Abstract
In this paper we discuss the solvability of boundary value problems for the Laplace operator on Lipschitz domains with arbitrary topology via boundary layers. An application to hydrodynamics is included.
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[CoMcMe] R. Coifman, A. McIntosh and Y. Meyer,L'intégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes Lipschitziennes, Ann. of Math.116 (1982), 361–387.
[Da] J. E. Dahlberg,On the Poisson integral for Lipschitz and C 1 domains, Studia Math.66 (1979), 13–24.
[DaKe] B. Dahlberg and C. Kenig,Hardy spaces and the L p-Neumann problem for Laplace's equation in a Lipschitz domain, Ann. of Math.125 (1987), 437–465.
[DaLi] R. Dautray and J.-L. Lions,Mathematical Analysis and Numerical Methods for Science and Technology, vol. 1–4, Springer-Verlag, Berlin, Heidelberg, 1990.
[EsFaVe] L. Escauriaza, E. Fabes and G. Verchota,On a regularity method for weak solutions to transmission problems with internal Lipschitz boundary, Proc. Amer. Math. Soc.115 (1992), 1069–1076.
[Fa] E. B. Fabes,Layer potential methods for boundary value problems on Lipschitz domains, Potential Theory, Surveys and Problems, J. Král et al eds., Springer-Verlag Lecture Notes in Math., No. 1344, 1988, pp. 55–80.
[FaJoRi] E. Fabes, M. Jodeit and N. Rivière,Potential techniques for boundary value problems on C 1 domains, Acta Math.141 (1978), 165–186.
[FaMeMi] E. Fabes, O. Mendez and M. Mitrea,Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains, preprint, (1997).
[Kel] W. T. B. Kelvin,Mathematical and Physical Papers, Cambridge University Press, 1910.
[JeKe] D. Jerison and C. Kenig,The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc.4 (1981), 203–207.
[MeCo] Y. Meyer, R. R. Coifman,Opérateurs multilinéaires Hermann, Paris, 1991.
[MiD] D. Mitrea,Layer potential operators and boundary value problems for differential forms on Lipschitz domains, Ph. D. Thesis, University of Minnesota, 1996.
[MiMiPi] D. Mitrea, M. Mitrea and J. Pipher, Vector potential theory on non-smooth domains in ℝ3 and applications to electro-magnetic scattering, J. of Fourier Analysis and Applications3 (1997), 131–192.
[MiMiTa] D. Mitrea, M. Mitrea and M. Taylor,Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds, preprint, (1997).
[MiM] M. Mitrea,The method of layer potentials in electro-magnetic scattering theory on non-smooth domains, Duke Math. J.77 (1995), 111–133.
[Ne] J. Nečas,Les méthodes directes en théorie des équations élliptique, Academia, Prague, 1967.
[Pi] R. PicardAn elementary proof for a compact embedding result in generalized electromagnetic theory, Math. Z.187 (1984), 151–164.
[Re] F. Rellich,Darstellung der Eigenwerte von Δu+λu durch ein Randintegral, Math. Z.46 (1940), 635–646.
[Tu] A. W. Tucker,A boundary-value theorem for harmonic tensors, Bull. Amer. Math. Soc.47 (1941), 714.
[Ve] G. Verchota,Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains, J. Funct. Anal.59 (1984), 572–611.
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Partially supported by a UMC Research Board grant and UMC Summer Research Fellowship
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Mitrea, D. The method of layer potentials for non-smooth domains with arbitrary topology. Integr equ oper theory 29, 320–338 (1997). https://doi.org/10.1007/BF01320705
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DOI: https://doi.org/10.1007/BF01320705