Über zwei spezielle inverse Probleme der Potentialtheorie. (On two special inverse problems of potential theory) (Q1101849)

The paper deals with two inverse problems relative to the Newtonian potential, where the density \(\rho\) is known. The first problem consists in determining the shape of a domain by the values of its potential on the boundary. In the second problem the values of the gradient are known on the boundary. The existence is proved in a neighbourhood of a given configuration. At first the linearized problem is studied. Then the inverse function theorem is applied to this problem. There are relations to investigations of L. Lichtenstein (1931, 1933) and the Molodensky problem. See also \textit{W. Keller} [Gerlands Beitr. Geophys. 96, No.3/4, 186-196 (1987)].