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 . 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 with respect to the surface measure. Furthermore we show that the second derivative of the solution is in 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.