Abstract
This article considers two weight estimates for the single layer potential — corresponding to the Laplace operator in R N+1 — on Lipschitz surfaces with small Lipschitz constant. We present conditions on the weights to obtain solvability and uniqueness results in weighted Lebesgue spaces and weighted homogeneous Sobolev spaces, where the weights are assumed to be radial and doubling. In the case when the weights are additionally assumed to be differentiable almost everywhere, simplified conditions in terms of the logarithmic derivative are presented, and as an application, we prove that the operator corresponding to the single layer potential in question is an isomorphism between certain weighted spaces of the type mentioned above. Furthermore, we consider several explicit weight functions. In particular, we present results for power exponential weights which generalize known results for the case when the single layer potential is reduced to a Riesz potential, which is the case when the Lipschitz surface is given by a hyperplane.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19(3), 613–626 (1988)
Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations, Applied Mathematical Sciences, vol. 164. Springer, Berlin (2008)
Kozlov, V., Maz’ya, V.: Differential Equations with Operator Coefficients. Springer, Berlin (1999)
Kozlov, V., Thim, J., Turesson, B.O.: Single Layer Potentials on Surfaces with Small Lipschitz Constant. J. Math. Anal. Appl. 418(2), 676–712 (2014)
Kozlov, V., Thim, J., Turesson, B.O.: A Fixed Point Theorem in Locally Convex Spaces. Collect. Math. 61(2), 223–239 (2010)
Kozlov, V., Thim, J., Turesson, B.O.: Riesz potential equations in local L p-spaces. Complex Var. Elliptic Equ. 54(2), 125–151 (2009)
Kozlov, V., Wendland, W., Goldberg, H.: The behaviour of elastic fields and boundary integral Mellin techniques near conical points. Math. Nachr. 180, 95–133 (1996)
Mayboroda, S., Mitrea, M.: Layer potentials and boundary value problems for Laplacian in Lipschitz domains with data in quasi-Banach Besov spaces. Ann. Mat. Pura Appl. (4) 185(2), 155–187 (2006)
Mazya, V.G.: Sobolev Spaces. Springer, Berlin (1985)
Muckenhoupt, B.: Hardy’s inequality with weights. Studia Math. 44, 31–38 (1972)
Riesz, M.: L’intégrale de Riemann-Liouville et le problème de Cauchy pour l’équation des ondes. Bull. Soc. Math. France 67, 153–170 (1939)
Riesz, M.: L’intégrale de Riemann-Liouville et le problème de Cauchy. Acta Math. 81(1), 1–222 (1949)
Rubin, B.: Fractional Integrals and Potentials. Addison Wesley Longman Limited, Harlow (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Thim, J. Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces with Small Lipschitz Constant. Potential Anal 43, 79–95 (2015). https://doi.org/10.1007/s11118-015-9464-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-015-9464-7