  .x/ D , we therefore obtain 1  mj x bj bj bi bi <   ; 1  mj x 1  mj x 1  mi x 1  mi x

8i 2 I:

Hence by continuity, there exists Nx an open neighborhood of x such that bj bi < 1  mj y 1  mi y

8i 2 I

8y 2 Nx :

bi , we get for y 2 Nx 1  mi z bj 0 bj , that is, .y/ D for some j 0 2 J that .y/ D minj 2J 1  mj y 1  mj 0 y bj 0 is a supporting ellipsoid to (depending also on y) which means that 1  mj 0 y Rb at y. Therefore Since the function defining Rb is .z/ D min1i N

 Nx  TRb [j 2J mj :  We then  have that every point x 2 TRb [j 2J mj has a neighborhood contained in TRb [j 2J mj , that is,    ı TRb [j 2J mj  TRb [j 2J mj ¤ : (8)    ı and TRb [j 2J Pj is closed, we get that TRb [j 2J mj n Since is connected TRb [j 2J mj is a non empty open set. This is a contradiction with the fact that Z f .x/ dx D TRb .[j 2J mj /

X j 2J

Z fj D

f .x/ dx; TRb  .[j 2J mj /

since f > 0 a.e.. t u From the lemma we deduce the uniqueness up to dilations in the discrete case. Let > 0. Notice that if E.m; b/ is a supporting ellipsoid to the refractor R with defining function .x/ at x0 if and only if E.m; b= / is a supporting ellipsoid to the refractor R with defining function .x/ at the point x0 . This implies that NR .x0 / D NR .x0 /, and consequently TR .m/ D TR .m/. Therefore, if R is a

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refractor solving the problem in Theorem 4.4, then R solves the same problem for any > 0. We now prove the uniqueness. Suppose Rb and Rb  are two solutions as in Theorem 4.4. Pick such that b1 D b1 . The refractor R b is also a solution to Theorem 4.4, and by Lemma 4.12 we obtain that bi D bi for all i . This means that Rb and Rb  are multiples one of each other and we obtain the uniqueness.

5 Maxwell’s Equations The electromagnetic field (EM) is a physical field produced by electrically charged objects. It extends indefinitely throughout space and describes the electromagnetic interaction. The field propagates by electromagnetic radiation; in order of increasing energy (decreasing wavelength) electromagnetic radiation comprises: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. The field (EM) can be viewed as the combination of an electric field E and a magnetic field H, that is, these are three-dimensional vector fields that have a value defined at every point of space and time: E D E.r; t/ and H D H.r; t/, where r represents a point in 3-d space r D .x; y; z/. The electric field is produced by stationary charges, and the magnetic field by moving charges (currents); these two are often described as the sources of the field. The way in which E and H interact is described by Maxwell’s equations (see [14, p. 22]): r ED

 ; 0

r H D 0; r ED

Gauss’s law

(1)

Gauss’s law for magnetism @H ; @t

Faraday’s law

r  H D 0 J C 0 0

@E ; @t

Amp`ere-Maxwell’s law:

Here r D .@x ; @y ; @z /  D .r; t/ 0 J D J.r; t/ 0

the gradient charge density permittivity of free space current density vector permeability of free space

p We have c D 1= 0 0 , the speed of light in vacuum.

(2) (3) (4)

Refraction Problems in Geometric Optics

131

5.1 General Case In several situations is necessary to consider a medium where the magnetic permeability  D .x; y; z/3 and the electric permittivity  D .x; y; z/4 are not constants. This is the case when the physical properties of the medium change from point to point, in particular, this happens in geometric optics when materials of different refractive indices are considered. In such case the Maxwell equations have the form:  @H ; c @t  @E 2 E C r H D c c @t r ED

r .E/ D 4 r .H/ D 0;

(5) (6) (7) (8)

c being the speed of light in vacuum. Recall that substances for which  ¤ 0 are conductors and if  is negligibly small, the substances are called insulators or dielectrics, see [1, Sect. 1.1.2]. Under certain assumptions on the field and the physical set up we have that J D E, see [1, Sect. 1.1.2, formula (9)]. It is important to notice that these equations are written in Gaussian units, and the Maxwell equations in the first section written in SI units.

5.2 Maxwell Equations in Integral Form Points in R4 are denoted by .x; y; z; t/, and suppose D  R4 is a domain for which the divergence theorem holds, for example, the boundary is piecewise smooth, that is, a finite union of C 1 surfaces. For a point P D .x; y; z; t/ on the boundary @D, the unit outer normal at P is denoted by  D .x ; y ; z ; t /. From equation (6) rH

2  @E D E: c @t c

(9)

Recall we assume  D .x; y; z/, and we want to derive an integral form of the last equation that does not require differentiability of the fields. In order to do that, we initially assume the fields are smooth and applying the divergence theorem we will obtain formulas independent of the derivatives of the fields. Set H D .H1 ; H2 ; H3 /. We have 3

For values of  for different substances see http://en.wikipedia.org/wiki/Permeability (electromagnetism)#Values for some common materials. 4 For relative permittivity of some substances see http://en.wikipedia.org/wiki/Relative permittivity.

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C.E. Guti´errez

Z r  H dxdyd zdt D

Z

Z

Di

Z

.@y H3  @z H2 /  j

.@x H3  @z H1 / C k

D

D

Z Di

Z

div .0; H3 ; H2 ; 0/  j D

Z

div .H3 ; 0; H1 ; 0/ C k D

Z Di

.@x H2  @y H1 / D

div .H2 ; H1 ; 0; 0/ D

.0; H3 ; H2 ; 0/ .x ; y ; z ; t / d @D

Z

j Z D

Z .H3 ; 0; H1 ; 0/ .x ; y ; z ; t / d C k

@D

.H2 ; H1 ; 0; 0/ .x ; y ; z ; t / d @D

.x ; y ; z /  .H1 ; H2 ; H3 / d: @D

So integrating (9) over D yields Z Z  Et dxdyd zdt .x ; y ; z /  .H1 ; H2 ; H3 / d  c @D D Z Z    E dxdyd zdt D .x ; y ; z /  .H1 ; H2 ; H3 / d  t @D D c Z Z  E t d .x ; y ; z /  .H1 ; H2 ; H3 / d  D c @D @D Z Z   2  .x ; y ; z /  .H1 ; H2 ; H3 /  E t d D E dxdyd zdt: D c @D D c Therefore the surface integral Z  Z   2 .x ; y ; z /  .H1 ; H2 ; H3 /  E t d D E dxdyd zdt; c @D D c

(10)

for each closed hyper-surface @D in R4 . Proceeding in the same way with (5) we obtain that the surface integral Z    .x ; y ; z /  .E1 ; E2 ; E3 / C H t d D 0; (11) c @D for each closed hyper-surface @D in R4 . Concerning (7) and (8), proceeding in the same way as before we obtain that Z Z E  d D 4  dxdyd zdt (12) Z

@D

H  d D 0; @D

D

(13)

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133

for each domain D  R4 for which the divergence theorem holds. These formulas make sense as long as the fields E; H and the coefficients  and  are piecewise continuous over @D and bounded. Equations (10)–(13) are Maxwell’s equations in integral form.

5.3 Boundary Conditions at a Surface of Discontinuity Let us consider a point P0 D .x0 ; y0 ; z0 ; t0 /, a hyper-surface 0 passing through P0 and suppose that the fields H and E, solutions to the Maxwell equations in integral form, as well as the functions  and , are discontinuous on 0 . Suppose that all these quantities are defined locally around P0 say in the 4-dimensional ball BR .P0 /. This situation is typical when we have two media with different indices of refraction and the surface 0 is the one separating the two media. The surface 0 divides the open ball BR .P0 / into two open pieces: BRC and BR . In order to make sense of the integrals we assume the surface 0 is C 1 , the fields E and H, and  and  are bounded in BR .P0 /, and all continuous on BR .P0 / n 0 . We assume also that for each Q 2 0 \ BR .P0 / the following limits exist and are finite: lim

E.P / D EC .Q/;

lim

E.P / D E .Q/;

C P !Q;P 2BR

 P !Q;P 2BR

lim

H.P / D HC .Q/

lim

H.P / D H .Q/;

C P !Q;P 2BR

 P !Q;P 2BR

and similar quantities for  and . Let us call C .R/ the spherical part of the boundary of BRC , and  .R/ the spherical part of the boundary of BR . If we let EC .Q/ D E.Q/ for Q 2 BRC and EC .Q/ D limP !Q;P 2B C E.P / for Q 2 0 \BR , R

and similarly HC , C , and  C , then all these functions are continuous in the closure of BRC . In a similar way, we define E ; H ;  and  in BR that are continuous in the closure of BR . Hence by a limiting argument we can apply (10) with D D BR˙ obtaining Z C .R/[.0 \BR .P0 //

C C .x ; y ; z /  .HC 1 ; H2 ; H3 / 

C C E t c

!

Z d D

C

BR

2 EC dxdyd zdt; c

(14) and Z  .R/[.0 \BR .P0 //



  .x ; y ; z /  .H 1 ; H2 ; H3 / 

  E t c



Z d D

 BR

2 E dxdyd zdt: c

(15)

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C.E. Guti´errez

Now   C C C C C E t d .x ; y ; z /  .H1 ; H2 ; H3 /  c C .R/[.0 \BR .P0 // Z    D .x ; y ; z /  .H1 ; H2 ; H3 /  E t d c C .R/   Z C C C C C E .x ; y ; z /  .HC d; ; H ; H /   t 1 2 3 c 0 \BR .P0 /

Z

(16)

where in the integral over 0 \ BR .P0 /,  WD .x ; y ; z ; t / is the downward unit normal to 0 ; and   Z      E t d .x ; y ; z /  .H1 ; H2 ; H3 /  (17) c  .R/[.0 \BR .P0 // Z    D .x ; y ; z /  .H1 ; H2 ; H3 /  E t d c  .R/   Z      C E t d; .x ; y ; z /  .H1 ; H2 ; H3 /  c 0 \BR .P0 / where in the integral over 0 \ BR .P0 /,  is the upward unit normal to 0 . Adding (16) and (17), and using (14) and (15) yields Z C

BR

2 EC dxdyd zdt C c

Z



D C .R/

BR

2 E dxdyd zdt c

.x ; y ; z /  .H1 ; H2 ; H3 / 

  E t d C c

Z



.x ; y ; z /  .H1 ; H2 ; H3 / 

 .R/

  E t d c

  1 C C .x ; y ; z /  .HC  H /   E    E t d: c 0 \BR .P0 /

Z

C

Z

On the other hand, applying (10) with D D BR yields Z

Z   2  .x ; y ; z /  .H1 ; H2 ; H3 /  E t d D E dxdyd zdt: c C .R/[ .R/ BR c

Since the field E is discontinuous only on 0 , which is a set of measure zero, we therefore obtain   Z 1 C C  E    E t d D 0 .x ; y ; z /  .HC  H /  c 0 \BR .P0 / for all R sufficiently small. Now letting R ! 0 we obtain the following equation valid at P0

Refraction Problems in Geometric Optics

1 .x ; y ; z /  .HC  H /  . C EC    E / t D 0; c

135

(18)

where  is the normal to the interface 0 at the point P0 . Suppose the interface is independent of time and is given by a function .x; y; z/ D 0, then the normal at a point is  D .x ; y ; z ; 0/, therefore (18) becomes r  .HC  H / D 0: ˙ ˙ We can write H˙ D H˙ tan C Hperp , where Hperp is the component in the direction of the normal r, and H˙ tan is the component perpendicular to the normal. We have ˙ ˙ r  H˙ D r  H˙ tan C r  Hperp D r  Htan . So  C  0 D r  .HC  H / D r  .HC tan  Htan / D jrj jHtan  Htan j;

since the vectors are perpendicular. So if r ¤ 0, then we obtain the important relation that  HC tan  Htan D 0;

that is, the tangential components of the magnetic field are continuous across the boundary. Proceeding in the same manner this time with (11) yields the equation 1 .x ; y ; z /  .EC  E / C .C HC   H / t D 0; c

(19)

where  is the normal to the interface 0 at the point P0 . If the interface is independent of t, proceeding exactly as before, we obtain r  .EC  E / D 0; and  EC tan  Etan D 0;

that is, also the tangential components of the electric field are continuous across the boundary. In regard to (12) and (13), we obtain similarly that . C EC    E /  D 0; and .C HC   H /  D 0: Since H˙  D H˙ perp , and similarly for E, assuming  D .x; y; z/ yields

136

C.E. Guti´errez   C C   0 D . C EC perp   Eperp / r D j Eperp   Eperp j jrj;

and   C C   0 D .C HC perp   Hperp / r D j Hperp   Hperp j jrj;

and therefore   C C   j C EC perp   Eperp j D j Hperp   Hperp j D 0:

Therefore the perpendicular components of the fields E and H are continuous across the interface.

5.4 Maxwell’s Equations in the Absence of Charges This is the case when  D 0 and J D 0. So the equations become r E D 0;

(20)

r H D 0;

(21)

@H ; @t @E r  H D 0 0 ; @t rED

(22) (23)

5.5 The Wave Equation Recall the following formula from vector analysis for a vector A D A.x; y; z/ D .Ax .x; y; z/; Ay .x; y; z/; Az .x; y; z//: r  .r  A/ D r.r A/  .r r/A:

(24)

Denote r r D r 2 , the Laplacian, and so r2A D  2   2   2  @2 Ay @2 Ay @ Ay @2 Ax @2 Ax @2 Az @2 Az @ Ax @ Az C C C C C C i C j C k: @x 2 @y 2 @z2 @x 2 @y 2 @z2 @x 2 @y 2 @z2

From Faraday’s law and Amp`ere’s law

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137

r  .r  E/ D 

@2 E @.r  H/ D 0 0 2 @t @t

and so from formula (24) we obtain that E satisfies the wave equation 0 0

@2 E D r 2 E: @t 2

Proceeding in the same manner for H we obtain 0 0

@2 H D r 2 H: @t 2

That is, both the electric and magnetic fields satisfy the wave equation. We have from physical considerations that 1 ; vDcD p 0 0 c being the speed of propagation of light in free space, which in this case is the velocity v of propagation. If free space is changed by a material with other values of 0 and 0 , the velocity v represent the speed of propagation of waves in this material.5

5.6 Dispersion Equation Suppose E and H solve the Maxwell equations (5)–(8) with  D 0,  D 0, and  and  constants. Then    r  .r  E/ D  r  Ht D  .r  H/t D  2 Ett : c c c On the other hand, from (24), r  .r  E/ D r 2 E and so r 2E 

 Ett D 0; c2

r 2H 

 Htt D 0: c2

and similarly

5

The relative permittivity is =0 and the relative permeability is =0 ; the index of refraction is 1 1 p defined by n D r r . The velocity of propagation v D p . Since c D p , we get that  0  0 n D c=v.

