Inverse Problems for Partial Differential Equations (Inverse & ill-posed problems series) 9067643580, 9789067643580

Table of contents :
Preface
Contents
Chapter 1. Auxiliary information from functional analysis and theory of differential equations
Chapter 2. The weak approximation method
Chapter 3. Identification problems for parabolic equations with Cauchy data
Chapter 4. The identification of the source function for a system of composite type and parabolic equation. The behavior of the problem’s solution under t —> + ∞
Chapter 5. The problem of determining the coefficient in a parabolic equation and some properties of its solution
Chapter 6. Two unknown coefficients of a parabolic type equation
Chapter 7. Some inverse boundary value problems
Bibliography

Citation preview

INVERSE AND ILL-POSED PROBLEMS SERIES

Inverse Problems for Partial Differential Equations

Also available in the Inverse and Ill-Posed Problems Series: Method of Spectral Mappings in the Inverse Problem Theory V.Yurko Theory of Linear Optimization I.I. Eremin Integral Geometry and Inverse Problems for Kinetic Equations A.Kh.Amirov Computer Modelling in Tomography and Ill-Posed Problems MM. Lavrent'ev, S.M. Zerkal and O.ETrofimov A n Introduction to Identification Problems via Functional Analysis A. Lorenzi Coefficient Inverse Problems for Parabolic Type Equations and Their Application P.G. Danilaev Inverse Problems for Kinetic and Other Evolution Equations Yu.E Anikonov Inverse Problems ofWave Processes A.S. Blagoveshchenskii Uniqueness Problems for Degenerating Equations and Nondassical Problems S.P. Shishatskii, A. Asanov and ER. Atamanov Uniqueness Questions in Reconstruction of Multidimensional Tomography-Type Projection Data V.P. Golubyatnikov Monte Carlo Method for Solving Inverse Problems of Radiation Transfer V.S.Antyufeev Introduction to theTheory of Inverse Problems A.L Bukhgeim Identification Problems ofWave Phenomena Theory and Numerics S.I. Kabanikhin and A. Lorenzi Inverse Problems of Electromagnetic Geophysical Fields P.S. Martyshko Composite Type Equations and Inverse Problems A.I. Kozhanov Inverse Problems of Vibrational Spectroscopy A.G.Yagola, I.V. Kochikov, G.M. Kuramshina andYuA Pentin Elements of theTheory of Inverse Problems A.M. Denisw Volterra Equations and Inverse Problems A.L Bughgeim Small Parameter Method in Multidimensional Inverse Problems A.S. Barashkov

Regularization, Uniqueness and Existence of Volterra Equations of the First Kind A. Asanov Methods for Solution of Nonlinear Operator Equations V.P.Tanana Inverse and Ill-Posed Sources Problems Yu.EAnikonov, BA. Bubnov and G.N. Erokhin Methods for Solving Operator Equations V.P.Tanana Nondassical and Inverse Problems for Pseudoparabolic Equations A.Asanov and ER.Atamanov Formulas in Inverse and Ill-Posed Problems Yu.EAnikonov Inverse Logarithmic Potential Problem V.G. Cherednichenko Multidimensional Inverse and Ill-Posed Problems for Differential Equations Yu.EAnikonov Ill-Posed Problems with A Priori Information V.V.Vasin andA.LAgeev Integral Geometry ofTensor Fields VA. Sharafutdinov Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin

INVERSE AND ILL-POSED PROBLEMS SERIES

Inverse Problems for Partial Differential Equations

Yu.Ya. Belov

IIIVSPIII UTRECHT · BOSTON · KÖLN · TOKYO

2002

VSP P.O. Box 346 3700 AH Zeist The Netherlands

Tel: +31 30 692 5790 Fax: +31 30 693 2081 [email protected] www.vsppub.com

© V S P 2002 First published in 2002 ISBN 90-6764-358-0 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands by Ridderprint bv, Ridderkerk.

Preface Included under the subject heading of inverse problems for differential equations we would consider problems of the determination of the input data for differential equations (coefficients), and also the determination problems of boundary and initial conditions by using the additional information about the solutions of these equations. Many important questions concerning elastic shift, electromagnetic oscillations, and diffusion processes lead to inverse problems. The area of basic research into such problems is constantly growing: Alifanov et al., 1988; Anikonov, 1995, 1997; Bukhgeim, 1988; Denisov, 1994; Lavrent'ev et al., 1980, 1986; Prilepko, 1991; Romanov, 1972, 1987, 1990, 1992, 1993; Romanov and Kabanikhin, 1991, 1994; Yakhno, 1990. This monograph is devoted to identification problems of coefficients in equations of mathematical physics. We investigate the existence and uniqueness of the solutions for identification coefficient problems in parabolic and hyperbolic equations and equation systems of composite type. We study the problems with the Cauchy data and equations in which the Fourier transform with respect to the chosen variable is supposed to occur. On the basis of the Fourier transform and overdetermination conditions the original inverse problems are reduced to direct integro-differential problems, that is as a rule, to nonlinear equations. We prove the solvability of such direct problems. The solutions of the original inverse problems are presented in the explicit form through the corresponding solutions of the direct problem. Differential properties of the solutions for the original direct problems and their behavior under great values of time are studied on the basis of solution properties for direct problems. The identification problems with one or two unknown coefficients are also investigated. In the case of the identification problem for a source function we analyze the convergence of the splitting method under the spli iAir ·^ of manydimensional inverse problems into inverse problems of lower dimension.

