An Asymptotic Approach to Simple Layer Potentials on Lipschitz Surfaces
(English)Manuscript (Other academic)
This paper is devoted to the equation
where the surface S is the graph of a Lipschitz function φ on RN, which has a small Lipschitz constant. The integral in the left-hand side is the simple layer potential of the Laplacian in RN+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 Lp-spaces and local Sobolev spaces, where 1 < p < ∞, and results are presented in terms of Λ. Estimates of solutions to the equation for the single layer potential are provided, which can be used to obtain knowledge about 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. Some examples are given when specific assumptions are placed on the function Λ.
Lipschitz surfaces, Simple Layer potentials, Singular integral operators
IdentifiersURN: urn:nbn:se:liu:diva-16538OAI: oai:DiVA.org:liu-16538DiVA: diva2:158261