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Multi-Layer Potentials and Boundary Problems pp 1–19Cite as

Introduction

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2063))

Abstract

One of the main goals of the present monograph is to develop the framework of a theory for the multiple layer (or multi-layer, for short) potential operators arising in the treatment of boundary value problems associated with a higher-order, matrix-valued (complex) constant coefficient, elliptic differential operator \(\begin{array}{rcl} Lu =\sum\limits_{\vert \alpha \vert =\vert \beta \vert =m}{\partial }^{\alpha }{A}_{ \alpha \beta }\,{\partial }^{\beta }u& &\end{array}\) (where \(m \in \mathbb{N}\)) in a Lipschitz domain \(\Omega \subset {\mathbb{R}}^{n}\).

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References

  1. V. Adolfsson, J. Pipher, The inhomogeneous Dirichlet problem for Δ 2 in Lipschitz domains. J. Funct. Anal. 159(1), 137–190 (1998)

    Article  MathSciNet  MATH  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, 179–239 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II. Comm. Pure Appl. Math. 17, 35–92 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  4. R.M. Brown, W. Hu, Boundary value problems for higher order parabolic equations. Trans. Am. Math. Soc. 353(2), 809–838 (2001) (electronic)

    Google Scholar 

  5. A.P. Calderón, Cauchy integrals on Lipschitz curves and related operators. Proc. Natl. Acad. Sci. USA 74(4), 1324–1327 (1977)

    Article  MATH  Google Scholar 

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

    Google Scholar 

  7. A.P. Calderón, C.P. Calderón, E. Fabes, M. Jodeit, N.M. Rivière, Applications of the Cauchy integral on Lipschitz curves. Bull. Am. Math. Soc. 84(2), 287–290 (1978)

    Article  MATH  Google Scholar 

  8. D.-C. Chang, S.G. Krantz, E.M. Stein, H p theory on a smooth domain in ℝ N and elliptic boundary value problems. J. Funct. Anal. 114(2), 286–347 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. D.-C. Chang, S.G. Krantz, E.M. Stein, Hardy spaces and elliptic boundary value problems, in The Madison Symposium on Complex Analysis, Madison, WI, 1991. Contemporary Mathematics, vol. 137 (American Mathematical Society, Providence, 1992), pp. 119–131

    Google Scholar 

  10. J. Cohen, BMO estimates for biharmonic multiple layer potentials. Studia Math. 91(2), 109–123 (1988)

    MathSciNet  MATH  Google Scholar 

  11. J. Cohen, J. Gosselin, The Dirichlet problem for the biharmonic equation in a C 1 domain in the plane. Indiana Univ. Math. J. 32(5), 635–685 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. R.R. Coifman, A. McIntosh, Y. Meyer, Lintégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes lipschitziennes. Ann. Math. 116, 361–387 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. B.E. Dahlberg, C.E. Kenig, G.C. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain. Ann. Inst. Fourier (Grenoble) 36(3), 109–135 (1986)

    Google Scholar 

  14. L. Diomeda, B. Lisena, Boundary problems for the biharmonic operator in a square with L p-data. SIAM J. Math. Anal. 15(6), 1153–1168 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Diomeda, B. Lisena, Dirichlet and oblique derivative problems for the Δ 2-operator in a plane sector. (Italian) Rend. Math. (7) 2(1), 189–217 (1982)

    Google Scholar 

  16. E.B. Fabes, M. Jodeit Jr., N.M. Rivière, Potential techniques for boundary value problems on C 1-domains. Acta Math. 141(3–4), 165–186 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Fabes, O. Mendez, M. Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159(2), 323–368 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. E.B. Fabes, C.E. Kenig, On the Hardy space H 1 of a C 1 domain. Ark. Math. 19(1), 1–22 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Franke, T. Runst, Regular elliptic boundary value problems in Besov-Triebel-Lizorkin spaces. Math. Nachr. 174, 113–149 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. L.I. Hedberg, Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem. Acta Math. 147, 237–264 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. D. Jerison, C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130(1), 161–219 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Jonsson, H. Wallin, Function Spaces on Subsets ofℝ n (Harwood Academic, New York, 1984)

    Google Scholar 

  23. C.E. Kenig, in Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. CBMS Regional Conference Series in Mathematics, vol. 83 (American Mathematical Society, Providence, 1994)

