On the boundary value problems of electro- and magnetostatics

Abstract

SynopsisThe classically well-known relation between the number of linearly independent solutions of the electro- and magnetostatic boundary value problems (harmonic Dirichlet and Neumann vector fields) and topological characteristics (genus and number of boundaries) of the underlying domain in 3-dimensional euclidean space is investigated in the framework of Hilbert space theory. It can be shown that this connection is still valid for a large class of domains with not necessarily smooth boundaries (segment property). As an application the inhomogeneous boundary value problems of electro- and magnetostatics are discussed.