Problem of conjugation of harmonic functions in three-dimensional domains with nonsmooth boundaries
The author deals with a problem that arises from the theory of continuum media: A finite number of disjoint, simply connected domains \(V_1, V_2,\dots, V_k\) in \(\mathbb R^3\) is given. These domains are bounded by closed surfaces \(S_1, S_2,\dots,S_k\) and each such surface contains a finite number of singular points (angular and conic points). The problem is to construct harmonic functions \(F_0, F_1, \dots, F_k\) such that \(F_j\) is harmonic in \(V_j\), \(j=1, 2,\dots, k\), and \(F_0\) is harmonic in \(\mathbb R^3\setminus \bigcup_{j=1}^k V_j\), and these functions must satisfy certain boundary conditions on the surfaces \(S_j\) (conditions of conjugation) of Dirichlet-Neumann-Robin type. The author investigates the solvability of the problem using single and double layer potentials.
scientific article; zbMATH DE number 1956429