Simple Layer Potentials on Lipschitz Surfaces: An Asymptotic Approach

Abstract

This work is devoted to the equation ˆ u(y) dS(y) = f(x), x ∈ S, (1) |x − y| N−1 S where S is the graph of a Lipschitz function ϕ on R N with small Lipschitz constant, and dS is the Euclidian surface measure. The integral in the left-hand side is referred to as a simple layer potential and f is a given function. The main objective is to find a solution u to this equation along with estimates for solutions near points on S. Our analysis is carried out in local L p-spaces and local Sobolev spaces, and the estimates are given in terms of seminorms. In Paper 1, we consider the case when S is a hyperplane. This gives rise to th