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Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces with Small Lipschitz Constant
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2015 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 1, 79-95 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Springer Verlag (Germany) , 2015. Vol. 43, no 1, 79-95 p.
Keyword [en]
Single layer potentials; Lipschitz surface; Singular integrals; Weighted spaces; Homogeneous Sobolev spaces
National Category
URN: urn:nbn:se:liu:diva-120220DOI: 10.1007/s11118-015-9464-7ISI: 000357132800004OAI: diva2:842683
Available from: 2015-07-21 Created: 2015-07-20 Last updated: 2015-07-21

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Thim, Johan
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Mathematics and Applied MathematicsFaculty of Science & Engineering
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