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C.E. Guti´errez

If E D A cos .r k C !t/, then we obtain the dispersion equation  ! 2 jkj2 D  : c

(25)

5.7 Plane Waves Let s be a unit vector. Any solution to the wave equation 1 2 @ V D r 2 V; v2 t of the form V .r; t/ D F .r s; t/ is called a plane wave, since at each time t, V is constant on each plane of the form r s=constant. That is, for each t the vector V .r; t/ is the same on each plane r s=constant. The plane wave propagates in the direction s. It can be proved that any solution to the wave equation of this form can be written as V .r; t/ D V1 .r s  vt/ C V2 .r s C vt/ where V1 ; V2 are arbitrary functions, see [1, Sect. 1.3.1]. Since the fields E and H both satisfy the wave equation, it is then natural to consider the case when E D E.r s  vt/;

H D H.r s  vt/;

that is, E and H are functions of the scalar variable r s  vt. We have @E D vE0 ; and r  E D s  E0 I @t and similarly for H under the assumption that J D 0. Thus, from the Faraday and 1 Amp`ere laws, and since v 2 D , we obtain the equations 0 0 s  E0 D vH0 1 s  H0 D  E0 : v Since s is a constant vector s  E0 D .s  E/0 , and so the equations are .s  E/0 D vH0 1 .s  H/0 D  E0 : v

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139

Integrating these equations and taking constants of integration zero (which amounts to neglect constant fields), we obtain the very important equations relating the electric and magnetic fields E D v.s  H/ HD

1 .s  E/: v

(26) (27)

This shows that s E D s H D 0, that means, the electric and magnetic field are always perpendicular to the direction of propagation s. In addition, E H D v.s  H/ H D 0, that is, E and H are always perpendicular. We also obtain taking absolute values that jEj D vjHj:

5.8 Fresnel Formulas We consider plane waves whose components have the form    r s C ı D a cos .!t  k r C ı/ ; a cos ! t  v ! that is, k D s and a; ı are real numbers. The quantity !t  k r C ı is called the v phase, and a is called the amplitude. Let si be the direction (unit) of an incident plane wave traveling for a while in medium I with velocity of propagation v1 that hits, at a point P , a boundary  between I and another medium II where the velocity of propagation is v2 (I and II are also called dielectrics as they are materials with zero conductivity, that is  D 0 and so the current density vector J D 0, see Sect. 5.1). Then the wave splits into two waves: a transmitted wave propagating in medium II and a reflected wave propagated back into medium I. We shall assume that these two waves are also plane. The plane determined by  and si is called the incidence plane. It is important to remark that for our analysis, we will choose a local system of coordinates around the point P . Indeed, we are going to write all fields as functions of r (position) and t, with r close to zero, such that the coordinates of P in this system are r D 0. In particular, the fields will be calculated near the point P . We choose a rectangular right-hand system of coordinates x; y; z such that the normal  is on the z-axis, and the x and y axes are on the plane perpendicular to  and in such a way that the vector si lies on the xz-plane. This means that the tangent plane to  at P is the xy-plane; in particular, P D .0; 0; 0/. So we assume that si D sin i i C cos i k

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C.E. Guti´errez

that is, si lives on the xz-plane and so the direction of propagation is perpendicular to the y-axis and i is the angle between the normal vector  to the boundary at P (the z-axis) and the incident direction si (as usual i; j; k are the unit coordinate vectors). The electric field corresponding to this incident field is  r si E .r; t/ D Ik cos i ; I? ; Ik sin i cos ! t  v1

!!

i

D

Ei0

r si cos ! t  v1

!! :

(28) Notice that E has this form because, as is was proved in Sect. 5.7, E is always perpendicular to the direction of propagation si . Notice also that the field Ei has a component that is perpendicular to the plane of incidence and a component that is parallel to this plane, indeed, we write Ei? and

   r si j; D I? cos ! t  v1

    r si Eik D Ik cos i i C Ik sin i k cos ! t  : v1

Also notice that      r si jEi j2 D Ik2 C I?2 cos2 ! t  v1 From (27), the magnetic field is then Hi .r; t / D

      1  r si r si D Hi0 cos ! t  : I? cos i ; Ik ; I? sin i cos ! t  v1 v1 v1

Let us now introduce st , the direction of propagation of the transmitted wave, and t the angle between the normal  and st , and similarly, sr is the direction of propagation of the reflected wave and r is the angle between the normal  and sr . We have from the Snell law that sr D sin r i C cos r k D sin i i  cos i k. Then the electric and magnetic fields corresponding to transmission are        r st r st D Et0 cos ! t  Et .r; t / D Tk cos t ; T? ; Tk sin t cos ! t  (29) v2 v2       1  r st r st D Ht0 cos ! t  I Ht .r; t / D T? cos t ; Tk ; T? sin t cos ! t  v2 v2 v2

and similarly the fields corresponding to reflection are

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141

       r sr r sr D Er0 cos ! t  Er .r; t / D Rk cos r ; R? ; Rk sin r cos ! t  (30) v1 v1       r sr 1  r sr R? cos r ; Rk ; R? sin r cos ! t  Hr .r; t / D D Hr0 cos ! t  : v1 v1 v1

Recall that from Snell’s law all vectors si ; st ; sr and  all live on the same plane, that is, the xz-plane. Each of the electric and magnetic fields can be decomposed uniquely as a sum of a component in the direction of the normal (normal component) or on the z-axis, plus another component perpendicular to the normal (tangential component) or on the xy-plane. From the integral form of Maxwell’s equations, as it shown in Sect. 5.3, the tangential components of E (and also of H if J D 0) at the interface are continuous (see also [1, Sect. 1.1.3, formula (23)]). Since the electric field on medium I near  equals Ei C Er , we get Eitan C Ertan D Ettan on , since Ettan is the transmitted electric field in medium II near . From the configuration we have, we can write Ei D Einormal k C Eitan , and so k  Ei D k  Eitan . Similarly, k  Er D k  Ertan and k  Et D k  Ettan . So k  Ei C k  Er D k  Et . Then          r si r sr r st C k  Er0 cos ! t  D k  Et0 cos ! t  ; k  Ei0 cos ! t  v1 v1 v2

for all r close to zero and all t. The interface point P is r D .0; 0; 0/, so in particular, we obtain k  Ei0 cos .!t/ C k  Er0 cos .!t/ D k  Et0 cos .!t/ for all t. Eliminating the cosines we get k  Ei0 C k  Er0 D k  Et0 :

(31)

Since we are assuming the current density vector J D 0, as it was mentioned earlier, the tangential component of the magnetic field is also continuous across the interface. So as before with the electric field, we have Hitan C Hrtan D Httan on , and so k  Hi0 C k  Hr0 D k  Ht0 :

(32)

From (31) we obtain the equations I? C R? D T? ;

cos i .Ik  Rk / D cos t Tk I

and from (32) we obtain Rk Tk Ik C D ; v1 v1 v2

 cos i

I? R?  v1 v1

 D cos t

T? : v2

We have n1 D c=v1 and n2 D c=v2 so solving the last two sets of equations yields

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C.E. Guti´errez

Tk D

2n1 cos i Ik n2 cos i C n1 cos t

T? D

2n1 cos i I? n1 cos i C n2 cos t

Rk D

n2 cos i  n1 cos t Ik n2 cos i C n1 cos t

R? D

n1 cos i  n2 cos t I? : n1 cos i C n2 cos t

These are the Fresnel equations expressing the amplitudes of the reflected and transmitted waves in terms of the amplitude of the incident wave.

5.9 Rewriting the Fresnel Equations We will replace si by x and st by m, and we also set  D n2 =n1 . Recall  is the normal to the interface. We have cos i D x  and cos t D m . In addition, from the Snell law x  m D , so the Fresnel equations have the form Tk D

2x  2x  2 x .x   m/ Ik D Ik D Ik x Cm  . x C m/  . x C m/ .x   m/

T? D

2x  2x  2 x .x   m/ I? D I? D I? x Cm  .x C  m/  .x C  m/ .x   m/

Rk D

x m  . x  m/  . x  m/ .x   m/ Ik D Ik D Ik x Cm  . x C m/  . x C m/ .x   m/

R? D

x m  .x   m/  .x   m/ .x   m/ I? D I? D I? : x Cm  .x C  m/  .x C  m/ .x   m/

Notice that the denominators of the perpendicular components are the same and likewise for the parallel components.

5.10 The Poynting Vector It is defined by SD

c E  H; 4

where c is the speed of light in free space. The vector S represents the flux of energy through a surface. Suppose dA is the area of a surface element at a point P and let

Refraction Problems in Geometric Optics

143

 be the normal at P . Then the flux of energy through dA at the point P is given by dF D S  dA: From (27) we get that SD

n c E  .s  E/ D jEj2 s: 4v 4

Using the form of the incident wave from the previous section, the amount of energy J i flowing through a unit area of the boundary per second at P , of the incident wave Ei given in (28), is then J i D jSi j cos i D

n1 i 2 jE j cos i :6 4 0

Similarly, the amount of energy in the reflected and transmitted waves (also given in the previous section) leaving a unit area of the boundary per second at P are given by n1 r 2 jE j cos i 4 0 n2 t 2 jE j cos t : J t D jSt j cos t D 4 0

J r D jSr j cos i D

The reflection and transmission coefficients are defined by 2   r 2 jE0 j Jt n2 cos t jEt0 j Jr ; and T D D : RD i D J Ji n1 cos i jEi0 j jEi0 j

(33)

By conservation of energy or by direct verification R C T D 1.

5.11 Polarization Polarization is a property of the field that describes the orientation of their oscillations. Since the electric vector is assumed a plane wave and as we showed it is perpendicular to the direction of propagation s, then for each r in the plane

Notice that from (28), the value of the field Ei at P is Ei .0; t / D Ei0 cos .! t /. Hence we n1 i 2 actually get J i D jE j cos i cos2 .! t /. Similarly, from (29), the value of the field Et at P is 4 0 t t E .0; t / D E0 cos .! t /, and from (30), the value of the field Er at P is Er .0; t / D Er0 cos .! t /. n1 r 2 n2 t 2 Therefore we have J r D jE j cos i cos2 .! t /, and J t D jE j cos t cos2 .! t /. So we 4 0 4 0 get formulas (33) because the factor cos2 .!t / cancels out.

6

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C.E. Guti´errez

r s D c and t fixed, the vector E.r; t/ is constant. We visualize E.r; t/ as a vector with origin at the intersection of the direction s with the plane r s D c. That is, as a vector with origin at the point .r s/s and terminal point .r s/s C E.r; t/. Then t is fixed and r runs over all space, the end point of this vector describes a curve in 3-d. If we now move t, this curve is shifted (and keeps the same shape) by changing the phase because of the presence of !t in the cos function. So when r s D c and t moves the vector E.r; t/ describes a curve in the plane r s D c. If this curve is an ellipse we say that the wave is elliptically polarized and when the ellipse is a circle we say the wave is circularly polarized, and if the ellipse degenerates to a segment we say the wave is linearly polarized. If the wave describing the incident field has components that have different phases, then this changes the sense of circulation and inclination of the ellipse (for elliptically polarized light), see [1, Sect. 1.4.2]. See http://en.wikipedia.org/wiki/Polarization for pictures. Suppose for example that the wave is linearly polarized perpendicularly to the plane of incidence. That is, Ik D 0. Then from Fresnel equations Tk D Rk D 0 and  RD

jEr0 j jEi0 j

2

 D

R? I?

2

 D

jx   mj2 1  2

2 ;

and T D D

m  x 



T? I?

2 D

m .x   m/ x .x   m/



2 x .x   m/ .x C  m/ .x   m/

2

4 .m .x   m// .x .x   m// : .1   2 /2

For the case when no polarization is assumed, that is, radiation has no particular preference for the direction in which it vibrates, we have from Fresnel’s equations that  2 jEr0 j2 D Rk2 CR? D

. x  m/ .x   m/ . x C m/ .x   m/

2

  .x   m/ .x   m/ 2 2 Ik2 C I? ; .x C  m/ .x   m/

and so  RD  D

D

jEr0 j jEi0 j

2 D

2 Rk2 C R?

Ik2 C I?2

. x  m/ .x   m/ . x C m/ .x   m/

1 .1   2 /2



2



Ik2 Ik2 C I?2

2  .1 C  2 / x m

2

C

.x   m/ .x   m/ .x C  m/ .x   m/

Ik2 Ik2 C I?2

2

I?2 Ik2 C I?2 ! 

I?2 2 2 C 12 x mC Ik2 CI?2

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145

which is a function only of x m. In principle the coefficients Ik and I? might depend on the direction x, in other words, for each direction x we would have a wave that changes its amplitude with the direction of propagation. The energy of the incident wave would be f .x/ D jEi0 j2 D Ik .x/2 C I? .x/2 . Notice that if the incidence is   1 2 which shows that even for radiation normal, that is, x D m, then R D 1C normal to the interface we lose energy by reflection. For example, if we go from air to glass, n1 D 1 and n2 D 1:5, we have  D 1:5 so R D :04, which means that 4% of the energy is lost in internal reflection.

5.12 Estimation of the Fresnel Coefficients For later purposes we need to estimate the function 1 .s/ D .1   2 /2



2  .1 C  2 / s

2

! 

2 2 ˛ C 1  2 s C  ˇ ;

(34)

have 0  ˛; ˇ  1 and ˛ C ˇ D 1. Set  g.t/ D so .t/ D

2  .1 C  2 / t

2 ;

2

h.t/ D 1  2 t C  2 ;

1 .g.t/ ˛ C h.t/ ˇ/. .1   2 /2

Case  < 1. Suppose  C  t  1. We have g 0 .t/ D 4 so g 0 .t/ > 0 for t >



 2 1  .1 C  2 / 2 , t t

2 2 , and g 0 .t/ < 0 for t < . Since  < 1, we have 1 C 2 1 C 2

2 < 1 for  > 0 small. Therefore, g decreases in the interval 1 C 2 2 2 , and g increases in the interval Π; 1. Hence ΠC ; 2 1C 1 C 2 C
g.1/ for  small, so max g.t/ D g. C /:

ŒC;1

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C.E. Guti´errez

 1 C 2 On the other hand, h0 .t/ D 4 1  2 t C  2 , and so h0 .t/ > 0 for t > 2 1 C 2 1 C 2 0 . Since > 1, the function h is decreasing in and h .t/ < 0 for t < 2 2 the interval ΠC ; 1 and so max h.t/ D h. C /:

ŒC;1

Therefore we obtain that max .t/ 

ŒC;1

1 .˛ g. C / C ˇ h. C // : .1   2 /2

It is easy to see that g. C / < .1   2 /2 ;

and h. C / < .1   2 /2

and so we obtain the bound max .t/  C < 1;

ŒC;1

with C D max

g. C / h. C / ; .1   2 /2 .1   2 /2

(35)

independent of ˛ and ˇ. We notice also that C ! 1 as  ! 0C , and C !   1 2 as  ! .1  / . Also notice that the function  in (34) is in general 1C not decreasing in the interval Œ C ; 1, that is, one can choose ˛ close to one and ˇ close to zero with ˛ C ˇ D 1, so that this is the case. 1 2 Case  > 1. For  small we have C  < < 1, so as before, g decreases  1 C 2 2 2 1 , and g increases in the interval Œ ; 1. Hence in the interval Œ C ; 2  1C 1 C 2  

1 max g.t/ D max g C  ; g.1/ : Œ.1=/C;1   1 C  , for  small, and so Since now  > 1 we have that g.1/ < g  

max

Œ.1=/C;1

g.t/ D g..1=/ C /:

Refraction Problems in Geometric Optics

Since we always have Œ.1=/ C ; 1 and so

147

1 C 2 > 1, the function h is decreasing in the interval 2 max

Œ.1=/C;1

h.t/ D h..1=/ C /:

Therefore we obtain that max

Œ.1=/C;1

.t/ 

1 .˛ g..1=/ C / C ˇ h..1=/ C // : .1   2 /2

It is clear that g..1=/ C / < .1   2 /2 , and h..1=/ C / < .1   2 /2 when 0 <  < 1  .1=/. So we obtain the bound max

Œ.1=/C;1

.t/  C < 1;

with

g..1=/ C / h..1=/ C / C D max ; .1   2 /2 .1   2 /2

(36)

independent of ˛ and ˇ.

