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Mapping Properties of Layer Potentials Associated with Higher-Order Elliptic Operators in Lipschitz Domains

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Topics in Operator Theory

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 203))

Abstract

The method of layer potentials has been applied with tremendous success in the treatment of boundary value problems for second-order differential operators for a very long time; the literature on this topic is enormous. By way of contrast this method is disproportionally underdeveloped in the case of higher-order operators; the difference between the higher-order and the second-order settings is striking in term of the scientific output. This paper presents new results which establish mapping properties of multiple layer potentials associated with higher-order elliptic operators in Lipschitz domains in ℝn.

This work was supported in part by the NSF Grant DMS 0547944 and by the Ruth Michler Prize from the Association of Women in Mathematics.

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References

  1. V. Adolfsson and J. Pipher, The inhomogeneous Dirichlet problem for Δ 2 in Lipschitz domains, J. Funct. Anal., 159 (1998), No. 1, 137–190.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Agmon, Multiple layer potentials and the Dirichlet problem for higher order elliptic equations in the plane. I., Comm. Pure Appl. Math., 10 (1957), 179-239.

    Google Scholar 

  3. A.P. Calderón, Commutators, singular integrals on Lipschitz curves and applications, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 85–96, Acad. Sci. Fennica, Helsinki, 1980.

    Google Scholar 

  4. J. Cohen, BMO estimates for biharmonic multiple layer potentials, Studia Math., 91 (1988), No. 2, 109–123.

    MATH  MathSciNet  Google Scholar 

  5. J. Cohen and J. Gosselin, The Dirichlet problem for the biharmonic equation in a C1 domain in the plane, Indiana Univ. Math. J., 32 (1983), No. 5, 635–685.

    Article  MATH  MathSciNet  Google Scholar 

  6. R.R. Coifman, A. McIntosh and Y. Meyer, L’intégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes lipschitziennes, Annals of Math., 116 (1982), 361–387.

    Article  MathSciNet  Google Scholar 

  7. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), No. 4, 569–645.

    Article  MATH  MathSciNet  Google Scholar 

  8. B.E. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal., 65 (1977), No. 3, 275–288.

    Article  MATH  MathSciNet  Google Scholar 

  9. B.E. Dahlberg, C.E. Kenig, J. Pipher and G.C. Verchota, Area integral estimates for higher order elliptic equations and systems, Ann. Inst. Fourier, (Grenoble) 47 (1997), No. 5, 1425–1461.

    MATH  MathSciNet  Google Scholar 

  10. B.E. Dahlberg and G.C. Verchota, Galerkin methods for the boundary integral equations of elliptic equations in nonsmooth domains, Harmonic analysis and partial differential equations (Boca Raton, FL, 1988), 39–60, Contemp. Math., 107, Amer. Math. Soc., Providence, RI, 1990.

    Google Scholar 

  11. E.B. Fabes, M. Jodeit Jr. and N.M. Rivière, Potential techniques for boundary value problems on C 1-domains, Acta Math., 141 (1978), No. 3–4, 165–186.

    Article  MATH  MathSciNet  Google Scholar 

  12. E.B. Fabes and C.E. Kenig, On the Hardy space H 1 of a C 1 domain, Ark. Mat. 19 (1981), No. 1, 1–22.

    Article  MATH  MathSciNet  Google Scholar 

  13. E.B. Fabes, O. Mendez, and M. Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains, J. Funct. Anal. 159 (1998), No. 2, 323–368.

    Article  MATH  MathSciNet  Google Scholar 

  14. C. Fefferman and E.M. Stein, H p spaces of several variables, Acta Math., 129 (1972), No. 3–4, 137–193.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal., Vol. 93 No. 1 (1990), 34–170.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series in Mathematics, Vol. 79, AMS, Providence, RI, 1991.

    Google Scholar 

  17. D. Goldberg, A local version of real Hardy spaces, Duke Math. J., 46 (1979), 27–42.

    Article  MATH  MathSciNet  Google Scholar 

  18. D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), No. 1, 161–219.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Jonsson and H. Wallin, Function spaces on subsets of ℝn, Math. Rep., Vol. 2, 1984.

