Single Layer Potentials on Surfaces with Small Lipschitz constant

Abstract

This paper considers to the equation [\int_{S} \frac{U(Q)}{|P-Q|^{N-1}} dS(Q) = F(P), P \in S,] where the surface S is the graph of a Lipschitz function \phi on R^N, which has a small Lipschitz constant. The integral in the left-hand side is the single layer potential corresponding to the Laplacian in R^{N+1}. Let \Lambda(r) be a Lipschitz constant of \phi on the ball centered at the origin with radius 2r. Our analysis is carried out in local L^p-spaces and local Sobolev spaces, where 1 < p < \infty, and results are presented in terms of \Lambda(r). Estimates of solutions to the equation are provided, which can be used to obtain knowledge about the behaviour of the solutions near a point on the surface. The estimates are given in terms of seminorms. Solutions are also shown to be unique if they are subject to certain growth conditions. Local estimates are provided and some applications are supplied.