5.13 Application to the Far Field Refractor Problem with Loss of Energy We have then seen that when radiation strikes a surface separating two homogeneous media I and II with different refractive indices, part of the radiation is transmitted through medium II and another part is reflected back into medium I. In fact, from Sect. 5.11 the percentage of internally reflected energy can be conveniently written for our purposes as 1 r.x/ D .1   2 /2



2  .1 C  2 / x m

2

Ik2 2 Ik2 C I?

i2 h C 1  2 x m C  2

2 I?

!

2 Ik2 C I?

where  D n2 =n1 . Therefore the percentage of energy transmitted is t.x/ D 1  r.x/. Here I? and Ik are the coefficients of the amplitude of the incident wave, which might depend on x in a continuous way. It is important to notice that from Snell’s law, x  m D , where  is unit normal to the surface at the striking point and > 0. This implies that the function r.x/ is a function only depending on x and the normal .

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We propose the following new model that takes into account the splitting of energy. Suppose we have f 2 L1 ./ and g 2 L1 . /, both ;  are domains in the sphere in R3 , the space with physical significance for our problem. The question is to find a surface R parameterized by f.x/x W x 2 g that separates media I and II such that each ray emanating from a point source, the origin, in the direction x 2  with intensity f .x/ is refracted into a direction m 2  and received with intensity g.m/. From the Fresnel formulas a surface R is only able to transmit in the direction x an amount of energy equal to f .x/tR .x/ where tR D 1  rR , since the amount f .x/rR .x/ is reflected back. As we said, the function tR .x/ depends of course on the surface R but only through x and the unit normal vector  D .x/ at the striking point. Since we will be seeking for refracting surfaces R, which, in particular, are convex or concave, the normal vector .x/ exists for almost every direction x. Also tR .x/ D G.x; .x//, with a function G.x; x 0 / continuous in    and so tR is defined for a.e. direction x. We then propose the following model: the refracting surface R is a solution to our problem if Z Z f .x/ tR .x/ dx  g.m/ dm (37) TR .F /

F

for each Borel subset F   . Here TR .F / is the collection of all directions x 2  that are refracted into a direction in the set F . We prove in [6] that if R is a refractor, then the function tR .x/ is continuous relative to the set  n S , where S is the set of directions where Rho is not differentiable, i.e., jS j D 0. Therefore tR .x/ is measurable and so (37) is well defined. Since a fraction of the energy is used in internal reflection, to be able to transmit and receive g.m/ a little extra energy will be needed at the outset. A refractor R will be admissible to transmit the amount g if Z Z f .x/ tR .x/ dx  g.m/ d m: (38) 



Since a priori we only know f , g and not R, we do not know if this is satisfied. In order to make sure this is the case, we proved in Sect. 5.12, that if for example n2 =n1 D  < 1, then r.x/  C < 1 for all x 2  such that x m   C ,7 where  > 0 and with C independent of R. So if we assume that the input energy is sufficiently larger than the output energy, then (38) holds. More precisely, if Z f .x/ dx  

1 1  C

Z 

g.m/ d m;

then (38) holds.

7

We recall that the physical constraint for refraction is that x m   for  < 1, see Lemma 2.1.

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Fig. 5 Refracted and reflected vectors

Fig. 6 Refracted vectors for an ellipse refracting into a fixed direction

Figure 5 represents an arc of ellipse separating glass and air,  D 2=3, where the refracted and reflected directions are multiplied by the Fresnel coefficients t.x/ and r.x/ respectively. Figure 6 represents all the refracted vectors in an ellipse having the uniform refraction property, i.e. all rays are refracted into a fix direction, where the refracted vectors are multiplied by the Fresnel coefficient t.x/. Notice that the size of the refracted vectors close to the critical angle, i.e., x m D  tend to zero. With this model it is proved in [6] existence of solutions, even for Radon measures  instead of g. The basic geometry of the refractors is described in [8] and depends on . Indeed, the surfaces having the uniform refracting property are semi ellipsoids if  < 1 and one sheet of hyperboloids of two sheets if  > 1, see Sect. 2.3. A difficulty in our case is the presence of the coefficient tR .x/ in (37). This prevents us from using the optimal transportation methods used in [8]. The route used was explained in Sect. 4, that is, to solve first the problem when the right hand side is a linear combination of delta functions and then proceed by approximation. To carry out this we need to understand how the Fresnel coefficients evolve when a sequence of refractors converge. When the measure  in the target is discrete, refractors always overshoot energy in one direction. When  is any Radon measure, it is proved in [6] that refractors transmit more energy in any a priori chosen direction mo 2  which lies in the support of , and there is one refractor overshooting the least amount of energy in the Rdirection of mo . That is, for each Borel set F   such that mo … F we have TR .F / f .x/ tR .x/ dx D R F g.m/ d m. To place our results in perspective, we finish enumerating some related results in this area. The refractor problem assuming energy conservation, i.e., tR .x/ D 1 in (37), was considered for the first time in [8] for the far field case, and in [7] and

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[9] for the near field case. For reflectors also assuming energy conservation, see [2–4, 15] for the far field problem, and [11], and [10] for the near field. Acknowledgements This material is based upon work partially supported by the National Science Foundation under Grants DMS-0901430 and DMS-1201401.

References 1. M. Born, E. Wolf, Principles of Optics, Electromagnetic Theory, Propagation, Interference and Diffraction of Light, 7th (expanded), 2006 edn. (Cambridge University Press, London, 1959) 2. L.A. Caffarelli, Q. Huang, Reflector problem in Rn endowed with non-Euclidean norm. Arch. Ration. Mech. Anal. 193(2), 445–473 (2009) 3. L.A. Caffarelli, V. Oliker, Weak solutions of one inverse problem in geometric optics. J. Math. Sci. 154(1), 37–46 (2008) 4. L.A. Caffarelli, C.E. Guti´errez, Q. Huang, On the regularity of reflector antennas. Ann. Math. 167, 299–323 (2008) 5. C.E. Guti´errez, The Monge–Amp`ere Equation (Birkh¨auser, Boston, 2001) 6. C.E. Guti´errez, H. Mawi, The far field refractor with loss of energy. Nonlinear Anal. Theor. Meth. Appl. 82, 12–46 (2013) 7. C.E. Guti´errez, Q. Huang, The near field refractor, in Geometric Methods in PDE’s, Conference for the 65th birthday of E. Lanconelli. Lecture Notes of Seminario Interdisciplinare di Matematica, vol. 7 (Universita‘ degli Studi della Basilicata, Potenza, 2008), pp. 175–188 8. C.E. Guti´errez, Q. Huang, The refractor problem in reshaping light beams. Arch. Ration. Mech. Anal. 193(2), 423–443 (2009) 9. C.E. Guti´errez, Q. Huang, The near field refractor. Annales de l’Institut Henri Poincar´e (C) Analyse Non Lin´eaire (to appear). https://math.temple.edu/gutierre/papers/nearfield.final. version.pdf (2013) 10. A. Karakhanyan, X.-J. Wang, On the reflector shape design. J. Differ. Geom. 84, 561–610 (2010) 11. S. Kochengin, V. Oliker, Determination of reflector surfaces from near-field scattering data. Inverse Probl. 13, 363–373 (1997) 12. R.K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964) 13. R. Schneider, Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 44 (Cambridge University Press, Cambridge, 1993) 14. A. Sommerfeld, Electrodynamics. Lectures on Theoretical Physics, vol. III (Academic, New York, 1952) 15. X.-J. Wang, On the design of a reflector antenna. Inverse Probl. 12, 351–375 (1996)

On the Levi Monge-Amp`ere Equation Annamaria Montanari

Abstract We are concerned with some notions of curvatures associated with pseudoconvexity and the Levi form as the classical Gauss and Mean curvatures are related to the convexity and to the Hessian matrix. In particular, given a prescribed non negative function K; the Levi Monge Amp`ere equation for the graph of a function u W R2nC1 ! R is det L D K.x; u/.1 C jDuj2 /

nC1 2

;

where L is the Levi form of the graph u and Du is the Euclidean gradient of u: More generally, we shall consider elementary symmetric functions of the eigenvalues of the Levi form L and we shall first show that these curvature equations contain information about the geometric feature of a closed hypersurface. Then, we shall show that the curvature operators lead to a new class of second order fully nonlinear equations whose characteristic form, when computed on generalized pseudoconvex functions, is nonnegative definite with kernel of dimension one. Thus, the equations are not elliptic at any point. However, they have the following redeeming feature: the missing ellipticity direction can be recovered by suitable commutation relations. We shall use this property to study existence, uniqueness and regularity of viscosity solutions of the Dirichlet problem for graphs with prescribed Levi curvature.

A. Montanari () Dipartimento di Matematica, Universit`a di Bologna, piazza di Porta S.Donato 5, 40127 Bologna, Italy e-mail: [email protected] L. Capogna et al., Fully Nonlinear PDEs in Real and Complex Geometry and Optics, Lecture Notes in Mathematics 2087, DOI 10.1007/978-3-319-00942-1 4, © Springer International Publishing Switzerland 2014

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1 Introduction In this note we present an introduction to the Levi curvature equations and we survey some existence, symmetry and regularity results for their solutions. Geometric theory of several complex variables leads to differential problems related to nonlinear second order PDE’s of degenerate elliptic type. In particular, in looking for domains of holomorphy, fully nonlinear PDE’s in nondivergence form appear, which are usually quoted as Levi curvature equations or Levi-Monge-Amp`ere equations. They are related to the pseudoconvexity as the classical Gauss curvature equations are related to the Euclidean convexity. These equations are of degenerate type but they have the following subelliptic property: when computed on strictly pseudoconvex functions, they become elliptic along certain directions, and the missing ones can be recovered by commutations. In the present paper we want to describe the state of the art of this subject. First of all, we will present the geometric arguments leading to the formal notion of Levi curvature, starting from the very beginning. Precisely, in Sect. 2 we introduce the Levi Form, which is the analogous to the Second Fundamental Form in differential geometry. The Levi form is the key differential geometric object which determines pseudoconvexity and many functions properties of a CR manifold. For example the holomorphic extendibility of CR functions. We also suggest the interested reader the monograph [15], that presents several differential geometric aspects in the theory of CR manifolds and tangential CauchyRiemann equations. Pseudoconvexity is a central concept in complex analysis because it relates to the very core of holomorphic (i.e. complex analytic) functions, which is linked to power series, the identity theorem, and analytic continuation. This concept is a higherdimensional phenomenon, because every open subset of the complex plane C is pseudoconvex. Let us recall that this is true also for Euclidean convexity: every open connected subset of R is convex. For the basic notion of several complex variables theory we refer to the monograph by S. Krantz [24] and to [35]. Let us now spend some words to the history of this concept. By following [36], pseudoconvexity has its roots in Friedrich Hartogs’s surprising discovery in 1906 of a simple domain H in C 2 with the property that every function that is holomorphic on H has a holomorphic extension to a strictly larger open set (see Example 4). In dimension one there is no such phenomenon because every domain in the complex plane is trivially a domain of holomorphy (see Remark 3) and it is elementary to show that every Euclidean convex domain D  Cn is a domain of holomorphy. Hartogs’s discovery raises the fundamental problem of characterizing those domains D  Cn for which holomorphic extension of all holomorphic functions on D does not hold. Such domains are called domains of holomorphy (see Definition 17). Just a few years after Hartogs’s discovery, E. E. Levi studied domains of holomorphy with differentiable boundaries and he found the following simple differential condition, which is very similar to the familiar differential characterization of Euclidean convexity. We assume that D \ U D fz 2 U W .z/ < 0g, where

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 is a C 2 real-valued function with d ¤ 0 on a neighborhood U of a point of the boundary. In 1910 Levi proved that, (a) If there exists a holomorphic function on D \ U which does not extend holomorphically to p (in particular, if D is a domain of holomorphy), then the Levi form at p is positive semidefinite. (b) If the Levi form is positive definite, then the neighborhood U can be chosen so that U \ D is a domain of holomorphy. Levi’s results showed that the restriction of the complex Hessian to the complex tangent plane (nowadays called the Levi form) plays a fundamental role in the characterization of domains of holomorphy. To distinguish Levi’s differential conditions from other formulations of pseudoconvexity, one refers to the condition in (a) as (Levi) pseudoconvexity. If the stronger version in (b) holds, one says that D is strictly or strongly pseudoconvex at p. By Levi’s result, if D is strictly pseudoconvex at every boundary point, then D is locally a domain of holomorphy. For many years the wished global version remained a central open problem, known as the Levi problem. Solutions were finally obtained in the early 1950s by K. Oka, H. Bremermann, and F. Norguet. The general version of the solution of Levi’s problem is then stated as: A domain in C n is a domain of holomorphy if and only if it is pseudoconvex. The extension to arbitrary domains requires an appropriate definition of pseudoconvexity. Many equivalent versions have been introduced over the years and the most elegant one is the formulation in Definition 16, which involves the notion of plurisubharmonic function, because general plurisubharmonic functions can be well approximated from above by C 2 or even C 1 plurisubharmonic functions. By following [5] in Sect. 2 we will first give the definition of the Levi form for an abstract CR structure, and then we will give more concrete representations of the Levi form for an immersed CR manifold. The Levi form for a real hypersurface will be discussed in some detail. Moreover, we will present the relationship between the Levi form and the second fundamental form of a real hypersurface. Then, by using the Levi form, we introduce the notion of pseudoconvexity, which is a most central concept in modern complex analysis (see for instance the beautiful expository article by Range in [36]). By following the expository article [25], in Sect. 3 we will show that the Levi problem and the construction of the envelopes of holomorphy lead to the notion of Levi curvature and to the Dirichlet problem for the prescribed Levi curvature equation. We will then introduce the prescribed Levi curvature equation for a real hypersurface of C2 and then we will discuss the solvability of Dirichlet problem for graphs in the class of viscosity solutions. The main result of this section is that different hypotheses on the zeros of the prescribed Levi curvature imply different regularity behavior of the solution. In particular, if the prescribed Levi curvature is smooth and strictly positive in [8] we proved the smooth solvability of the Dirichlet problem, while in the Levi flat case in [10] we proved a Frobenius type result.

