Advertisement

Potential Analysis

, Volume 43, Issue 1, pp 79–95 | Cite as

Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces with Small Lipschitz Constant

  • Johan Thim
Article
  • 77 Downloads

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.

Keywords

Single layer potentials Lipschitz surface Singular integrals Weighted spaces Homogeneous Sobolev spaces 

Mathematics Subject Classification (2010)

45Exx 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19(3), 613–626 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations, Applied Mathematical Sciences, vol. 164. Springer, Berlin (2008)CrossRefGoogle Scholar
  3. 3.
    Kozlov, V., Maz’ya, V.: Differential Equations with Operator Coefficients. Springer, Berlin (1999)zbMATHCrossRefGoogle Scholar
  4. 4.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Kozlov, V., Thim, J., Turesson, B.O.: A Fixed Point Theorem in Locally Convex Spaces. Collect. Math. 61(2), 223–239 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Kozlov, V., Thim, J., Turesson, B.O.: Riesz potential equations in local L p-spaces. Complex Var. Elliptic Equ. 54(2), 125–151 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Mazya, V.G.: Sobolev Spaces. Springer, Berlin (1985)Google Scholar
  10. 10.
    Muckenhoupt, B.: Hardy’s inequality with weights. Studia Math. 44, 31–38 (1972)zbMATHMathSciNetGoogle Scholar
  11. 11.
    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)MathSciNetGoogle Scholar
  12. 12.
    Riesz, M.: L’intégrale de Riemann-Liouville et le problème de Cauchy. Acta Math. 81(1), 1–222 (1949)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Rubin, B.: Fractional Integrals and Potentials. Addison Wesley Longman Limited, Harlow (1996)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Linköpings UniversitetLinköpingSweden

Personalised recommendations