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Necessary and Sufficient Conditions for the Solvability of the L p Dirichlet Problem on Lipschitz Domains

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Abstract

We study the homogeneous elliptic systems of order \(2\ell\) with real constant coefficients on Lipschitz domains in\(\mathbb{R}^n\), \(n\ge 4\). For any fixed p  >  2, we show that a reverse Hölder condition with exponent p is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in L p. We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the L p Dirichlet problem for \(n\ge 4\) and \(2-\epsilon< p<\frac{2(n-1)}{n-3} +\epsilon\). The range of p is known to be sharp if \(\ell\ge 2\) and \(4\le n\le 2\ell + 1\). For the polyharmonic equation, the sharp range of p is also found in the case n  =  6, 7 if \(\ell=2\), and \(n=2\ell+2\) if \(\ell\ge 3\).

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References

  1. Adams R.A. (1975) Sobolev Spaces. Academic, New York

    MATH  Google Scholar 

  2. Caffarelli L.A., Peral I. (1998) On W 1,p estimates for elliptic equations in divergence form. Comm. Pure Appl. Math. 51, 1–21

    Article  MathSciNet  MATH  Google Scholar 

  3. Dahlberg B. (1977) On estimates for harmonic measure. Arch. Rat. Mech. Anal. 65, 273–288

    Article  MathSciNet  Google Scholar 

  4. Dahlberg B. (1979) On the Poisson integral for Lipschitz and C 1 domains. Studia Math. 66, 13–24

    MathSciNet  MATH  Google Scholar 

  5. Dahlberg B., Kenig C. (1987) Hardy spaces and the Neumann problem in L p for Laplace’s equation in Lipschitz domains. Ann. Math. 125, 437–466

    Article  MathSciNet  Google Scholar 

  6. Dahlberg, B., Kenig, C. L p estimates for the three-dimensional systems of elastostatics on Lipschitz domains. In: Lecture Notes in Pure and Applied Mathematics, vol.122, pp. 621–634 (1990)

  7. Dahlberg B., Kenig C., Pipher J., Verchota G. (1997) Area integral estimates for higher order elliptic equations and systems. Ann. Inst. Fourier (Grenoble) 47, 1425–1461

    MathSciNet  MATH  Google Scholar 

  8. Dahlberg B., Kenig C., Verchota G. (1986) The Dirichlet problem for the biharmonic equation in a Lipschitz domain. Ann. Inst. Fourier (Grenoble) 36, 109–135

    MathSciNet  MATH  Google Scholar 

  9. Dahlberg B., Kenig C., Verchota G. (1988) Boundary value problems for the systems of elastostatics in Lipschitz domains. Duke Math. J. 57, 795–818

    Article  MathSciNet  MATH  Google Scholar 

  10. Douglis A., Nirenberg L. (1955) Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math. 8, 503–538

    Article  MathSciNet  MATH  Google Scholar 

  11. Fabes, E. Layer potential methods for boundary value problems on Lipschitz domains. In: Lecture Notes in Mathematics, vol. 1344, pp. 55–80 (1988)

  12. Fabes E., Kenig C., Verchota G. (1988) Boundary value problems for the Stokes system on Lipschitz domains. Duke Math. J. 57, 769–793

    Article  MathSciNet  MATH  Google Scholar 

  13. Gao W. (1991) Boundary value problems on Lipschitz domains for general elliptic systems. J. Funct. Anal. 95, 377–399

    Article  MathSciNet  MATH  Google Scholar 

  14. Giaquinta, M. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. In: Annals of Mathematica Studies, vol. 105. Princeton University Press, Princeton (1983)

  15. Gilbarg D., Trudinger N.S. (1983) Elliptic Partial Differential Equations of Second Order. 2nd edition. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  16. Hörmander L. (1983) The Analysis of Linear Partial Differential Operators I. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  17. Jerison D., Kenig C. (1981) The Neumann problem in Lipschitz domains. Bull. Am. Math. Soc. 4, 203–207

    Article  MathSciNet  MATH  Google Scholar 

  18. John F. (1955) Planes Waves and Spherical Means. Interscience Publisher, Inc., New York

    Google Scholar 

  19. Kenig, C. Elliptic boundary value problems on Lipschitz domains. In: Beijing Lectures in Harmonic Analysis, Annals of Mathematical Studies, vol. 112, pp. 131–183 (1986)

  20. Kenig, C. Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. In: CBMS Regional Conference Series in Mathematics, vol. 83 AMS, Providence (1994)

