Single layer potentials on surfaces with small Lipschitz constants
2014 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 418, no 2, 676-712 p.Article in journal (Refereed) Published
This paper considers to the equation integral(S) U(Q)/vertical bar P - Q vertical bar(N-1) dS(Q) = F(P), P is an element of S, where the surface S is the graph of a Lipschitz function phi 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 RN+1. Let Lambda(r) be the 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 less than p less than infinity, and results are presented in terms of Lambda. 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.
Place, publisher, year, edition, pages
Elsevier , 2014. Vol. 418, no 2, 676-712 p.
Single layer potential; Lipschitz surface; Local estimates
IdentifiersURN: urn:nbn:se:liu:diva-108788DOI: 10.1016/j.jmaa.2014.04.013ISI: 000336887700007OAI: oai:DiVA.org:liu-108788DiVA: diva2:732924