    Google Scholar 

  20. C.E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, Vol. 83, AMS, Providence, RI, 1994.

    Google Scholar 

  21. S. Mayboroda and M. Mitrea, Green potential estimates and the Poisson problem on Lipschitz domains, preprint, 2005.

    Google Scholar 

  22. V. Maz’ya, M. Mitrea and T. Shaposhnikova, The Dirichlet problem in Lipschitz domains with boundary data in Besov spaces for higher order elliptic systems with rough coefficients, preprint, 2005.

    Google Scholar 

  23. M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds, J. Funct. Anal. 163 (1999), No. 2, 181–251.

    Article  MathSciNet  Google Scholar 

  24. J. Pipher and G. Verchota, The Dirichlet problem in L p for the biharmonic equation on Lipschitz domains, Amer. J. Math., 114 (1992), No. 5, 923–972.

    Article  MATH  MathSciNet  Google Scholar 

  25. J. Pipher and G.C. Verchota, Maximum principles for the polyharmonic equation on Lipschitz domains, Potential Anal., 4 (1995), No. 6, 615–636.

    Article  MATH  MathSciNet  Google Scholar 

  26. J. Pipher and G. Verchota, A maximum principle for biharmonic functions in Lipschitz and C 1 domains, Comment. Math. Helv., 68 (1993), No. 3, 385–414.

    Article  MATH  MathSciNet  Google Scholar 

  27. J. Pipher and G.C. Verchota, Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators, Ann. of Math., (2) 142 (1995), no. 1, 1–38.

    Google Scholar 

  28. T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Operators, de Gruyter, Berlin, New York, 1996.

    MATH  Google Scholar 

  29. V. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains, J. London Math. Soc., (2) 60 (1999), no. 1, 237–257.

    Google Scholar 

  30. Z. Shen, The L p Dirichlet Problem for Elliptic Systems on Lipschitz Domains, Math. Research Letters, 13 (2006), 143–159.

    MATH  Google Scholar 

  31. Z. Shen, Necessary and Sufficient Conditions for the Solvability of the L p Dirichlet Problem on Lipschitz Domains, to appear in Math. Ann., (2006).

    Google Scholar 

  32. Z. Shen, The L p Boundary Value Problems on Lipschitz Domains, preprint (2006).

    Google Scholar 

  33. E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970.

    Google Scholar 

  34. H. Triebel, Theory of function spaces. II, Monographs in Mathematics, 84. Birkhäuser Verlag, Basel, 1992. viii+370 pp.

    Google Scholar 

  35. G. Verchota, The biharmonic Neumann problem in Lipschitz domains, Acta Math., 194 (2005), 217–279.

    Article  MATH  MathSciNet  Google Scholar 

  36. G. Verchota, The Dirichlet problem for the biharmonic equation in C 1 domains, Indiana Univ. Math. J., 36 (1987), No. 4, 867–895.

    Article  MathSciNet  Google Scholar 

  37. G.C. Verchota, Potentials for the Dirichlet problem in Lipschitz domains, pp. 167–187 in Potential Theory — ICPT 94, de Gruyter, Berlin, 1996.

    Google Scholar 

  38. G. Verchota, The Dirichlet problem for the polyharmonic equation in Lipschitz domains, Indiana Univ. Math. J., 39 (1990), No. 3, 671–702.

    Article  MATH  MathSciNet  Google Scholar 

  39. G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal., 59 (1984), No. 3, 572–611.

    Article  MATH  MathSciNet  Google Scholar 

  40. H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), No. 1, 63–89.

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA, 01609-2280, USA

    Irina Mitrea

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  1. Irina Mitrea
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    Joseph A. Ball

  2. Department of Mathematics, College of William and Mary, P. O. Box 8795, Williamsburg, VA, 23187-8795, USA

    Vladimir Bolotnikov , Leiba Rodman  & Ilya M. Spitkovsky ,  & 

  3. Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093-0112

    J. William Helton

Additional information

Communicated by J.A. Ball.

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© 2010 Birkhäuser Verlag Basel/Switzerland

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Mitrea, I. (2010). Mapping Properties of Layer Potentials Associated with Higher-Order Elliptic Operators in Lipschitz Domains. In: Ball, J.A., Bolotnikov, V., Rodman, L., Spitkovsky, I.M., Helton, J.W. (eds) Topics in Operator Theory. Operator Theory: Advances and Applications, vol 203. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0161-0_15

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