Inverse source problems. (English) Zbl 0721.31002

Isakov, Victor

Mathematical Surveys and Monographs, 34. Providence, RI: American Mathematical Society (AMS). xiv, 193 p. (1990).

The book is a review of the results on the inverse problem of potential theory (Chapters 2-5) supplemented by a discussion of some inverse problems for parabolic and hyperbolic equations (Chapters 6, 7). In Chapter 1 some auxiliary material is given. In Chapter 2 the results are described. In Chapter 3 uniqueness theorems are given. They are based on the idea of P. Novikov (1938). In Chapter 4 singularities of the exterior potentials are studied: relations between smoothness of the continuation of the potential across \(\Gamma\) and smoothness of \(\Gamma\), provided that the source and the coefficients of the operator are smooth. Some results of V. K. Ivanov are given. Chapter 5 deals with the existence of the mass distribution which generates a given harmonic function as an exterior potential. This is largely an open problem. Local existence results are described. An example of nonexistence is given. Chapters 6 and 7 deal with some inverse problems of parabolic and hyperbolic equations. Typically the extra data are given on the lateral surface of the cylinder or on its upper base.
[A detailed comment of the reviewer concerning omissions of other work and inaccuracies in the text of this book may be obtained from the editor on special request.]

Reviewer: A.G.Ramm (Manhatten)

MSC:

31B20	Boundary value and inverse problems for harmonic functions in higher dimensions
35R30	Inverse problems for PDEs
31-02	Research exposition (monographs, survey articles) pertaining to potential theory
31B05	Harmonic, subharmonic, superharmonic functions in higher dimensions
35Kxx	Parabolic equations and parabolic systems
35Lxx	Hyperbolic equations and hyperbolic systems
35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations

Keywords:

uniqueness; exterior potential

Citations:

Zbl 0632.35075; Zbl 0636.35066; Zbl 0602.35015; Zbl 0604.31004; Zbl 0608.35077; Zbl 0603.45012; Zbl 0424.31007; Zbl 0415.31004; Zbl 0645.35098