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In Sect. 4 we shall explicitly compute the Levi curvatures for a real hypersurface in CnC1 : We then shall study the Dirichlet problem for the prescribed Gauss-Levi curvature equation in the class of Lipschitz continuous viscosity solutions, under very natural conditions on the prescribed curvature K and on the boundary data. To state our result we need a suitable notion of a generalized Levi pseudoconvex solution to the prescribed Gauss Levi curvature equation, which is a fully non linear and a non elliptic pde. This is provided by the notion of viscosity solution in the sense of Crandall et al. [12]. The modifications needed to adapt the theory of viscosity to equations of Levi Monge Amp`ere have been studied in the paper [13] and we will recall them for reader convenience. We will also use a comparison principle for viscosity solution, Theorem 14, and existence results of continuous and of Lipschitz continuous viscosity solutions of the prescribed Gauss Levi curvature equation, Theorem 15, which have been proved in [13]. The problem of classical regularity of Lipschitz continuous viscosity solutions is widely still open for n > 1: However, we shall show that the fully nonlinear PDE of the prescribed GaussLevi curvature equation has the following hypoellipticity property, which has been proved in [32]: every classical solution is smooth, if the prescribed Gauss-Levi curvature K ¤ 0 at any point and K 2 C 1 : In Sect. 5 we show the existence of nonsmooth viscosity solutions for the Levi Monge Amp`ere equation for n > 1: This generalizes a celebrated result of Pogorelov for the Monge Amp`ere equation (we refer to the book [17] for a deep study of the Monge Amp`ere equation in Rn ). In 1971 Pogorelov [34] showed that convex generalized solutions of the Monge Amp`ere equation det D 2 u D f .x/

(1)

in a domain ˝  RnC1 ; n > 1; need not be of class C 2 ; even if f is positive and smooth. Urbas in [44] proved that this absence of classical regularity is not confined to equations of Monge Amp`ere type, but in fact occurs for the m-th elementary symmetric functions of the eigenvalues of the Hessian and the equation of prescribed m-th curvature, where in each case m  3: Recently, in [18] we proved that a similar result holds for the Levi Monge Amp`ere equation. In Sect. 6 we discuss symmetry properties for compact real hypersurfaces with some constant Levi curvatures. First of all we sketch the proof of the strong comparison principle proved in [31] and which leads to symmetry theorems for domains with constant curvatures. These results suggested the following question: are spheres the unique compact hypersurfaces with constant Levi curvatures? Klingenberg in [23] gave a first positive answer to this problem by showing that a compact and strictly pseudoconvex real hypersurface M  CnC1 is isometric to a sphere, provided that M has constant horizontal mean curvature, the Levi form is diagonal and the characteristic direction of the hypersurface is a geodesic. Later on in [29] we relaxed Klingerberg conditions and by using differential geometry techniques we proved that if the characteristic direction is a geodesic, then Klingerberg’s Theorem holds for compact hypersurfaces with positive constant Levi mean curvature (see Theorem 24).

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The study of surfaces in the Euclidean space with either constant Gauss curvature or constant mean curvature received in the past a great amount of attention. In 1899 Liebmann [26] proved that the spheres are the only compact surfaces in R3 with constant Gauss curvature. In 1952 S¨uss [41] extended the Liebmann result showing that a compact convex hypersurface in the Euclidean space must be a sphere, provided that for some j the j -th elementary symmetric polynomial in the principal curvatures is constant. In 1954 Hsiung [22] proved that the “convexity” assumption can be relaxed to the “star-shapedness” one. The proofs of the above results are based on the Minkowski formulae. A breakthrough for this sort of problems was made by Alexandrov [1] in 1956; who proved that the sphere is the only compact hypersurface embedded into the Euclidean space with constant mean curvature. Alexandrov method is completely different from the Liebmann-S¨uss method, and it is based on the moving plane technique, on the interior maximum principle for elliptic equations and on the boundary maximum principle of Hopf type for uniformly elliptic equations. In 1978 Reilly [37] obtained another proof of the Alexandrov theorem combining the Minkowski formulae with some new elegant arguments. In Sect. 6, by following [28], we will show how to use the null Lagrangian property for elementary symmetric functions of the eigenvalues of the complex Hessian matrix and the classical divergence theorem to prove an integral formula (59) for a closed hypersurface in term of the j -th Levi curvature. Then, we will follow the Reilly approach to prove an isoperimetric inequality Theorem 22 and we will use the Minkowski formula for the classical mean curvature to conclude an Alexandrov type result. The problem of characterizing hypersurfaces with constant Levi curvature has been recently studied by many authors. Let us mention that Hounie and Lanconelli in [20] showed the result for Reinhardt domain of C2 ; i.e. for domains D such that if .z1 ; z2 / 2 D then .e i 1 z1 ; e i 2 z2 / 2 D for all real 1 ; 2 : Their technique has then been used in [21] to prove an Alexandrov Theorem for Reinhardt domains in CnC1 with spherical symmetry for every n: Then in [33] Monti and Morbidelli proved a Darboux-type theorem for n  2: the unique Levi umbelical hypersurfaces in CnC1 with all constant Levi curvatures are spheres or cylinders. However, to the best of our knowledge, nowadays it is still not clear if balls are the only compact hypersurfaces with constant Gauss Levi curvature.

2 The Levi Form In this first section we introduce the Levi Form, which is the analogous to the Second Fundamental Form in differential geometry. The Levi form is the key differential geometric object which determines many functions theoretic properties of a CR manifold. For example the holomorphic extendibility of CR functions. By following [5] we will first give the definition of the Levi form for an abstract CR structure, and then we will give more concrete representations of the Levi

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form for an immersed CR manifold. The Levi form for a real hypersurface will be discussed in some detail. Moreover, we will present the relationship between the Levi form and the second fundamental form of a real hypersurface. We start with notation

2.1 Complexified Vectors By starting from the very beginning, we define objects such as @ 1 D @zj 2

@ @ i @xj @yj

! ;

(2)

which are often encountered in complex analysis. This is a complexified vector due p to presence of i D 1: We now make this notion precise. Suppose V is a real vector space. The complexification of V is the tensor product V ˝ C D fX C iY I X; Y 2 V g

(3)

If V is a vector space of real dimension N then dimR V ˝C D 2N; dimC V ˝C D N:

2.2 Complex Structure Definition 1. Suppose V is a real vector space. A linear map J W V ! V is a complex structure if J ı J D Id: A complex structure can only be defined on an even-dimensional real vector space because .det J /2 D .1/N ; with N D dimR V: 2.2.1 Standard Complex Structure Example 1. Let V D Tp .R2n / be the space of vector fields in R2n : We give R2n the coordinates .x1 ; y1 ; : : : ; xn ; yn /: The standard complex structure J for Tp .R2n / is defined by setting for all j D 1; : : : ; n     @ @ @ @ J D D J (4) @xj @yj @yj @xj and then by extending J to all Tp .R2n / by linearity. This complex structure map is designed to simulate multiplication by i D It is an isometry with respect to the Euclidean metric on R2n :

p 1:

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2.2.2 Eigenvalues and Eigenspaces of the Standard Complex Structure Since J ı J D Id on V; the same holds true on V ˝ C. Therefore, J W V ˝ C ! V ˝ C has eigenvalues Ci and i with corresponding eigenspaces denoted by V 1;0 and V 0;1 : From elementary linear algebra we have V ˝ C D V 1;0 ˚ V 0;1 : As an example, let V D Tp .R2n / ' Tp .Cn /: Define the vector fields @ 1 D @zj 2

@ @ i @xj @yj

!

1 @ D @zNj 2

@ @ Ci @xj @yj

! (5)

A basis for Tp1;0 .Cn / is given by f@z1 ; : : : ; @zn g and a basis for Tp0;1 .Cn / is given by f@zN1 ; : : : ; @zNn g:

2.3 Immersed CR Manifold We now give the definition of an immersed CR manifold, which is the simplest case of CR manifold. For a smooth submanifold M of Cn ; we denote by Tp .M / the real tangent space of M at a point p 2 M:

2.3.1 Holomorphic Tangent Space In general Tp .M / is not invariant under the standard complex structure J for Tp .Cn / ' Tp .R2n /: Therefore we give special designation to the largest J -invariant subspace of Tp .M /: Definition 2. For a point p 2 M; the complex tangent space of M at p is the vector space Hp .M / D Tp .M / \ J.Tp .M //:

(6)

Remark that Hp .M / must be an even-dimensional real vector space, because J ı J jHp .M / D Id and therefore j det J jHp .M / j2 D .1/m with m D dim Hp .M /:

2.3.2 Totally Real Part of the Tangent Space We also give special designation for the other directions in Tp .M / which do not lie in Hp .M /: Definition 3. The totally real part of the tangent space of M is the quotient space Xp .M / D Tp .M /=Hp .M /:

(7)

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Using the Euclidean inner product on Tp .R2n / we can identify Xp .M / with the orthogonal complement of Hp .M / and we have Tp .M / D Hp .M / ˚ Xp .M /: Remark that J.Xp .M // is orthogonal to Hp .M /: Therefore, it is transverse to Tp .M /: From elementary linear algebra we have Lemma 1. Suppose M is a real submanifold of Cn of real dimension 2n  d; then 2n  2d  dimR Hp .M /  2n  d;

0  dimR Xp .M /  d:

(8)

The dimensions of Hp .M / and of Xp .M / are of crucial importance, as we will see in a moment. If M is a real hypersurface, then d D 1; and the only possibility is dimR Hp .M / D 2n  2: In particular the dimension of Hp .M / never changes. If d > 1 there are more possibilities.

2.3.3 Immersed CR Manifold Definition 4. A submanifold M of Cn is called an immersed CR manifold if dimR Hp .M / is independent of p 2 M: Since Hp .M / is J -invariant, the complex structure map J on Tp .R2n / ˝ C D fX C iY W X; Y 2 Tp .R2n /g restricts to a complex structure on Hp .M / ˝ C and Hp .M / ˝ C admits a decomposition as direct sum of Hp1;0 .M / and Hp0;1 .M /; which are the Ci and i eigenspaces of J; respectively. It will be useful to have a way of identifying the above spaces in terms of a local defining system for M: Lemma 2. Let M be a smooth manifold of Cn defined near a point p 2 M by M D fz 2 Cn W 1 .z/ D d .z/ D 0g where 1 ; d are smooth real valued functions with d1 .z/ ^ ^ dd .z/ ¤ 0 near p: W D

n X

wj @zj 2 Hp1;0 .M / ” W .k /.p/ D 0; 1  k  d

j D1

W D

n X

(9) wj @zNj 2

Hp0;1 .M /

” W .k /.p/ D 0; 1  k  d:

j D1

Definition 5. We say that a subbundle V is involutive if it is closed under the Lie bracket, that is ŒX1 ; X2  belongs to V whenever X1 ; X2 2 V: Lemma 3. Suppose M is a CR submanifold of Cn : Then • Hp1;0 .M / \ Hp0;1 .M / D f0g for each p 2 M • Hp1;0 .M / and Hp0;1 .M / are involutive.

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2.4 CR Structure Let T C .M / denote the complexified tangent bundle whose fibers at each point p 2 M are Tp .M / ˝ C: Definition 6. Let M be a smooth manifold and suppose L is a subbundle of T C .M /: The pair .M; L/ is called a CR structure (or an abstract CR manifold) if • Lp \ Lp D f0g for each p 2 M • L is involutive. It is clear from the above discussion that if M is a CR submanifold of Cn then the pair .M; H 1;0 .M // is a CR structure.

2.4.1 The Levi Form for a CR Structure Given a CR structure .M; L/ the subbundle L ˚ L  T C .M / is not necessarily involutive. The Levi form for M is defined so that it measures the degree to which L ˚ L fails to be involutive. For p 2 M let p W Tp .M / ˝ C ! .Tp .M / ˝ C/=.L ˚ L/ be the natural projection map. Definition 7. The Levi form at a point p 2 M is the map Lp W LL ! .Tp .M /˝ C/=.L ˚ L/ defined by Lp .Lp ; W p / D

1 p .ŒLp ; W p /; 2i

8 Lp 2 Lp ; W p 2 Lp

(10)

where L; W are any vector fields in L; L respectively, which equal Lp ; W p at p: Remark 1. Let us remark the following facts: • The definition is independent of the L vector field extension of the vector Lp 2 Lp • The vector field ŒL; W  lies in T C .M / because T C .M / is involutive. • The Levi Form measures the piece of 2i1 .ŒL; W p / that lies outside of Lp ˚ Lp : • The factor 2i1 is introduced to make the Levi form real valued as a quadratic form, i.e. Lp .Lp ; Lp / D Lp .Lp ; Lp /; for all Lp 2 Lp :

2.4.2 Levi Flat Definition 8. We say that a CR structure .M; L/ is Levi Flat if the Levi form of M vanishes at each point in M:

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Example 2. Let M D f.z; w/ 2 Ch  Cn1 iW I m z D 0g: A basis for L D H 1;0 .M / is given by @w@ 1 ; : : : ; @w@n1 : Since @w@ j ; @w@N k D 0 then M is Levi Flat. Also note that M is foliated by the complex manifolds Mx D f.x; w/ 2 C  Cn1 g;

x 2 R:

(11)

The complexified tangent bundle at each Mx is given by L ˚ L: Theorem 1. Suppose .M; L/ is a Levi Flat CR structure. Then M is locally foliated by complex manifold whose complexified tangent bundle is given by L ˚ L: The idea of the proof is to prove that L ˚ L and its underlying real bundle H.M / D fL C L W L 2 Lg are involutive. Then the foliation is obtained by the real Frobenius Theorem.

2.5 The Levi Form for an Immersed CR Manifold For an immersed CR manifold M L D H 1;0 .M /; L D H 0;1 .M /;

(12)

and the quotient space T C .M /=L ˚ L is identified with the totally real part of the real tangent space. Definition 9. The Levi form of M at p 2 M is the map

Lp .Lp ; W p / D

Lp W H 1;0 .M /  H 0;1 .M / ! Xp .M /

(13)

1 p ŒLp ; W p ; 2i

(14)

Lp 2 H 1;0 .M /; W p 2 H 0;1 .M /

where p W Tp .M / ! Xp .M / is the orthogonal projection.

