About Solutions of Poisson's Equation with Transition Condition in Non-Smooth Domains

Abstract

Starting from integral representations of solutions of Poisson's equation with transition condition, we study the first and second derivatives of these solutions for all dimensions $d\ge 2$. This involves derivatives of single layer potentials and Newton potentials, which we regularize smoothly. On smooth parts of the boundary of the non-smooth domains under consideration, the convergence of the first derivative of the solution is uniform; this is well known in the literature for regularizations using a sharp cut-off by balls. Close to corners etc.\ we prove convergence in $L^1$ with respect to the surface measure. Furthermore we show that the second derivative of the solution is in $L^1$ on the bulk.

The interface problem studied in this article is obtained from the stationary Maxwell equations in magnetostatics and was initiated by work on magnetic forces.