Abstract
In this article, we introduce higher order conjugate Poisson and Poisson kernels, which are higher order analogues of the classical conjugate Poisson and Poisson kernels, as well as the polyharmonic fundamental solutions, and define multi-layer potentials in terms of the Poisson field and the polyharmonic fundamental solutions, in which the former is formed by the higher order conjugate Poisson and the Poisson kernels. Then by the multi-layer potentials, we solve three classes of boundary value problems (i.e., Dirichlet, Neumann and regularity problems) with Lp boundary data for polyharmonic equations in Lipschitz domains and give integral representation (or potential) solutions of these problems.
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Andrews G E, Askey R, Roy R. Special Functions. Cambridge: Cambridge University Press, 1999
Barton A, Hofmann S, Mayboroda S. Square function estimates on layer potentials for higher-order elliptic equations. Math Nachr, 2017, 290: 2459–2511
Barton A, Hofmann S, Mayboroda S. The Neumann problem for higher order elliptic equations with symmetric coefficients. Math Ann, 2018, 371: 297–336
Begehr H, Du J, Wang Y. A Dirichlet problem for polyharmonic functions. Ann Mat Pura Appl (4), 2008, 187: 435–457
Begehr H, Du Z, Wang N. Dirichlet problems for inhomogeneous complex mixed-partial differential equations of higher order in the unit disc: New view. Oper Theory Adv Appl, 2009, 205: 101–128
Begehr H, Gaertner E. A Dirichlet problem for the inhomogeneous polyharmonic equations in the upper half plane. Georgian Math J, 2007, 14: 33–52
Begehr H, Schmersau D. The Schwarz problem for polyanalytic functions. Z Anal Anwend, 2005, 24: 341–351
Calderón A. Cauchy integrals on Lipschitz curves and related operators. Proc Natl Acad Sci USA, 1977, 74: 1324–1327
Coifman R, Mcintosh A, Meyer Y. L’integrale de Cauchy definite un operateur borne sur L2 pour les courbes lipschitziennes. Ann of Math (2), 1982, 116: 361–387
Dahlberg B. Estimates of harmonic measure. Arch Ration Mech Anal, 1977, 65: 275–288
Dahlberg B. On the Poisson integral for Lipschitz and C1 domains. Studia Math, 1979, 66: 13–24
Dahlberg B. Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain. Studia Math, 1980, 67: 297–314
Dahlberg B, Kenig C. Harmonic Analysis and Partial Differential Equations. Göteborg: University of Göteborg, 1985
Dahlberg B, Kenig C. Hardy spaces and the Neumann problem in Lp for Laplace’s equation in Lipschitz domains. Ann of Math (2), 1987, 125: 437–465
Dahlberg B, Kenig C, Pipher J, et al. Area integral estimates for higher order elliptic equations and systems. Ann inst Fourier (Grenoble), 1997, 47: 1425–1461
Dahlberg B, Kenig C, Verchota G. The Dirichlet problem for the biharmonic equation in a Lipschitz domain. Ann inst Fourier (Grenoble), 1986, 36: 109–135
Dindos M. Hardy Spaces and Potential Theory on C1 Domains in Riemannian Manifolds. Memoirs of the American Mathematical Society. Providence: Amer Math Soc, 2008
Du J, Wang Y. On boundary value problems of polyanalytic functions on the real axis. Complex Var Theory Appl, 2003, 48: 527–542
Du Z. Boundary value problems for higher order complex differential equations. Doctoral Dissertation. Berlin: Freie Universität Berlin, 2008
Du Z, Guo G, Wang N. Decompositions of functions and Dirichlet problems in the unit disc. J Math Anal Appl, 2010, 362: 1–16
Du Z, Kou K, Wang J. Lp polyharmonic Dirichlet problems in regular domains I: The unit disc. Complex Var Elliptic Equ, 2013, 58: 1387–1405
Du Z, Qian T, Wang J. Lp polyharmonic Dirichlet problems in regular domains II: The upper-half plane. J Differential Equations, 2012, 252: 1789–1812
Du Z, Qian T, Wang J. Lp polyharmonic Dirichlet problems in regular domains IV: The upper-half space. J Differential Equations, 2013, 255: 779–795
Du Z, Qian T, Wang J. Lp polyharmonic Dirichlet problems in regular domains III: The unit ball. Complex Var Elliptic Equ, 2014, 59: 947–965
Fabes E, Jodeit Jr M, Riviere N. Potential techniques for boundary value problems on C1 domains. Acta Math, 1978, 141: 165–186
Folland G. Introduction to Partial Differential Equations. Princeton: Princeton University Press, 1995