2.6 Levi Form of a Real Hypersurface Let M be a real hypersurface in Cn that separates R2n in two open sets D and R2n n D: Since dim Xp .M / D 1 we can forget about the vectorial structure of the Levi form. Let Np denote the unit inward normal to D at p 2 M: Definition 10. Tp D J.Np / is called the characteristic direction of M at p: We have SpanR fTp g D Xp .M /:

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Definition 11. The Levi form of M at p 2 M is the C-bilinear Hermitian form Lp W H 1;0 .M /  H 0;1 .M / ! C; Lp .Lp ; W p / D 2i1 hŒLp ; W p ; Tp i; Lp 2 H 1;0 .M /; W p 2 H 0;1 .M / where h ; i is the C-linear extension of the Euclidean inner product. Recall that @zj D 12 .@xj  i @yj /; @zNk D 12 .@xk C i @yk / and h@zj ; @zNk i D

1 ıj k ; 2

h@zj ; @zk i D 0:

(15)

2.7 Comparison with the Second Fundamental Form To compare the Levi Form with the Second fundamental form it is convenient to introduce the Levi Civita connection. P Definition 12 (Levi Civita connection). Suppose W D wj @xj is a vector field N N N on R and let Vp 2 Tp .R /: Define rVp W 2 Tp .R / rVp W D

N X

Vp .wj /@xj :

(16)

j D1

Remark that rVp W is the derivative of W in the direction of Vp and that ŒV; W p D rVp W  rWp V: Definition 13. The second fundamental form is the map IIp .M / W Tp .M /  Tp .M / ! R defined by IIp .Vp :Wp / D hrVp N ; Wp i;

Vp :Wp 2 Tp .M /

where h ; i is the Euclidean inner product. We now derive a formula for IIp in term of the real Hessian of the following defining function for M .x/ D

dist.x; M / if x 2 D dist.x; M / if x 2 RN n D

(17)

k If M is C k ; kP  2 then  is C P near M in RN and r D N on M: Let Vp D vj @xj ; Wp D wk @xk 2 Tp .M /: Then

II.Vp ; Wp / D

X @2 .p/ vj wk @xj @xk

(18)

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Moreover, we have IIp .Vp ; Wp / D hrVp W; Np i

(19)

We now extend the Levi Civita connection to Cn by C-linearity and we get Theorem 2. For all Z 2 H 1;0 .M /; W 2 H 0;1 .M / L .Z; W / D

1 hŒZ; W ; T i D hrZ W ; N i D hrZ N ; W i 2i

(20)

Proof. By using the fact that ŒZ; W  D rZ W  rW Z is a tangent vector fields, we get hŒZ; W ; T i D hrZ W  rW Z; T i D hrZ J.W /  rW J.Z/; J.T /i

(21)

D i hrZ W C rW Z; N i D 2i hrZ W ; N i D 2i hrZ N ; W i Theorem 3. For all Z D X  iY 2 H

1;0

t u

.M / with Y D J.X /; we have

L .Z; Z/ D II.X; X / C II.Y; Y /

(22)

Proof. Since ŒX; Y  D rX Y  rY X we have 1 1 hŒZ; Z; T i D hŒX  iY; X C iY ; T i D hŒX; Y ; T i 2i 2i D hrX Y  rY X; T i D hrX J.Y /  rY J.X /; J.T /i

L .Z; Z/ D

D hrX X  rY Y; N i D hrX X; N i C hrY Y; N i:

(23)

t u

Theorem 4 (LEVI FORM OF A REAL HYPERSURFACE). Let M D fz 2 Cn W .z/ D 0g be a smooth real oriented hypersurface. If p 2 M and @.p/ D .@z1 .p/; : : : ; @zn .p// ¤ 0; then Lp .V; W / D

n X @2  1 .p/vj wN k 2j@.p/j @zj @Nzk

(24)

j;kD1

for V D

Pn kD1

vk @zk ; W D

Pn kD1

wk @zk 2 Hp1;0 .M /:

Proof. The inward unit normal vector is N D 

Nz` @z` Cz` @Nz` j@j

D

x` @x` Cy` @y` jrj

On the Levi Monge-Amp`ere Equation

163

Lp .V; W / D hrV N ; W i D

n X

 vj wN k h@zj

j;kD1

 `N @z` ; @zNk i j@j

  n n @zj kN kN 1 X 1 X D D vj wN k @zj vj wN k 2 j@j 2 j@j j;kD1

(25)

t u

j;kD1

For this reason the Levi form is identified with the restriction of the complex Hessian of  to Hp1;0 .M /: It is important to note that if Q is another defining function for M with d Q ¤ 0 on M then the map .v1 ; : : : ; vn ; wN 1 ; : : : ; wN n ; / 2 Hp1;0 .M /  Hp0;1 .M / 7!

n X j;kD1

@2 Q .p/vj wN k (26) @zj @Nzk

is a nonzero multiple of the Levi Form at p: To see this, write Q D ˛ for some smooth function ˛ W Cn ! R which is nonzero near M: 2.7.1 Levi Form of a Real Hypersurface in C2 Example 3. Let Z D z2 @z1  z1 @z2 : Since Z./ D 0 we have Z 2 H 1;0 .M / and 1  jz2 j2 z1 Nz1  z1 zN2 z2 zN1  z2 zN1 z1 zN2 C jz1 j2 z2 zN2 2j@j 0 1 0 z1 z2 1 D det @ zN1 z1 zN1 z2 Nz1 A : jrj zN2 z1 zN2 z2 Nz2

L .Z; Z/ D

(27)

Here r is the Euclidean gradient vector.

2.8 Strictly Pseudoconvexity Definition 14. An oriented real hypersurface M is called strictly pseudoconvex at a point p 2 M if the Levi form at p is positive definite, i.e. if there exists a defining function  for M so that n X j;kD1

for all W D

Pn

kD1 wk @zk

@2  .p/wj wN k > 0 @zj @Nzk

2 Hp1;0 .M /:

(28)

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The above inequality is invariant under biholomorphic change of coordinates. This follows by explicitly computing the complex Hessian of  ı F where F is a biholomorphism. Strictly pseudoconvex hypersurfaces have the following beautiful property Theorem 5 (Narasimhan). Suppose M is strictly pseudoconvex at a point p 2 M: Then there is a biholomorphic map F defined in a neighborhood U of p so that F .M \ U / is strictly convex in F .U /: It is elementary, but nontrivial, to show that a domain is strictly pseudoconvex near p if and only if it is strictly Euclidean convex (i.e., the relevant matrix of second-order partial derivatives is positive definite) with respect to suitable local holomorphic coordinates centered at p: Stated differently, strict pseudoconvexity is locally simply the biholomorphically invariant version of strict convexity. Unfortunately, this neat characterization breaks down if the Levi form is only positive semidefinite for each p 2 M; as shown by an example discovered by J. J. Kohn and L. Nirenberg in 1972.

2.9 Pseudoconvex Domains Definition 15. The domain D D fz 2 Cn W f .z/ < 0g; with f 2 C 2 ; is called pseudoconvex if Lp  0;

8 p 2 @D:

An equivalent definition (see for instance [24] for the proof), which will be useful in the sequel, is the following : Definition 16. D is pseudoconvex if D 3 z 7!  log.dist.z; @D//

is plurisubharmonic.

Remark 2. This last definition does not require any regularity of @D and it is called Hartog’s pseudoconvexity. Note that it is a continuous function which tends to C1 as z ! @D: One verifies that convex domains are pseudoconvex and that a domain with C 2 boundary is pseudoconvex according to this definition if and only if it is Levi pseudoconvex. Also, any pseudoconvex domain is the increasing union of strictly (Levi) pseudoconvex domains with C 1 boundaries. Let us recall that a domain D is (Euclidean) convex if and only if  log dist.z; @D/ is a convex function.

On the Levi Monge-Amp`ere Equation

165

3 The Levi Problem and Levi Curvature for a Real Hypersurface In this section we will talk about the Levi problem, which leads to the notion of Levi curvature and to the Dirichlet problem for the prescribed Levi curvature equation. We will introduce the prescribed Levi curvature equation for a real hypersurface of C2 and then we will discuss the solvability of Dirichlet problem for graphs in the class of viscosity solutions. The main result of this section is that different hypotheses on the zeros of the prescribed Levi curvature imply different regularity behavior of the solution. Let us begin with some classical things about domains of holomorphy. Definition 17. D  Cn is a domain of holomorphy if for every p 2 @D there exists Fp W D ! C;

holomorphic

which does not extend to any neighborhood of p Remark 3. Every domain D  C is a domain of holomorphy: just take Fp .z/ D

1 : zp

It is clearly holomorphic in D, but surely it has no holomorphic extension to any neighborhood of p. This sort of simple construction does not extend to more than one variable, as the zeroes and singularities of holomorphic functions are not isolated in the case of two or more variables. Example 4 (Hartogs’s hammer (1906)). There exists a domain D  C2 which is not a domain of holomorphy:



1 1 H D .z1 ; z2 / 2 C W jz1 j < ; jz2 j < 1 [ jz1 j < 1; < jz2 j < 1 2 2

2

Every holomorphic function F W H ! C can be extended to a holomorphic function F W HQ ! C, where HQ D fjz1 j < 1; jz2 j < 1g strictly contains H Proof. Let f W H ! C be holomorphic and fix r with 1=2 < r < 1: The function 1 F .z1 ; z2 / D 2 i

Z jwjDr

f .z1 ; w/ dw w  z2

is easily seen to be holomorphic on G D fjz1 j < 1; jz2 j < rg: Observe that for fixed zQ1 with jQz1 j < 1=2 the function z2 ! f .Qz1 ; z2 / is holomorphic on the disc fjz2 j < 1g

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and hence by the Cauchy integral formula f .Qz1 ; z2 / D F .Qz1 ; z2 / for jz2 j < r: Thus f F on fjz1 j < 1=2; jz2 j < rg; which implies f F on H \ G by the identity theorem, so that F provides the holomorphic extension of f from H to HQ D H [ G: t u In 1910 E.E. Levi tried to characterize domains of holomorphy in term of a differential property of the boundary. He showed that every domain of holomorphy is pseudoconvex and that, if p 2 @D and Lp > 0, then D \ U is a domain of holomorphy, for a suitable neighborhood U of p: Comparison between pseudoconvexity and domains of holomorphy was called the Levi problem and it was completely solved in 1954 by Oka, Bremmerman and Norgouet: D  Cn is a domain of holomorphy iff D is pseudoconvex: A natural subsequent problem is the following: given D  Cn look for its holomorphic hull, i.e, the smallest domain of holomorphy containing D. This problem led to the notion of Levi curvature and to the Dirichlet problem for the prescribed Levi curvature equation.

3.1 Levi Curvature for a Real Hypersurface in C2 Let D D f.z1 ; z2 / 2 C2 W f .z1 ; z2 / < 0g; @p f ¤ 0 if f .p/ D 0. Consider the Levi form of M D @D at a point p 2 @D L .Z; Z/ for Z 2 Hp1;0 .@D/ Definition 18. We define the Levi curvature of @D at the point p as K@D .p/ WD L .Z; Z/;

Z 2 Hp1;0 .@D/; hZ; Zi D 1

We can explicitly write it in term of a defining function f for the domain D: Indeed, p take Z D j@f2j .f2 @1  f1 @2 /: Then hZ; Zi D 1 and K@D .p/ D L .Z; Z/ D

p .f / j@p f j3

where p .f / D .fz2 fz2 /fz1 z1  .fz2 fz1 /fz1 z2  .fz1 fz2 /fz2 z1 C .fz1 fz1 /fz2 z2 The Levi curvature of @D was first introduced by Bedford-Gaveau [2] and Tomassini [42] by analogy with the real case. However, a posteriori we are able to recognize its geometric meaning of complex curvature.

On the Levi Monge-Amp`ere Equation

167

Remark 4. Let us remark that • K@D .p/ is real and independent of the defining function f of D • K@D .p/ is invariant with respect to unitary biholomorphic transformation of C2

3.1.1 Levi Curvature of a Ball Example 5. Let BR D fz D .z1 ; z2 / 2 C2 W jzj < Rg: Then at every p 2 @BR K@BR .p/ D

1 : R

Proof. We can choose f D jz1 j2 C jz2 j2  R2 as a defining function of BR : We have j@f j2 Djz1 j2 C jz2 j2 D R2 p .f / D.fz2 fz2 /fz1 z1  .fz2 fz1 /fz1 z2  .fz1 fz2 /fz2 z1 C .fz2 fz1 /fz2 z2 Djz2 j2 C jz1 j2 D R2 K@BR .p/ D

p .f / R2 1 D 3 D 3 j@p f j R R

t u

3.1.2 Levi Curvature of a Graph in C2 Let u W ˝ ! R be a C 2 -function on the open set ˝  R3 . The graph of u; M WD f.x; y; t; / 2 R4 W  D u.x; y; t/g is part of the boundary of D D f.x; y; t; / 2 R4 W  > u.x; y; t/g We identify R4 with C2 and denote z 2 C2 as z D .z1 ; z2 /; z1 D x C iy; z2 D t C i : Then D D f.z1 ; z2 / 2 C2 W f .z1 ; z2 / < 0g, with f .z1 ; z2 / D f .x; y; t; / WD u.x; y; t/   so that fz1 D

1 1 1 .fx  ify / D .ux  i uy / and fz2 D .ft  if / D .ut C i /: 2 2 2

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At a point p D .; u.// 2 M;  D .x; y; t/ 2 ˝, one has KM .p/ D

p .f / 1 C u2t D L .u/ ; 0 3 j@p f j3 .1 C jDuj2 / 2

D D .@x ; @y ; @t /

where L0 .u/ WD uxx C uyy C 2 a uxt C 2 b uyt C .a2 C b 2 /utt and a D a.Du/ D

uy  ux ut ; 1 C u2t

b D b.Du/ D 

ux C uy ut : 1 C u2t

Then, given a function K D K.; r/, the PDE 3

L0 .u/ D K.; u/

.1 C jDuj2 / 2 ; 1 C u2t

2˝

is the prescribed Levi curvature equation The characteristic form of L0 : qL0 ./ WD 12 C 22 C 2a 1 3 C 2b 2 3 C .a2 C b 2 / 32 is the quadratic form related to the matrix 1 10 a A A WD @ 0 1 b 2 2 a b a Cb 0

whose eigenvalues are 1 D 1; 2 D 1 C a2 C b 2 ; 3 D 0. • Then L0 is a quasilinear, degenerate elliptic operator, which is not elliptic at any point. Debiard and Gaveau [14] showed that L0 is not variational, i.e. cannot be written in divergence form.

