Source supports in electrostatics
The paper deals with the inverse source problem of electrostatic in a bounded and convex domain in \({\mathbb R}^2\) with compactly supported source, i.e., the problem of reconstructing the source term in the Poisson equation supplemented by the homogeneous Neumann boundary conditions, from the given Dirichlet boundary values. By comparing the dimensions, it is obvious that the knowledge of the Dirichlet data does not suffice to reconstruct uniquely the source function. Moreover, not even the support of the source can be determined. In the paper, the authors try to determine the smallest set that carries the source which is compatible with the given data. For this, the concept of the convex scattering support developed in the series of works by Kusiak and Sylvester is adopted for the case of electrostatics in a bounded domain. Namely, the authors introduce the notion of discoidal source support, i.e., the smallest set made up by the intersection of discs within the domain, which carries a source compatible with the given data. Numerical examples are provided in the case of the unit disk.
scientific article; zbMATH DE number 5343404