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On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries


Authors: L. Escauriaza, E. B. Fabes and G. Verchota
Journal: Proc. Amer. Math. Soc. 115 (1992), 1069-1076
MSC: Primary 35B65; Secondary 35J15
DOI: https://doi.org/10.1090/S0002-9939-1992-1092919-1
MathSciNet review: 1092919
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Abstract: We show that if $ u$ is a weak solution to $ \operatorname{div} (A\nabla u) = 0$ on an open set $ \Omega $ containing a Lipschitz domain $ D$, where $ A = kI{\chi _D} + I{\chi _{\Omega /D}}(k > 0,k \ne 1)$. Then, the nontangential maximal function of the gradient of $ u$ lies in $ {L^2}(\partial D)$.


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  • [1] R. R. Coifman, A. McIntosh, and Y. Meyer, Líntegrale de Cauchy definit un operateau borne sur $ {L^2}$ pour les courbes lioschitziennes, Ann. of Math. (2) 116 (1982), 361-387. MR 672839 (84m:42027)
  • [2] E. B. Fabes, M. Jodeit, Jr., and N. M. Riviere, Potential techniques for boundary value problems on $ {C^1}$ domains, Acta Math. 141 (1978), 165-186. MR 501367 (80b:31006)
  • [3] D. S. Jerison and C. E. Kenig, The Neuman problem in Lipschitz domains, Bull. Amer. Math. Soc. 4 (1981), 203-207. MR 598688 (84a:35064)
  • [4] J. Neças, Les methodes directes en theorie des equations elliptiques, Academia, Prague, 1967.
  • [5] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. MR 0290095 (44:7280)
  • [6] G. C. Verchota, Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), 572-611. MR 769382 (86e:35038)
  • [7] -, Layer potentials and boundary value problems for Laplace's equation on Lipschitz domains, Thesis, Univ. of Minnesota, June 1982, Appendix.
  • [8] M. Vogelius and A. Friedman, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence, Arch. Rational Mech. Anal. 105 (1989), 299-326. MR 973245 (90c:35198)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1092919-1
Article copyright: © Copyright 1992 American Mathematical Society

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