3.2 The Dirichlet Problem for the Prescribed Levi Curvature Equation Motivated by the following problems: • construction of real surfaces of C2 with given boundary and prescribed Levi curvature • construction of the envelopes of holomorphy

On the Levi Monge-Amp`ere Equation

169

we now study the Dirichlet problem for the Levi equation (

3 2 2

/ L0 .u/ D K.; u/ .1CjDuj ;2˝ 1Cu2 t

uD'

on @˝

This problem has received increasing attention since the early 1980s. The first existence results were obtained in the case K 0, with the analytic disc approach, by Bedford-Gaveau [3] and Bedford-Klingenberg [4]. However, to the best of our knowledge, their techniques do not work if K ¤ 0:

3.3 The Prescribed Non Vanishing Levi Curvature Equations In [39] Slodkowski and Tomassini introduced the use of PDE’s techniques in studying (

3 2 2

/ L0 .u/ D K.; u/ .1CjDuj ;  2 ˝  R3 1Cu2

uD'

t

on @˝

when K ¤ 0. However the C 1;˛ techniques usually used for quasilinear elliptic equations do not work for the Levi equations. Nevertheless, Slodkowski and Tomassini, were able to prove • L1 a-priori estimates for the gradient of the solutions Then, using a Comparison Principle for L0 ; they proved the existence of u 2 Lip.˝/ solving the equation in the weak viscosity sense of Crandall-Lions, and s.t. u D ' on @˝:

3.4 Regularity of the Viscosity Solutions: Classical Solvability of the Dirichlet Problem In [8] Citti et al. proved that: every Lipschitz-continuous viscosity solution to the K-prescribed Levi curvature equation is of class C 1 if K 2 C 1 and K > 0. Thus, [39] plus [8] give the smooth solvability of the Dirichlet problem (

3 2 2

/ L0 .u/ D K.; u/ .1CjDuj ;  2 ˝  R3 1Cu2

uD' if K is smooth and strictly positive.

t

on @˝

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This kind of hypoellipticity result comes from the sub-Riemannian structure underlying L0 .

3.5

L0 ’s Sub-Riemannian Structure and the Smoothness Result

We identify X and Y with the vector fields (i.e. first order PDO) X D @x C a @t

and Y D @y C b @t

The following crucial identities hold L0 u D .X 2 u C Y 2 u/ .1 C u2t /

ŒX ; Y  D 

(29)

L0 u @t 1 C u2t

(30)

Then, the K prescribed Levi curvature equation can be written as 3

X 2 u C Y 2 u D K.; u/

.1 C jDuj2 / 2 .1 C u2t /2

Moreover 3

ŒX ; Y  D q @t ;

q WD K.; u/

.1 C jDuj2 / 2 .1 C u2t /2

Summing up: u solves the K-prescribed Levi-curvature equation if 3

X 2 u C Y 2 u D K.; u/

.1 C jDuj2 / 2 .1 C u2t /2

and, if K ¤ 0 everywhere (, q ¤ 0 everywhere), X D @x C a @t

Y D @y C b @t ;

ŒX ; Y  D q @t

are linearly independent at any point. In geometric language: rank LiefX; Y g./ D 3

at any point  2 ˝ . R3 /

On the Levi Monge-Amp`ere Equation

171

If the vector fields X and Y were linear and smooth, this condition would imply the hypoellipticity of X 2 C Y 2 , i.e. the smoothness of the weak distributional solutions to X 2 u C Y 2 u D f; f 2 C 1 : In [8] we proved that a similar regularity result holds for the nonlinear Levi vector fields: Theorem 6 (G.Citti-E.Lanconelli-A.M. (2002)). If u 2 Lip.˝/ solves the Kprescribed Levi-curvature equation 3

X 2 u C Y 2 u D K.; u/

.1 C jDuj2 / 2 ; .1 C u2t /2

in ˝  R3 ;

in the weak viscosity sense, then u 2 C 1 if K 2 C 1 ; K > 0. Then : • The Dirichlet problem for the K-prescribed Levi curvature equation in an open set ˝  R3 has a smooth solution if K 2 C 1 ; K > 0 • Every Lipschitz-continuous domain of C2 whose boundary has smooth and strictly positive Levi-curvature, is actually smooth. Very recently, jointly with C. Gutierrez and E. Lanconelli in [18] we discovered that these results have no counterpart in higher dimension. In next subsections we exploit the steps and the history of the proof of Theorem 6.

3.6 Strong Solutions By differentiation of the equation and integration by parts, in [9] we obtained the following first regularity achievement Theorem 7 (Citti, M. 1999). If K 2 Lip.˝  R/; every viscosity solution u 2 Lip.˝/ of the prescribed Levi curvature equation X 2 u C Y 2 u D K.; u/

.1 C jDuj2 /3=2 .1 C .@t u/2 /2

1 is a strong solution, i.e. X u; Y u 2 Hloc .˝/ and u pointwise solves the equation a.e.

3.7 Sketch of the Proof of Theorem 7 In an open set ˝ we fix a solution u of the regularized equation L" u D K.; u/

.1 C jDuj2 /3=2 .1 C .@t u/2 /2

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where L" denotes the elliptic operator (with minimum eigenvalue "2 ) L" u WD L u C "2

utt 1 C u2t

and L u WD X 2 u C Y 2 u: The uniform limit as " goes to 0 of a sequence of solutions of the regularized equation is a viscosity solution of the Levi equation in the weak sense of CrandallIshii-Lions.

3.7.1 Linear Operator In the sequel we shall denote by L" a linear elliptic operator formally defined as L" : L" D X 2 C Y 2 C T"2 ; where T" D ".1 C u2t /1=2 @t , and the coefficients of the vector fields X and Y depend on the fixed solution u. Then we prove that the coefficients a and b of the vector fields and the two functions ! D @t u;

v D arctg.ut /

and

are solutions of L" z D f; with different functions f . The proof of the regularity Theorem 7 is based on the regularity of the solutions of this linear equation in some Sobolev Spaces W"1;2 naturally defined in terms of the vector fields X , Y and T" .

3.7.2 Caccioppoli-Type Estimates Lemma 4. For every  2 C01 .˝/ and ı 2 .0; 1/ there is Cı > 0 s.t. Z

Z .jX zj2 C jY zj2 C jT" zj2 / 2  ı ˝

.f 2 C j@t aj2 C j@t bj2 C jT" vj2 / 2 ˝

Z CCı

.jXj2 C jY j2 C jT" j2 C .1 C ! 2 / 2 /z2 ˝

On the Levi Monge-Amp`ere Equation

173

Proof. Integrate by parts and recall that X  z D X z  @t a z; Y  z D Y z  @t b z; T" z D T" z C !T" v z t u

Then use Cauchy Schwarz inequality

3.7.3 Elegant Equalities With the previous notation, we have Lemma 5. @t a D Y v  !Xv;

@t b D Xv  !Y v:

Proof. Since a D Y u; b D X u ( because we are computing tangential vector fields on the graph u) we have @t a D@t Y u D Œ@t ; Y u C Y @t u D !@t b C Y !; @t b D  @t X u D Œ@t ; X u C X @t u D !@t a  X! !X! C Y ! D !Xv C Y v; 1 C !2 !Y !  X! @t b D D D !Y v  Xv 1 C !2

@t a D D

3.7.4 Caccioppoli-Type Estimates Lemma 6. For every  2 C01 .˝/ Z

Z .jXaj2 C jYaj2 C jT" aj2 / 2 C

Z

˝

.jXbj2 C jY bj2 C jT" bj2 / 2 ˝

Z

.jXvj C jY vj C jT" vj /  C1 2

˝

2

2

.jrKj2 C K 2 / 2

2

˝

Z

CC2

.jXj2 CjY j2 C jT" j2 C .1 C ! 2 / 2 /.a2 C b 2 C v 2 / ˝

In particular krakL2 ; krbkL2 are uniformly bounded in ": loc

loc

t u

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3.8 Levi Flat In [10] we proved the following Frobenius type result for strong solutions of the Levi flat equation. Theorem 8 (Citti, M. 2001). Every strong solution u of the Levi equation with K 0; is foliated by analytic curves and u is analytic on every leaf. The key Lemma in the proof is the following Lemma 7. Since X 2 u C Y 2 u D 0 we have ŒX; Y  D 0 and X 2 v C Y 2 v D 0 Proof. We formally differentiate the equation X 2 u C Y 2 u D 0 with respect to @t and we get 0 D@t .X 2 u C Y 2 u/ DŒ@t ; X X u C X Œ@t ; X u C X 2 ! C Œ@t ; Y Y u C Y Œ@t ; Y u C Y 2 ! D@t a@t X u C X.@t a!/ C X 2 ! C @t b@t Y u C Y .@t b!/ C Y 2 ! D  @t a@t b C X.Y v!  Xv! 2 / C X 2 ! C @t b@t a C Y .Xv!  Y v! 2 / C Y 2 ! D  .X 2 v C Y 2 v/! 2  2!.XvX! C Y vY !/ C X 2 ! C Y 2 ! D  .X 2 v C Y 2 v/! 2  2!.XvX! C Y vY !/ C X.Xv.1 C ! 2 // C Y .Y v.1 C ! 2 // D X 2 v C Y 2 v

t u

Let D D .X; Y; T" /: By Caccioppoli-type estimates we get that for every multiindex I and for every p the norms kD I ukLp ; kD I vkLp are uniformly bounded in ": loc loc In particular the coefficients a; b of the vector fields X; Y are Lipschitz continuous and we can conclude by Frobenius theorem.

3.9 Classical Solution If K ¤ 0 at every point, in a joint work with Citti and Lanconelli [8] we obtained the following regularity result. Theorem 9 (Citti, Lanconelli, M. 2002). Every strong solution u of the Levi equation, with K ¤ 0; K 2 C 1 .˝/; is smooth.

On the Levi Monge-Amp`ere Equation

175

3.10 Sketch of the Proof of Theorem 9 Since the minimum eigenvalue of 0

1 10 a A A D @0 1 b a b a2 C b 2 is equal to zero for every p 2 R3 ; the operator L0 in (29) is not elliptic at any point and the regularity results for viscosity solutions of non-linear elliptic and parabolic equations cannot be applied to our case. We have to introduce a completely different procedure, based on the particular structure of the Levi equation.

3.10.1 Lie Bracket Let us recall that in this situation the Lie-bracket of the first order differential operators X and Y is ŒX; Y  D 

L0 u @t : 1 C u2t

3.10.2 Interpolation Inequalities The classical elliptic regularization procedure is based on Sobolev inequalities and on a priori estimates of Caccioppoli type. In the present situation neither the classical Caccioppoli inequality holds, since the vector fields are not selfadjoint, nor the Sobolev inequality, since the coefficients of the vector fields are only bounded. To overcome these difficulties we first prove an interpolation inequality, which will play a role similar to the Sobolev one in the classical setting. 3.10.3 Interpolation Inequalities Lemma 8. Let C be a positive constant such that jjajj1 C jjbjj1 C jjvjj1  C; then for every function z 2 C 1 ,  2 C01 , we have Z Z  Z  3 6 2 6 jD" j6 C  6 .1 C z6 / jX zj   c jD" .X z/j  C c where D" is the vector .X; Y; T" / and the constant c > 0 only depends on C and K. An analogous inequality is also satisfied if we replace X z with Y z or T" z.

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3.10.4 Linear Operator • We first establish some a priori estimates in the intrinsic directions X and Y , weaker than the classical Caccioppoli one. • Using these inequalities together with the interpolation ones, we prove a priori m;p estimates in W" , for solutions z; which hold under very general assumptions on the commutators of the vector fields, but requires some strong a priori estimates on the derivative @t z. • To estimate the derivative @t , we recall that it can be expressed in term of the commutator of the vector fields.

3.10.5 Non Linear Equation • We then use in an essential way the nonlinearity of the equation: we apply the interpolation and Caccioppoli inequalities to the derivative @t v to obtain a L2 estimate for Xvt and Y vt . utt • Since vt D 1Cu 2 ; then vt has to be considered a derivative of weight 4 of u, while t Xvt and Y vt are derivatives of weight 5 of the same function. Once proved the summability of these derivatives with respect to t, we obtain analogous estimates for any derivation of weight 5 and 4: 3.10.6 Estimate of @t v Lemma 9. If K ¤ 0 then for every  2 C01 Z

Z .j@t vj C jD" @t vj /  C 3

2

6

.j@t j2 C jD" @t j2 /.v 2 C jD" vj2 /

Proof. Write @t D  q1 ŒX; Y ; integrate by parts and use Cauchy Schwarz

t u

In particular the coefficients a and b of the vector fields are now regular, and we can apply a Sobolev type inequality.

3.10.7 Non Linear Equation By applying the previous results to the non linear equation we get that • the derivatives of weight 5 belong to L2 • the derivatives of weight 4 belong to L4 , because the homogeneous dimension in this case is Q D 4 and 12  Q1 D 14 • the derivatives of weight 3 belong to Lp for every p,

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177

• the derivatives of weight 2 belong to suitable classes C ˛ for every ˛ 20; 1Œ and we will write u 2 CX2;˛ Now, we can freeze the coefficients a; b 2 CX1;˛ of the vector fields. 3.10.8 Freezing Method Let 0 D .x0 ; y0 ; t0 / 2 ˝: Given a 2 CX1 we define the first order Taylor polynomial of a in the horizontal directions as P0 a./ D a.0 / C Xa.0 /.x  x0 / C Ya.0 /.y  y0 /;

 D .x; y; t/:

Now we define the frozen vector fields X0 D @x C P0 a./@t ;

Y0 D @y C P0 b./@t :

We have ŒX0 ; Y0  D .Xb  Ya/.0 /@t and .Xb  Ya/.0 / D .X 2 u C Y 2 u/.0 / D q.0 / ¤ 0: Then the vector fields X0 ; Y0 satisfy H¨ormander condition and the linear operator L0 D X20 C Y20 is hypoelliptic. By using integral formulas for u in terms of the fundamental solution of L0 we deduce that u 2 C 2;˛ . By a bootstrap argument we then conclude that u 2 C 1 : In [11] a freezing method with second order Taylor polynomials has been introduced to prove the following regularity theorem. Theorem 10 (Citti, M. 2002). If the prescribed curvature K is smooth and it has some first order zeros, then classical solutions of the prescribed Levi curvature equation are C 1 :

4 Levi Curvatures in Higher Dimension In this section we shall explicitly compute the Levi curvatures for a real hypersurface in CnC1 : We then shall study the Dirichlet problem for the prescribed Gauss-Levi curvature equation in the class of Lipschitz continuous viscosity solutions, under very natural conditions on the prescribed curvature K and on the boundary data. To state our result we need a suitable notion of a generalized Levi pseudoconvex solution to the prescribed Gauss Levi curvature equation, which is a fully non linear and a non elliptic pde. This is provided by the notion of viscosity solution in the sense of Crandall et al. [12]. The modifications needed to adapt the theory

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of viscosity to equations of Levi Monge Amp`ere have been studied in the paper [13] and we will recall them for reader convenience. We will also use a comparison principle for viscosity solution, Theorem 14, and existence results of continuous and of Lipschitz continuous viscosity solution of the prescribed Gauss Levi curvature equation, Theorem 15, which have been proved in [13]. The problem of classical regularity of Lipschitz continuous viscosity solutions is widely still open. However, we shall show that the fully nonlinear PDE of the prescribed Gauss-Levi curvature equation has the following hypoellipticity property: every classical solution is smooth, if K ¤ 0 at any point and K 2 C 1 :

4.1 Normalized Levi Form If M is represented in a neighborhood of p by f D 0, rf ¤ 0 and the orientation of M is so that rf =jrf j is the outward pointing normal, then the normalized Levi form at p is the Hermitian form on Hp1;0 M  Hp0;1 M Lp .Zj ; ZN k / D h

N .p/ @@f Zj ; ZN k i; j@f j

fZ1 ; : : : ; Zn g orthonormal basis of Hp1;0 M: Let us recall that Lp contains less information on the shape of M than IIp because N D IIp .X; X / C IIp .Y; Y / ; Lp .Z; Z/ 2

8Z D

X  iY 2 Hp1;0 M p 2

Remark 5. In this situation • M D @D D fz 2 CnC1 W f .z/ D 0g is a real manifold of CnC1 R2nC2 of dim 2n C 1 • Hp1;0 .@D/ D fh 2 CnC1 W hh; @p f i D 0g and dim Hp1;0 .@D/ D n, as complex vector space dim Hp1;0 .@D/ D 2n, as real vector space • in passing from @D to Hp1;0 .@D/ we lose a real dimension

4.2 Gauss-Levi Curvature for a Real Hypersurface in CnC1 Definition 19. A domain D is pseudoconvex if the Levi form of its boundary is positive semidefinite at every point of @D: The Gauss Levi Curvature of @D in p 2 @D is the real number

On the Levi Monge-Amp`ere Equation

179 n Y

K@D .p/ D

j ;

j D1

where j D are the eigenvalues of the normalized Levi form. By a linear algebra argument we can write it as in [31] K@D .p/ D



1 LMA.f /; j@p f jnC2

LMA.f / WD  det

0 fzN fz fz;Nz

 :

(31)

The operator LMA is called the Levi Monge-Amp`ere operator.

