Single layer potentials on surfaces with small Lipschitz constants

This paper considers to the equation ∫ S U ( Q ) | P − Q | N − 1 d S ( Q ) = F ( P ) , P ∈ S , where the surface S is the graph of a Lipschitz function φ on R N , which has a small Lipschitz constant. The integral on the left-hand side is the single layer potential corresponding to the Laplacian in R N + 1 . Let Λ ( r ) be the Lipschitz constant of φ 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 < ∞ , and results are presented in terms of Λ. Estimates of solutions to the equation are provided, which can be used to obtain knowledge about the behavior of the solutions near a point on the surface. These 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.