Yu. Ya. Belov. Inverse problems for partial differential equations For initial boundary value conditions we study linear and nonlinear parabolic equations. We investigate the identification problem of a source function for the hyperbolic equation having a small parameter under the higher derivative with respect to time. The material of the book was used for delivering special lecture courses at the faculty of mathematics of Krasnoyarsk State University. The author would like to express his gratitude to Professor Yu.E. Anikonov the team-work who promoted the publication of this book, and to Professor V.S. Belonosov and Professor V.N. Shelukhin for consultations on special questions. Great appreciation is expressed to Doctor T.N. Shipina for her help in preparation of the book manuscript for publication. This work was supported by the Russian Foundation for Fundamental Research, grant number 99-01-00993.

Contents

Chapter 1. Auxiliary information from functional analysis and theory of differential equations 1.1. The basic notions and notations 1.2. Inequalities 1.3. Some concepts and theorems of functional analysis 1.4. Linear partial differential equation of the first order 1.5. The maximum principle for parabolic equations of second order . Chapter 2. The weak approximation method 2.1. Examples reducing to the concept of the weak approximation method 2.2. General formulation of the weak approximation method 2.3. Two theorems 2.4. An example. The linear partial differential equation Chapter 3. Identification problems for parabolic equations with Cauchy data 3.1. The unknown source function 3.2. The unknown lowest coefficient 3.3. The unknown coefficient to the first order derivative 3.4. An unknown coefficient to the time derivative 3.5. Inverse problem for a semilinear parabolic equation 3.6. Equations of Burgers type 3.7. The splitting of one many-dimensional inverse problem into problems of lower dimension

1 1 2 3 5 5 11 11 18 21 27

33 33 49 57 66 77 85 96

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Yu. Ya. Belov. Inverse problems for partial differential equations

Chapter 4. The identification of the source function for a system of composite type and parabolic equation. The behavior of the problem's solution under t ->• +00 4.1. The statement of the problem 4.2. Theorems of existence and uniqueness "on the whole" 4.3. The behavior of solution by t ->· +00 4.4. Stationary problem 4.5. Convergence to the stationary problem solution 4.6. Unique solvability of the problem of identifying the source function for a parabolic equation 4.7. The stabilization of the solution to the inverse problem

103 104 107 115 127 130 133 136

Chapter 5. The problem of determining the coefficient in a parabolic equation and some properties of its solution 5.1. Statement of the problem 5.2. Theorems of existence and uniqueness "on the whole" 5.3. Properties of solution under t +00

139 139 141 146

Chapter 6. Two unknown coefficients of a parabolic type equation 6.1. Uniform boundary conditions 6.2. Inhomogeneous conditions of over-determination 6.3. Input data of the special form

159 160 171 175

Chapter 7. Some inverse boundary value problems 7.1. Unknown source function 7.2. Nonlinear heat equation 7.3. Hyperbolic equation with small parameter

183 183 185 193

Bibliography

201

Chapter 1. Auxiliary information from functional analysis and theory of differential equations

1.1.

THE BASIC NOTIONS A N D NOTATIONS

Let Ω be a bounded domain in Euclidean space En, and π > 1. By χ = (si, •••,xn) we denote a point in En. Symbol dü denotes the boundary of a domain Ω. By Qt we denote a cylinder (0,Τ) χ Ω. Let a = (αϊ,..., a n ) be a multi - index, ati, where i = l , n , are nonnegative integers and |a| = αϊ + ... + an. Let = d^/dx?, D« = . k k Below we denote by C (ü) (C (Ω)) the space of functions that in Ω (Ω) have all continuous derivatives up to order k including: Ck (Ω) (Ck (Ω)) = {/|£>°/are continuous ίηΩ(Ω),|α| < k}. If we introduce in Ck (Ω) the norm 11/11= ^ max \Daf(x)\ 0 < | α | < Α :

then the space Ck (Ω) is a Banach space. If k = 0 instead of C? (Ω) we shall use €(Ω). L p (Ω) where 1 < ρ < oo is a Banach space which consists of the classes of functions defined in Ω and integrating with the p-th degree according to the Lebesque definition. The functions which are equal everywhere belong

2

Yu. Ya. Belov. Inverse problems for partial differential equations

to one and the same class. The norm in this space is defined by the formula 11/11 = ( / n \f(x)\p dx) 1 ^, ρ < oo. The norm in Ζ-οο(Ω) is defined by the formula ||/||oo = esssup|/(x)|. xen Further W p (ü) = { / | u 6l V ϋ ρ (Ω),|α| < 1} is the Sobolev space \' ° with the norm I ^ ^ Ι , ρ > 1 - const. is the closure \N