    Google Scholar 

  24. J. Kilty, Z. Shen, A bilinear estimate for biharmonic functions in Lipschitz domains. Math. Ann. 349, 367-394 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. V.A. Kozlov, V.G. Maz’ya, J. Rossmann, in Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Mathematical Surveys and Monographs, vol. 85 (American Mathematical Society, Providence, 2001)

    Google Scholar 

  26. S. Mayboroda, M. Mitrea, Sharp estimates for Green potentials on non-smooth domains. Math. Res. Lett. 11(4), 481-492 (2004)

    MathSciNet  MATH  Google Scholar 

  27. S. Mayboroda, V. Maz’ya, Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation. Inventiones. Math. 175(2), 287–334 (2009)

    Google Scholar 

  28. V. Maz’ya, M. Mitrea, T. Shaposhnikova, The Dirichlet problem in Lipschitz domains with boundary data in Besov spaces for higher order elliptic systems with rough coefficients. J. d’Analyse Math. 110(1), 167–239 (2010)

    Google Scholar 

  29. I. Mitrea, M. Mitrea, M. Wright, Optimal estimates for the inhomogeneous problem for the bi-Laplacian in three-dimensional Lipschitz domains. J. Math. Sci. 172(1), 24–134 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. I. Mitrea, G. Verchota, The Spectrum of Biharmonic Layer Potentials on Curvilinear Polygons (in preparation)

    Google Scholar 

  31. M. Mitrea, M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem. J. Funct. Anal. 176(1), 1–79 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. J. Nečas, in Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris (Academia, Éditeurs, Prague, 1967)

    MATH  Google Scholar 

  33. J. Pipher, G.C. Verchota, Maximum principles for the polyharmonic equation on Lipschitz domains. Potential Anal. 4 (6), 615–636 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Pipher, G. Verchota, A maximum principle for biharmonic functions in Lipschitz and C 1 domains. Comment. Math. Helv. 68(3), 385–414 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Pipher, G.C. Verchota, Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators. Ann. Math. (2) 142(1), 1–38 (1995)

    Google Scholar 

  36. R. Pisani, M. Tucci, Dirichlet problem for the operator Δ N in a plane sector. (Italian) Rend. Acad. Sci. Fis. Mat. Napoli (4) 48, 317–327 (1982) (1980/1981)

    Google Scholar 

  37. N. Rivière, Some open questions. Proc. Symp. Pure Appl. Math. XXXV(Part 1), xvii (1978)

    Google Scholar 

  38. T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Operators (de Gruyter, Berlin, 1996)

    Google Scholar 

  39. V. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. (2) 60(1), 237–257 (1999)

    Google Scholar 

  40. R. Selvaggi, I. Sisto, An existence theorem for the Dirichlet problem with respect to the operator Δ 2 in certain domains of class C 1. (Italian) Boll. Un. Mat. Ital. B (5) 18 (2), 473–483 (1981)

    Google Scholar 

  41. J.K. Seo, Regularity for solutions of biharmonic equation on Lipschitz domain. Bull. Korean Math. Soc. 33(1), 17–28 (1996)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  43. Z. Shen, The L p Dirichlet problem for elliptic systems on Lipschitz domains. Math. Res. Lett. 13(1), 143-159 (2006)

    MathSciNet  MATH  Google Scholar 

  44. Z. Shen, The L p Boundary value problems on Lipschitz domains. Adv. Math. 216, 212–254 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  45. G. Verchota, The Dirichlet problem for the biharmonic equation in C 1 domains. Indiana Univ. Math. J. 36(4), 867–895 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  46. G. Verchota, The Dirichlet problem for the polyharmonic equation in Lipschitz domains. Indiana Univ. Math. J. 39(3), 671–702 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  49. H. Whitney, Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36(1), 63–89 (1934)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first named author was supported in part by the US NSF Grant DMS 1201736.

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Authors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, PA, USA

    Irina Mitrea

  2. Department of Mathematics, University of Missouri, Columbia, MO, USA

    Marius Mitrea

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  1. Irina Mitrea
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Mitrea, I., Mitrea, M. (2013). Introduction. In: Multi-Layer Potentials and Boundary Problems. Lecture Notes in Mathematics, vol 2063. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32666-0_1

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