Gaertner E. Basic complex boundary value problems in the upper half plane. Doctoral Dissertation. Berlin: Freie Universitäat Berlin, 2006
Grafakos L. Modern Fourier Analysis. Graduate Texts in Mathematics, vol. 250. Berlin: Springer, 2009
Helms L. Potential Theory. London: Springer, 2009
Jerison D, Kenig C. An identity with applications to harmonic measure. Bull Amer Math Soc, 1980, 2: 447–451
Jerison D, Kenig C. The Dirichlet problem in non-smooth domains. Ann of Math (2), 1981, 113: 367–382
Jerison J, Kenig C. The Neumann problem in Lipschitz domains. Bull Amer Math Soc, 1981, 4: 203–207
Kenig C. Harmonic Techniques for Second Order Elliptic Boundary Value Problems. CBMS Regional Conference Series in Mathematics, No. 83. Providence: Amer Math Soc, 1994
Lanzani L, Capogna L, Brown R. The mixed problem in Lp for some two-dimensional Lipschitz domains. Math Ann, 2008, 342: 91–124
Lanzani L, Shen Z. On the Robin boundary condition for Laplace’s equation in Lipschitz domains. Comm Partial Differential Equations, 2004, 29: 91–109
Mayboroda S, Maz’ya V. Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation. Invent Math, 2009, 175: 287–334
Maz’ya V, Rossmann J. Elliptic Equations in Polyhedral Domains. Mathematical Surveys and Monographs, vol. 162. Providence: Amer Math Soc, 2010
Mitrea I, Mitrea M. Multi-Layer Potentials and Boundary Problems. Lecture Notes in Mathematics, vol. 2063. Berlin: Springer, 2013
Mitrea M, Taylor M. Boundary layer methods for Lipschitz domains in Riemannian manifolds. J Funct Anal, 1999, 163: 181–251
Mitrea M, Taylor M. Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem. J Funct Anal, 2000, 176: 1–79
Mitrea M, Taylor M. Potential theory on Lipschitz domains in Riemannian manifolds: Lp, Hardy and Höolder type results. Comm Anal Geom, 2001, 9: 369–421
Nečas J. Direct Methods in the Theory of Elliptic Equations. Springer Monographs in Mathematics. Berlin: Springer, 2012
Ott K, Brown R. The mixed problem for the Laplacian in Lipschitz domains. Potential Anal, 2013, 38: 1333–1364
Pipher J, Verchota G. Area integral estimates for the biharmonic operator in Lipschitz domains. Trans Amer Math Soc, 1991, 327: 903–917
Pipher J, Verchota G. The Dirichlet problem in Lp for the biharmonic equation on Lipschitz domains. Amer J Math, 1992, 114: 923–972
Pipher J, Verchota G. A maximum principle for the biharmonic equation in Lipschitz and C1 domains. Comment Math Helv, 1993, 68: 385–414
Pipher J, Verchota G. Dilation invariant estimates and the boundary Garding inequality for higher order elliptic operators. Ann of Math (2), 1995, 142: 1–38
Rellich F. Darstellung der Eigenwerte von Δu + λu durch ein Randintegral. Math Z, 1940, 46: 635–646
Shen Z. The Lp Dirichlet problem for elliptic systems on Lipschitz domains. Math Res Lett, 2006, 13: 143–159
Stein E. Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton: Princeton University Press, 1993
Stein E, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Princeton University Press, 1971
Szegö G. Orthogonal Polynomials. American Mathematical Society Colloquium Publications, vol. 23. Providence: Amer Math Soc, 1975
Verchota G. Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J Funct Anal, 1984, 59: 572–611
Verchota G. The Dirichlet problem for the biharmonic equation in C1 domains. Indiana Univ Math J, 1987, 36: 867–895
Verchota G. The Dirichlet problem for the polyharmonic equation in Lipschitz domains. Indiana Univ Math J, 1990, 39: 671–702
Verchota G. The biharmonic Neumann problem in Lipschitz domains. Acta Math, 2005, 194: 217–279
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11401254). The author greatly expresses his gratitude to the referees for their careful reading, questions and suggestion, which are helpful to improve this paper.
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Du, Z. Higher order Poisson kernels and Lp polyharmonic boundary value problems in Lipschitz domains. Sci. China Math. 63, 1065–1106 (2020). https://doi.org/10.1007/s11425-018-9383-0
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DOI: https://doi.org/10.1007/s11425-018-9383-0
Keywords
- polyharmonic equations
- boundary value problems
- higher order Poisson and conjugate Poisson kernels
- integral representation