## Inverse source problems.(English)Zbl 0721.31002

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.]

### 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