Inverse problems of potential theory and flows in porous media with time-dependent free boundary

Abstract

It is known that fluid flows in a homogeneous porous medium with time-dependent free boundary have the property that integrals over the fluid domain of harmonic functions vary linearly in time [1]. This property relates these flows to the classical inverse problem of Newtonian potentials [2]. It is possible to apply this relation to construction of finite or infinite sets of distinct domains homeomorphic to a ball in R d and having the same potential outside. Considering flows in a non-homogeneous medium, one obtains examples of non-uniqueness of solutions of the inverse potential problem for arbitrary charge density and space dielectric permeability.