Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems

1. Front Matter

   Pages i-xii

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2. Basics of Carleman Estimates

   • Mourad Bellassoued, Masahiro Yamamoto

   Pages 1-50

3. Basic Tools of Riemannian Geometry

   • Mourad Bellassoued, Masahiro Yamamoto

   Pages 51-62

4. Well-Posedness and Regularity for the Wave Equation with Variable Coefficients

   • Mourad Bellassoued, Masahiro Yamamoto

   Pages 63-80

5. Carleman Estimate for the Wave Equation on a Riemannian Manifold

   • Mourad Bellassoued, Masahiro Yamamoto

   Pages 81-110

6. Inverse Problems for Wave Equations on a Riemannian Manifold

   • Mourad Bellassoued, Masahiro Yamamoto

   Pages 111-166

7. Realization of the Convexity of the Weight Function

   • Mourad Bellassoued, Masahiro Yamamoto

   Pages 167-182

8. Carleman Estimates for Some Thermoelasticity Systems

   • Mourad Bellassoued, Masahiro Yamamoto

   Pages 183-210

9. Inverse Heat Source Problem for the Thermoelasticity System

   • Mourad Bellassoued, Masahiro Yamamoto

   Pages 211-223

10. Inverse Problem for a Hyperbolic Equation with a Finite Set of Boundary Data

    • Mourad Bellassoued, Masahiro Yamamoto

    Pages 225-239

11. Supplementary Research Problems

    • Mourad Bellassoued, Masahiro Yamamoto

    Pages 241-251

12. Back Matter

    Pages 253-260

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This book is a self-contained account of the method based on Carleman estimates for inverse problems of determining spatially varying functions of differential equations of the hyperbolic type by non-overdetermining data of solutions. The formulation is different from that of Dirichlet-to-Neumann maps and can often prove the global uniqueness and Lipschitz stability even with a single measurement. These types of inverse problems include coefficient inverse problems of determining physical parameters in inhomogeneous media that appear in many applications related to electromagnetism, elasticity, and related phenomena. Although the methodology was created in 1981 by Bukhgeim and Klibanov, its comprehensive development has been accomplished only recently. In spite of the wide applicability of the method, there are few monographs focusing on combined accounts of Carleman estimates and applications to inverse problems. The aim in this book is to fill that gap. The basic tool is Carleman estimates, the theory of which has been established within a very general framework, so that the method using Carleman estimates for inverse problems is misunderstood as being very difficult. The main purpose of the book is to provide an accessible approach to the methodology. To accomplish that goal, the authors include a direct derivation of Carleman estimates, the derivation being based essentially on elementary calculus working flexibly for various equations. Because the inverse problem depends heavily on respective equations, too general and abstract an approach may not be balanced. Thus a direct and concrete means was chosen not only because it is friendly to readers but also is much more relevant. By practical necessity, there is surely a wide range of inverse problems and the method delineated here can solve them. The intention is for readers to learn that method and then apply it to solving new inverse problems.

• Cauchy problem
• Riemannian geometry
• coefficient invers problem
• conditional stability
• hand-made derivation of Carleman estimate
• partial differential equations

“The book under review is devoted to Carleman estimates and their applications to get stability estimates for inverse problems of determining spatially varying coefficients or source terms in hyperbolic systems with finitely many measurements. … This is a nice book on applications of Carleman estimates to inverse problems for hyperbolic systems.” (Dinh Nho Hào, zbMATH 1412.35002, 2019)

• Department of Mathematics, ENIT—LAMSIN, University of Tunis El Manar, Tunis, Tunisia

  Mourad Bellassoued

• Department of Mathematical Sciences, The University of Tokyo, Tokyo, Japan

  Masahiro Yamamoto

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