Reconstruction of Solutions to a Generalized Moisil–Teodorescu System in Jordan Domains with Rectifiable Boundary

Abstract

In this paper we consider the problem of reconstructing solutions to a generalized Moisil-Teodorescu system in Jordan domains of R 3 with rectifiable boundary. In order to determine conditions for existence of solutions to the problem we embed the system in an appropriate generalized quaternionic setting.

Generalized Laplacian decomposition of vector fields on fractal surfaces

“…Then the integral on the right side of (3.1) exists and represents a continuous function in the whole R 3 (see [12,Theorem 2.3.9]). Hence, the functions F ± possess continuous extensions to the closures of the domains Ω± and they satisfy that F + Γ − F − Γ = f .…”

“…By properties of the Teodorescu operator (see [13,Theorem 4.17]), ψ θ D[F + ] = 0 and ψ θ D[F − ] = 0 in the domains Ω±, respectively. The uniqueness of F ± can be proved as in [12,Theorem 5.1. ].…”

“…In the case of a rectifiable surface Γ (the Lipschitz image of some bounded subset of R 2 ) these problems have been investigated in [12].…”