ABSTRACT
The first section of this chapter deals with inverse problems in potential theory and places special emphasis on questions of existence, uniqueness and stability along with further development of efficient methods for solving them. As to the question of existence, we are unaware of any criterion providing its global solution. There are a number of the existence theorems "in the small" for inverse problems related to a body differing only slightly from a given one as it were. And even in that case the problems were not completely solved because of insufficient development of the theory of nonlinear equations capable of describing inverse problems. That is why, it is natural from the viewpoint of applications to preassume in most cases the existence of global solutions beforehand and pass to deeper study of the questions of uniqueness and stability. Quite often, solutions of inverse problems turn out to be nonunique, thus causing difficulties. It would be most interesting to learn about extra restrictions on solutions if we want to ensure their uniqueness. The main difficulty involved in proving uniqueness lies, as a rule, in the fact that the inverse problems of interest
are equivalent to integral equations of the first kind with the Urysohn-type kernel for which the usual ways of solving are unacceptable. The problem of uniqueness is intimately connected with the problem of stability of the inverse problem solutions. For the inverse problems in view, because they are stated by means of first kind equations, arbitrarily small perturbations of the right-hand side function may, generally speaking, be responded by a finite variation of a solution. The requirement of well-posedness necessitates imposing additional restrictions on the behavior of a solution.
