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Zeitschrift für Analysis und ihre Anwendungen

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Volume 27, Issue 3, 2008, pp. 253–281
DOI: 10.4171/ZAA/1354

Published online: 2008-09-30

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

Anja Schlömerkemper[1]

(1) Mathematik in den Naturwissenschaften, Leipzig, Germany

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 $d\geq 2$. 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.

Keywords: Poisson equation with transition condition, integral representations of solutions, derivatives of single layer potentials, regularization of potentials, magnetostatics

Schlömerkemper Anja: About Solutions of Poisson's Equation with Transition Condition in Non-Smooth Domains. Z. Anal. Anwend. 27 (2008), 253-281. doi: 10.4171/ZAA/1354