  21. Kozlov V.A., Maz’ya V.G. (1992) On the spectrum of the operator pencil generated by the Dirichlet problem in a cone. Math. USSR Sbornik 73, 27–48

    Article  MathSciNet  Google Scholar 

  22. Maz’ya, V.G. Behavior of solutions to the Dirichlet problem for the biharmonic operator at a boundary point. In: Equadiff IV, Lecture Notes in Mathematics, vol. 703, pp. 250–262 (1979)

  23. Maz’ya, V.G. Unsolved problems connected with the Wiener criterion. In: The Legacy of Norbert Wiener: A Centennial Symposium, Proceedings of the Symposia in Pure Math., vol. 60 pp. 199–208 (1997)

  24. Maz’ya V.G. (2002) The Wiener test for higher order elliptic equations. Duke Math. J. 115, 479–512

    Article  MathSciNet  MATH  Google Scholar 

  25. Maz’ya, V.G., Donchev, T. On the Wiener regularity of a boundary point for the polyharmonic operator. Dokl. Bolg. AN 36 (2), 177–179 (1983); English transl. in Am. Math. Soc. Transl. 137, 53–55 (1987)

  26. Maz’ya V.G., Nazarov S.A., Plameneskii B.A. (1985) On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone. Math. USSR Sbornik 50, 415–437

    Article  MATH  Google Scholar 

  27. Mitrea D., Mitrea M. (2001) General second order, strongly elliptic systems in low dimensional nonsmooth manifold. Contemporary Math. 277, 61–86

    MathSciNet  Google Scholar 

  28. Pipher J., Verchota G. (1992) The Dirichlet problem in L p for the biharmonic equation on Lipschitz domains. Am. J. Math. 114, 923–972

    Article  MathSciNet  MATH  Google Scholar 

  29. Pipher J., Verchota G. (1993) A maximum principle for biharmonic functions in Lipschitz and C 1 domains. Commen. Math. Helv. 68, 385–414

    Article  MathSciNet  MATH  Google Scholar 

  30. Pipher J., Verchota G. (1995) Dilation invariant estimates and the boundary Garding inequality for higher order elliptic operators. Ann. Math. 142, 1–38

    Article  MathSciNet  MATH  Google Scholar 

  31. Pipher J., Verchota G. (1995) Maximum principle for the polyharmonic equation on Lipschitz domains. Potential Anal. 4, 615–636

    Article  MathSciNet  MATH  Google Scholar 

  32. Shen Z. (1995) Resolvent estimates in L p for elliptic systems in Lipschitz domains. J. Funct. Anal. 133, 224–251

    Article  MathSciNet  MATH  Google Scholar 

  33. Shen Z. (1995) A note on the Dirichlet problem for the Stokes system in Lipschitz domains. Proc. Am. Math. Soc. 123, 801–811

    Article  MATH  Google Scholar 

  34. Shen Z. (2003) Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains. Am. J. Math. 125, 1079–1115

    Article  MATH  Google Scholar 

  35. Shen, Z. Weighted estimates for elliptic systems in Lipschitz domains. Indiana Univ. Math. J. (in press)

  36. Shen Z. (2006) The L p Dirichlet problem for elliptic systems on Lipschitz domains. Math. Res. Lett. 13, 143–159

    MathSciNet  MATH  Google Scholar 

  37. Stein E.M. (1970) Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton

    MATH  Google Scholar 

  38. Verchota G. (1984) Layer potentials and regularity for the Dirichlet problem for Laplace’s equation. J. Funct. Anal. 59, 572–611

    Article  MathSciNet  MATH  Google Scholar 

  39. Verchota G. (1987) The Dirichlet problem for the biharmonic equation in C 1 domains. Indiana Univ. Math. J. 36, 867–895

    Article  MathSciNet  MATH  Google Scholar 

  40. Verchota G. (1990) The Dirichlet problem for the polyharmonic equation in Lipschitz domains. Indiana Univ. Math. J. 39, 671–702

    Article  MathSciNet  MATH  Google Scholar 

  41. Verchota, G. Potentials for the Dirichlet problem in Lipschitz domains. Potential Theory-ICPT94, 167–187

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  1. Department of Mathematics, University of Kentucky, Lexington, 40506, KY, USA

    Zhongwei Shen

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  1. Zhongwei Shen
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Correspondence to Zhongwei Shen.

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Research supported in part by the NSF.

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Shen, Z. Necessary and Sufficient Conditions for the Solvability of the L p Dirichlet Problem on Lipschitz Domains. Math. Ann. 336, 697–725 (2006). https://doi.org/10.1007/s00208-006-0022-x

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