4.3 m-Pseudoconvexity Definition 20. A domain D of CnC1 is (strictly) m-pseudoconvex if, for every j 2 f1; : : : ; mg; X

 .j / . 1 ; : : : ; n / WD

i1 ij  0 .> 0/

1i1 < 1

4.7.3 Some Results From the subelliptic properties of L .m/ one gets: Theorem 11 (STRONG COMPARISON PRINCIPLE). Let u; v W ˝ ! R, ˝ R2nC1 open and connected. Assume u and v strictly m-pseudoconvex and (i) u  v in ˝, u.x0 / D v.x0 / at x0 2 ˝ (ii) L .m/ .u/  L .m/ .v/ in ˝ Then

u D v in ˝.

The strong comparison principle has been proved in [6] for n D 1 and in [31] for the general case.

On the Levi Monge-Amp`ere Equation

183

Theorem 12 (SMOOTHNESS OF C 2;˛ -SOLUTIONS). Let u 2 C 2;˛ .˝/ be a strictly m-pseudoconvex solution to the K-prescribed Levi curvature equation L .m/ .u/ D K. ; u; Du/ in ˝ If K is strictly positive and C 1 in its domain, then u 2 C 1 .˝/ The smoothness of C 2;˛ -solutions has been proved in [7] for n D m D 1, in [32] for n D m  1 and in [30] for the general case 1  m  n:

4.8 Hypoellipticity Precisely, in [32] we proved Theorem 13 (Lascialfari, M. (2004)). If u 2 C 2;˛ .˝/ is a pseudoconvex solution of LMA.u/ D q. ; u; Du/ where ˝  R2nC1 and q 2 C 1 .˝  R  R2nC1 /; q > 0; then u 2 C 1 .˝/: Here LMA.u/ D LM a.uy/ is the Levi Monge-Amp`ere operator in (31) for the graph u:

4.9 Sketch of the Proof of Theorem 13 If u is a strictly Levi pseudoconvex solution, LMA.u/ D

2n X

aij .x/Zi Zj u;

i;j D1 2 where  Zj D 2@j C aj .x/@x2nC1 ; aj .x/ D aj .Du.x// and aij .x/ D aij .Z u/ D aij Du.x/; D u.x/ : For all K  ˝ there exists C > 0 such that 2n X i;j D1

aij .x/i j  C

2n X

j2 ;

8 2 R2n ;

8x 2 K:

j D1

The LMA operator is elliptic in 2n linearly independent directions and we have the lack of ellipticity in one direction.

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4.9.1 Frozen Operator If u is a fixed C 2 strictly Levi pseudoconvex solution and x0 2 ˝, we define the linear operator Lx0 D

2n X

aij .x0 /Zx0 ;i Zx0 ;j ;

i;j D1

where, with the notation of subsection 3.10.8, Zx0 ;j D @j C Px0 aj .x/@x2nC1 ; P 2 Px0 aj .x/ D aj .x0 / C 2n i D1 Zi aj .x0 /.xi  x0;i / and aij .x0 / D aij .Z u.x0 //: The linear second order operator Lx0 is elliptic in 2n linearly independent directions and we have the lack of ellipticity in one direction.

4.9.2 Brackets A direct computation shows that ŒZx0 ;i Zx0 ;j  D qij .x0 /@x2nC1 : Since u is a strictly pseudoconvex solution we get • there is .i; j / such that qij .x0 / ¤ 0: Hence  ˚  di m Span Zx0 ;i ; ŒZx0 ;i ; Zx0 ;j ; i; j D 1; : : : ; 2n D 2n C 1 at every point ˝: • The linear operator Lx0 is hypoelliptic. • By using integral formulas for u 2 C 2;˛ in terms of the fundamental solution of 3;ˇ Lx0 we deduce that u 2 Cloc , for every 0 < ˇ < ˛: By a bootstrap argument we 1 can conclude that u 2 C :

4.10 A More Explicit Form of the Gauss-Levi Curvature Equation In order to deal with viscosity solutions we shall write the Gauss-Levi curvature equation in a more explicit form as a degenerate elliptic fully nonlinear PDE. A function u solves the K-prescribed Gauss-Levi curvature equation if L .KI u/ WD LMA.u/  K.x; u/ F .Du/ D 0;

LMA.u/ D det A.Du; D 2 u/

Du D gradient of u, D 2 u D Hessian of u with respect to all the variables x 2 R2nC1 ;

On the Levi Monge-Amp`ere Equation

185

nC2 2 u/2 2nC1 2/

• F .Du/ D 2n .1CjDuj 1C.@x

• A.Du; D 2 u/ WD ˙ D 2 u ˙N T is an n  n matrix with complex entries • ˙ D .In ; iIn ; a  ib/, a and b are vectors with components

a` WD a` .Du/ D

@x2` u  @x2`1 u @x2nC1 u 1 C .@x2nC1 u/2

b` WD b` .Du/ D

@x2`1 u  @x2` u @x2nC1 u ; 1 C .@x2nC1 u/2

4.11 A More Explicit Form of Pseudoconvexity Remark 6. u is pseudoconvex iff A.Du; D 2 u/  0: Remark 7. If u is pseudoconvex then LMA.u/ D det A.Du; D 2 u/ is a degenerate elliptic operator.

4.12 The Dirichlet Problem for LMA Viscosity techniques work very well to study the Dirichlet problem

det A.Du; D 2 u/ D K.x; u/ F .Du/ x 2 ˝  R2nC1 uD on @˝

(32)

Under quite natural assumptions, this problem has a Lipschitz continuous viscosity solution (see [13, 40]).

4.13 Viscosity Solutions to the Prescribed Levi Curvature Equation By following [13] we give the following definitions Definition 21. Given u and ' real functions in ˝, we say that ' touches u from below (above) at x0 2 ˝ if u.x0 / D '.x0 / and '.x/  ./u.x/ for all x in some neighborhood of x0 . Definition 22. u W ˝ ! R is a viscosity subsolution if u is in BUS C.˝/, and for every ' 2 C 2 .˝/ and for every x0 2 ˝ such that ' touches u from above at x0 ; we have A.D'; D 2 '/.x0 /  0 and L .K; '/.x0 /  0.

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Definition 23. u W ˝ ! R is a viscosity supersolution if u is in BLS C.˝/, and for every ' 2 C 2 .˝/ and for every x0 2 ˝ such that ' touches u from below at x0 and A.D'; D 2 '/.x0 /  0 we have L .K; '/.x0 /  0. Definition 24. A function u W ˝ ! R is a viscosity solution if it is both a viscosity subsolution and a viscosity supersolution.

4.13.1 Hartogs’s Pseudoconvexity Remark 8. The epigraph of a viscosity solution u is a pseudoconvex domain of CnC1 in the Hartogs’s generalized sense of Definition 16. The proof of this statement is implicitly contained in [19, Theorem 4.1.27]) and we leave details to the reader. We express this property by simply saying that u is pseudoconvex.

4.14 Hypotheses on K In order to study the Dirichlet problem (32) we assume that the prescribed function KW ˝  R ! Œ0; C1/ is bounded and continuous and it satisfies the following hypotheses (H1) STRICTLY MONOTONICITY for every R > 0; there is `R > 0; such that, for all x 2 ˝; and R  v  u  R `R .u  v/  K. ; u/  K. ; v/ (H2) MODULUS OF CONTINUITY for every R > 0; there is !R such that !R .s/ ! 0 for s ! 0C and jK.x; u/  K.y; u/j  !R .jx  yj/ for every .x; y/ 2 ˝ and juj  R: (H3) BOUNDARY CONDITION ˝  i R is strictly pseudoconvex and its curvature K@˝i R satisfies sup˝R K < K@˝i R .x0 / for every x0 2 @˝

4.15 Comparison Principle Theorem 14 (COMPARISON PRINCIPLE). Assume (H1)–(H3), u sub- and v supersolution of (32), then uv

N in ˝:

Proof. Use Ishii method. We refer to [13] for the adaptation of this method to the Levi Monge-Amp`ere equation t u

On the Levi Monge-Amp`ere Equation

187

4.16 Existence and Uniqueness Theorem 15 (UNIQUENESS AND EXISTENCE THEOREMS). If (H1)–(H3) hold, then 8 2 C.@˝/ 9Šu viscosity solution of (32). Moreover, if  2 Lip.@˝/ and K 2 Lip.˝  R/ then u 2 Lip.˝/. Proof. The ingredients of the proof in [13] are • Perron method C comparison principle C existence of a particular subsolution and of a particular supersolution • Interior gradient estimates C boundary gradient estimates • To prove boundary gradient estimates, choose such that u D   d  u   C d D u near the boundary, where d is the boundary distance.

t u

4.17 The Dirichlet Problem for LMA Summing up: • Viscosity techniques work to study the Dirichlet problem (32) and under quite natural assumptions, the problem has a Lipschitz continuous viscosity solution by [13]. • In dimension n > 1 the problem of the C 2;˛ solvability of this problem is still widely open. • The fully nonlinear elliptic techniques do not work for the Levi equations • In dimension n > 1 Lipschitz continuous viscosity solutions are not expected to be smooth (even if the data K is smooth)

5 A Negative Regularity Result for LMA In this section we prove the existence of nonsmooth viscosity solutions for the Levi Monge Amp`ere equation for n > 1: This generalizes a result of Pogorelov for the Monge Amp`ere equation (see [17]). In 1971 Pogorelov [34] showed that convex generalized solutions of the Monge Amp`ere equation det D 2 u D f .x/

(33)

in a domain ˝  RnC1 ; n > 1; need not be of class C 2 ; even if f is positive and smooth. Urbas in [44] proved that this absence of classical regularity is not confined

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to equations of Monge Amp`ere type, but in fact occurs for the m-th elementary symmetric functions in the Hessian and the equation of prescribed m-th curvature, where in each case m  3: Recently, in [18] we showed that a similar result holds for the Levi Monge Amp`ere equation. Let us remark the following facts: • It is well known [8] that for n D 1; if the prescribed Levi curvature K is smooth and positive, then every Lipschitz continuous viscosity solution of the prescribed Levi curvature equation is smooth. Moreover, if K is strictly monotone increasing with respect to u; then by the interior gradient bound in [27] every continuous viscosity solution is locally Lipschitz continuous • If the prescribed Gauss Levi curvature is smooth and positive, the smoothness of classical solutions is non trivial because the equation is non elliptic, and it was proved in [32] The main result of this section is Theorem 16, where we show the existence of a viscosity solution of the Dirichlet problem in a small ball B" for the prescribed Gauss Levi curvature equation (34) with appropriate boundary data of class C 1;12=n and by using suitable comparable functions of Pogorelov type to show that the solution cannot have better regularity than this in the interior. The low degree of regularity of the boundary data is necessary for our proof and by using our technique we cannot obtain (nor do we expect) a similar result for boundary data of class C 1;1˛ for ˛ > 1  2=n:

5.1 Notation and Main Theorem Let ˝  R2nC1 be open. The function u W ˝ ! R is a classical solution to the K-prescribed Levi curvature equation if u 2 C 2 .˝/ and satisfies the pde L .K; u/ WD det A.Du; D 2 u/  K.; u/ F .Du/ D 0;

(34)

for all  2 ˝, where Du and D 2 u denote the gradient and Hessian of u in all the variables x; y; t, respectively, nC2

F .Du/ D 2n

.1 C jDuj2 / 2 ; 1 C .@t u/2

and the matrix A.Du; D 2 u/ is an n  n matrix defined as follows. For ` D 1; ; n, let a` WD a` .Du/ D

@y` u  @x` u @t u ; 1 C .@t u/2

b` WD b` .Du/ D

@x` u  @y` u @t u ; 1 C .@t u/2

(35)

On the Levi Monge-Amp`ere Equation

189

and let a be the column vector with components a` , and b the column vector with components b` . Let ˙ be the n  .2n C 1/ complex matrix defined by ˙ D .In ; iIn ; a  ib/; where In is the n  n identity matrix. Then the matrix A.Du; D 2 u/ is defined by A.Du; D 2 u/ WD ˙ D 2 u ˙N T :

(36)

Equation (34) geometrically means that the hypersurface Mu , graph of the solution u, has Levi curvature equal K, agreeing to let Mu WD fz 2 CnC1 W z D .x C iy; t C iu.x; y; t//; .x; y; t/ 2 ˝g: Let Br D Euclidean ball in R2nC1 centered at the origin with radius r and assume K 2 C 1 .B1  R/ is strictly positive and s 7! K. ; s/ is increasing. The main result of this section is Theorem 16. There exists r 2 .0; 1/ and a pseudoconvex function u 2 Lip.BNr / solving det A.Du; D 2 u/ D K.; u/ F .Du/ in

Br ;

in the weak viscosity sense, and such that • u 2 C 1 .Br / if n D 1 • u2 6 C 1 .Br / if n D 2, • u 62 C 1;ˇ for any ˇ > 1  n2 ; if n  2. Remark 9. In every dimension n  1 : if u were C 2;˛ then u would be C 1 . Remark 10. Comments on the case n D 2 • In the case n D 2 the classical Pogorelov counterexample for det D 2 u D f does not hold, and you have a C 1;˛ regularity theory for convex viscosity solutions. • In the case n D 2 for every K smooth and positive, we will build a Lipschitz continuous viscosity solutions of the LMA equation such that u 62 C 1 .Br / In order to guarantee the existence of a viscosity solution on BR we argue as follows • We fix 0 < R < 1 such that sup K
0, only depending on r and supBR jD'j, such that L .K; u / > 0 in Br ;

for every >  :

