Some special inverse problems for the Laplace equation and the Helmholtz equation (Q800580)

The paper deals with a special inverse source problem relative to the Laplace equation. An unknown mass distribution (measure) concentrated on a given domain is to be determined from the boundary values of its Newtonian potential or from the boundary values of its gradient. In Theorem 1 we prove that the second problem can be reduced to the first one. There exist infinitely many positive mass distributions satisfying the condition under consideration. Therefore an equivalence relation on the set of all positive mass distributions is introduced. In earlier papers the first author studied this equivalence relation from a systematic point of view [see: Die moderne Potentialtheorie als Grundlage des inversen Problems in der Geophysik, Geod. geophys. Veröff., R. III, H.45, 15-95 (1980; Zbl 0448.31006)]. In order to determine the unknown mass distribution uniquely additional conditions are necessary. Here we study the case in which the density of a volume distribution is a harmonic function and prove concrete formulas for a ball. The analytic continuation plays an important role and generalizes results of C. Neumann (1909) and G. Herglotz (1914). Further, we prove that an identification problem for the Helmholtz equation can be transformed into the special inverse problem considered.