(39)

5.3 Existence Result Lemma 11. If 0 < r < R < 1 and ' 2 C 2 .B1 / is a convex function such that L .K; '/  0 in B1 , then the Dirichlet problem

On the Levi Monge-Amp`ere Equation

L .K; u/ D 0; in Br ;

191

u D ' on @Br

(40)

has a viscosity solution u 2 Lip .Br / satisfying kukL1 .Br / C kukLip .Br /  C

(41)

where C only depends on r, k'kL1 .BR / , kD'kL1 .BR / , and kDKkL1 .BR / . Remark 11. C is independent of kD 2 'kL1 Proof. Let u D '  d be the function given by the previous Lemma 10 with

>  . Then u 2 C 2 .Br / and is a classical subsolution to L .K; u/ D 0 in Br . Moreover, u D ' on @Br . On the other hand, since L .K; '/  0 in B1 , ' is a classical supersolution to L .K; u/ D 0 in Br . Then, the Dirichlet problem has a viscosity solution u 2 C.Br / verifying u  u  ' in Br . Hence supBr juj  supBr j'j C r. By Lemma 10 supBr jDu j  C with C only depending on r and supBR jD'j, by the interior gradient estimates in [13], we can conclude that u 2 Lip.Br / with kukLip.Br / only depending on r, k'kL1 .BR / ; kD'kL1 .BR / , and kDKkL1 .BR / . u t

5.4 Stability of Viscosity Solutions It is a well known fact that the notion of viscosity solutions is stable with respect to uniform convergence, however we shall prove this stability property for our equation for reader convenience. Lemma 12. Let K" uniformly converge to K as " goes to zero and let fu" g be a sequence of viscosity solutions for the problem L .K" ; u" / D 0 which uniform converges to u as " goes to zero, then u is a viscosity solution of the problem L .K; u/ D 0. Proof. We need to prove that u is either a viscosity sub-solution and super-solution. First we prove that u is a viscosity sub-solution. Let then 0 2 ˝ and let ' be a C 2 function in a neighborhood of 0 such that ' touches u from above at 0 . We can choose a sequence " ! 0 , as " approaches zero, such that ' touches u" from above at " . Since u" is a viscosity sub-solution for L .K" ; u" / D 0, then it holds A.D'; D 2 '/." /  0 and L .K" ; u" /." /  0: Passing to the limit, as " goes to zero, we get A.D'; D 2 '/.0 /  0 and L .K; u/.0 /  0: Therefore u is a viscosity sub-solution for L .K; u/ D 0. Now, let 0 2 ˝ and let ' be a C 2 function in a neighborhood of 0 such that ' touches u from below at 0 . We can choose a sequence " ! 0 , as " approaches zero, such that ' touches u" from below at " :

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In this situation, since u" is a viscosity super-solution for L .K" ; u" / D 0, we have to distinguish two cases. • if for all " > 0 A.D'; D 2 '/." /  0, then we have L .K" ; u" /." /  0: Again passing to the limit as " goes to zero, we get A.D'; D 2 '/.0 /  0 and L .K" ; u/.0 /  0: Therefore u is a viscosity super-solution for L .K; u/ D 0. • there exists a " > 0 such that A.D'; D 2 '/." / is not positive semidefinite. Three situations can occur passing to the limit: (i) A.D'; D 2 '/.0 / > 0, (ii) A.D'; D 2 '/.0 / is not positive semidefinite. (iii) A.D'; D 2 '/.0 / has at least one null eigenvalue In (i) we can choose " > 0 small enough such that A.D'; D 2 '/." / > 0 and we have again the previous situation. In (ii) u is a viscosity super-solution for L .K; u/ D 0 at 0 by definition. In (iii), since det A.D'; D 2 '/.0 / D 0 and K; F > 0; we have L .K; '/.0 / D K.0 ; '/F .D'/ < 0: Therefore u is a viscosity super-solution for L .K; u/ D 0:

t u

5.5 Proof of Theorem 16 The case n D 1 is contained in Theorem 6. Hence, we assume n  2: We denote by x D .x1 ; x 0 /; x 0 D .x2 ; : : : ; xn /, y D .y1 ; y 0 /; y 0 D .y2 ; : : : ; yn / with x; y 2 Rn , and  D .x; y; t/ is a point in R2nC1 . We fix R 20; 1Πsuch that sup K
0, and independent of x1 ; y1 and t. From (36) we then obtain 1 0 2 2  Dxy  0 Dxx C B A.D ; D 2  / D ˙ @ D 2  D 2  0 A ˙N T yx

0

yy

0

0

On the Levi Monge-Amp`ere Equation

193

and since the first column and the first row of this matrix are null vectors we get det A.D ; D 2  / D 0. Therefore: L .K;  / D K. ;  /F .D / < 0

in B1 ;

8  20; 1Œ:

(43)

Thus, applying Lemma 40, the Dirichlet problem L .K; u/ D 0

in Br ;

u D 

on @Br ;

with  20; 1Πand 0 < r < R, has a viscosity solution u such that k u kL1 .Br / C k u kLip .Br /  C.r; ; M / with C.r; ; M / depending on  only through C. /WDk kL1 .Br /CkD kL1 .Br / . On the other hand, an elementary computation shows that C. /  8M . Then, we can choose C.r; ; M / independent of , and so k u kL1 .Br / C k u kLip .Br /  C.r; M /:

(44)

We now define a supersolution. Let w .x; y/ D w .x1 ; x 0 ; y1 ; y 0 / WD .r 2 C x12 C y12 /. C jx 0 j2 C jy 0 j2 /˛ ;

(45)

where ˛ D 1  n1 ; and  ./ D  .x; y; t/ WD M w .x; y/; with M a positive constant that will be determined in a moment. Since  ./ D  .x; y; t/ WD 2M. C jx 0 j2 C jy 0 j2 /˛ ; we have 0     ; in B1 : Claim. We claim that, if 0 < r < R, we can fix M D M.r/ such that L .K;

/

> 0 in Br ;

8  20; r 2 Œ:

(46)

Assuming this claim for a moment, we can use the Comparison Principle to compare u with  and  . Indeed, by the claim,  and  are, respectively, classical supersolution and subsolution to L .K; u/ D 0 in Br . On the other hand    in B1 , in particular, on @Br . Thus, by the Comparison Principle, 

 u   

in Br ; 8 20; r 2 Œ:

(47)

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The uniform estimate (44) implies the existence of a sequence j & 0 such that .uj /j 2N uniformly converges to a viscosity solution u 2 Lip.Br / to the Dirichlet problem L .K; u/ D 0 in Br ;

u D 0 on @Br :

Moreover, from (47), we get 0

 u  0

in Br :

In particular M r 2 jx2 j2˛  u.0; x2 ; 0; : : : ; 0/  2M jx2 j2˛ :

(48)

These inequalities imply: • u … C 1 ; if 2˛ D 1 .i.e. n D 2/ • u … C 1;ˇ ; for every ˇ > 2˛  1 D 1 

2 n

if 2˛ > 1 .i.e. n > 2/:

The first statement is trivial. For the second one we only have to remark that if 2˛ > 1, then @x2 u.0; 0; : : : ; 0/ D 0 D u.0; 0; : : : ; 0/ so that, if u would be C 1;ˇ , with ˇ > 2˛  1, we would have u.0; x2 ; : : : ; 0/  C jx2 j1Cˇ for a suitable C > 0 and for every x2 sufficiently small. Hence, by the first inequality in (48), it would be ˇ  2˛  1, a contradiction. Proof (Proof of the claim). Elementary direct computations show that  jDw j2 D4 .x12 C y12 /. C jx 0 j2 C jy 0 j2 /2˛ C ˛ 2 .jx 0 j2 C jy 0 j2 /.r 2 C x12 C y12 /2 . C jx 0 j2 C jy 0 j2 /2.˛1/



and det A.Dw ; D 2 w / D 22n f

(49)

with f D ˛ n .r 2 C x12 C y12 /n2

r 2 .˛ 1  C jx 0 j2 C jy 0 j2 / C ˛ 1 .jx 0 j2 C jy 0 j2 / : . C jx 0 j2 C jy 0 j2 /

Then, for every  20; r 2 Œ, jDw j2  22˛C3 r 4˛C2

in Br ;

(50)

On the Levi Monge-Amp`ere Equation

195

and f  ˛ n r 2.n1/ Keeping in mind that



in Br :

(51)

D M w is independent of t, we get

det A.D  ; D 2 F .D  /

/

D

.2M /nf .1 C M 2 jDw j2 /

nC2 2

:

Therefore from (51) and (50), we obtain det A.D  ; D 2 F .D  /

/



.2M˛/n r 2.n1/ .1 C 22˛C3 r 4˛C2 M 2 /

nC2 2

in Br :

(52)

Choosing M D 2˛.3=2/ r 2˛1 , the right hand side of (52) equals .2˛/n 2.3=2/nC1 r n 2

nC2 2

D

C.n/ 1 > ; n r 2Rn

if r < .2C.n//1=n R:

Then from (42) we obtain L .K;

/

D det A.D > F .D

/

;D



2

/

 K. ;

1  K. ; 2Rn

/



 /F .D

/

> 0:

This proves the claim (46) and completes the proof of the theorem.

t u

6 Symmetry Results The study of surfaces in the Euclidean space with either constant Gauss curvature or constant mean curvature received in the past a great amount of attention. In 1899 Liebmann [26] proved that the spheres are the only compact surfaces in R3 with constant Gauss curvature. In 1952 S¨uss [41] extended the Liebmann result showing that a compact convex hypersurface in the Euclidean space must be a sphere, provided that for some j the j -th elementary symmetric polynomial in the principal curvatures is constant. In 1954 Hsiung [22] proved that the “convexity” assumption can be relaxed to the “star-shapedness” one. The proofs of the above results are based on the Minkowski formulae. A breakthrough for this sort of problems was made by Alexandrov [1] in 1956; who proved that the sphere is the only compact hypersurface embedded into the Euclidean space with constant mean curvature. Alexandrov method is completely different from the Liebmann-S¨uss method, and is based on the moving plane technique, on the interior maximum principle for elliptic

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equations and on the boundary maximum principle of Hopf type for uniformly elliptic equations. In 1978 Reilly [37] obtained another proof of the Alexandrov theorem combining the Minkowski formulae with some new elegant arguments. In a joint paper with Lanconelli [31] we proved a strong comparison principle, leading to symmetry theorems for domains with constant curvatures. Precisely: Theorem 17. Let D CnC1 be a strictly j -pseudoconvex domain with connected boundary, 1  j  n: Let BR .z0 / D be a tangent sphere to @D at some point .j / p 2 @D: If K@D .z/ is the j -th Levi curvature of @D at z 2 @D and .j /

K@D .z/  1=Rj ;

8z 2 @D;

then D D BR .z0 /: In [31] we also proved that if ˝ is a bounded domain of CnC1 ; with boundary a real hypersurface of class C 2 ; then the j -th Levi curvature of @˝ at z D .z1 ; : : : ; znC1 / 2 @˝ writes in term of defining function f of ˝ D ff .z/ < 0g as 1 1 .j / K@˝ .z/ D    n j@f jj C2 j for all j D 1; : : : ; n, where j@f j D

X

(53)

1 fi j C1 fi1 ;i j C1 C C C :: C A : fij C1 ;i j C1

(54)

qP

nC1 2 j D1 jfj j

0

0 Bf B i1 .i1 ; ;ij C1 / .f / D det B B :: @: fij C1 2

.i1 ; ;ij C1 / .f /

1i1 < 0 such that D D BR .˛/ ‹ If yes : c D

 1 n R

>0Š

Remark 12. The boundedness assumption is necessary. Example 7 (Cylinders with constant and positive Gauss-Levi curvature). Let us now consider the cylinder CR D f.z1 ; ; znC1 / 2 CnC1 W .Re z1 /2 C C .Re znC1 /2 < R2 g: A defining function of CR is f .z/ D .Re z1 /2 C C .Re znC1 /2  R2 : The Gauss-Levi curvature is: K@CR .p/ D

1 ; 8p 2 @CR : .2R/n

6.2 Ingredients of Alexandrov Method Alexandrov proved that every direction is a direction of symmetry by using the following ingredients: 1. The prescribed mean curvature is invariant with respect to hyperplane reflections. 2. The equation is uniformly elliptic and the strong comparison principle holds. 3. Boundary comparison principle.

6.3 An Obstacle to the Boundary Comparison Principle In general sums of squares of H¨ormander vector fields do not satisfy Hopf Lemma, as the following example shows

On the Levi Monge-Amp`ere Equation

199

Example 8 (Hounie, Lanconelli). In R2 consider L D X 2 C Y 2 X Dy

@ ; @x

Y D

@ @ @ y ) ŒX; Y  D  : @y @x @x

The function u.x; y/ D y 4  6.x C y 2 =2/2 satisfies 1. Lu D 0 2. u < 0 in the half plane x  0 except the origin and u.0; 0/ D 0: Then the restriction of u to the half plane x  0 has a strict maximum at the origin 3. ru.0; 0/ D 0: Hence, L does not satisfy Hopf Lemma.

6.4 Strong Comparison Principle Even if the prescribed Levi curvature equations are not elliptic at any point, in [31] we proved the following strong comparison principle. Theorem 19 (STRONG COMPARISON PRINCIPLE). Let D and D 0 be strictly mpseudoconvex domains of CnC1 with connected boundaries. If 1. D 0 D and @D \ @D 0 ¤ ¿ .m/ .m/ 2. K@D 0 .p 0 /  K@D .p/; 8 p 2 @D and p 0 2 @D 0 then D 0 D D. Proof. We just give a sketch of the proof and we suggest the interested reader to see the original proof in [31] for details. We locally write @D and @D 0 as the graph of u and v 2 C 2 ; respectively. Then, the function w D u  v  0 in an open set ˝ and with the notation of Sect. 4.7.1 L m .w/ D

2n X

aij .x/Zi Zj w C bj .x/Zj w  0;

i;j D1

where 1. the matrix aij .x/ D aji .x/ and it is positive definite as a quadratic form at every point x 2 ˝ 2. there is a couple .i; j / such that ŒZi ; Zj  ¤ 0: t u We then conclude by the following Maximum principle of Bony type Theorem 20 (MAXIMUM PRINCIPLE). Let ˝0 ˝ be an open and connected set. Let w 2 C 2 .˝0 ; R/ such that

200

A. Montanari



L m w  0 in ˝0 w  0 in ˝0

Then w < 0 in ˝0 or w 0 in ˝0 : By the strong comparison principle we get the proof of the identification result Theorem 17.

6.5 Null Lagrangian Property for Elementary Symmetric Functions in the Eigenvalues of the Complex Hessian Matrix Given a Hermitian matrix A; let j .A/ be the j -th elementary symmetric function in the eigenvalues of A. Let us recall that if A is the .nC1/.nC1/ matrix with eigenvalues 1 ; : : : ; nC1 then X j .A/ D

i1 ij : 1i1