Skip to main content
SpringerLink
Log in

Boundary-value Problems for Higher-order Elliptic Equations in Non-smooth Domains

  • Conference paper
  • First Online:
Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 236))

Abstract

This paper presents a survey of recent results, methods, and open problems in the theory of higher-order elliptic boundary value problems on Lipschitz and more general non-smooth domains. The main topics include the maximum principle and pointwise estimates on solutions in arbitrary domains, analogues of the Wiener test governing continuity of solutions and their derivatives at a boundary point, and well-posedness of boundary value problems in domains with Lipschitz boundaries.

Mathematics Subject Classification (2010). Primary 35-02; Secondary 35B60, 35B65, 35J40, 35J55.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Auscher and A. Axelsson, Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I, Invent. Math. 184 (2011), no. 1, 47– 115.

    Google Scholar 

  2. M.A. Alfonseca, P. Auscher, A. Axelsson, S. Hofmann, and S. Kim, Analyticity of layer potentials and 2 solvability of boundary value problems for divergence form elliptic equations with complex ∞ coefficients, Adv. Math. 226 (2011), no. 5, 4533–4606.

    Google Scholar 

  3. P. Auscher, A. Axelsson, and S. Hofmann, Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems, J. Funct. Anal. 255 (2008), no. 2, 374–448.

    Google Scholar 

  4. P. Auscher, A. Axelsson, and A. McIntosh, Solvability of elliptic systems with square integrable boundary data, Ark. Mat. 48 (2010), no. 2, 253–287.

    Google Scholar 

  5. D.R. Adams, Potential and capacity before and after Wiener, Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, MI, 1994) (Providence, RI), Proc. Sympos. Appl. Math., vol. 52, Amer. Math. Soc., 1997, pp. 63–83.

    Google Scholar 

  6. 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 

  7. , Maximum theorems for solutions of higher order elliptic equations, Bull. Amer. Math. Soc. 66 (1960), 77–80.

    Google Scholar 

  8. M.S. Agranovich, On the theory of Dirichlet and Neumann problems for linear strongly elliptic systems with Lipschitzdom ains, Funktsional. Anal. I Prilozhen. 41 (2007), no. 4, 1–21, 96. English translation: Funct. Anal. Appl. 41 (2007), no. 4, 247–263.

    Google Scholar 

  9. D.R. Adams and L.I. Hedberg, Function spaces and potential theory, Grundlehren der MathematischenWissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer Verlag, Berlin, 1996.

    Google Scholar 

  10. P. Auscher, S. Hofmann, A. McIntosh, and P. Tchamitchian, The Kato square root problem for higher order elliptic operators and systems on , J. Evol. Equ. 1 (2001), no. 4, 361–385.

    Google Scholar 

  11. V. Adolfsson and J. Pipher, The inhomogeneous Dirichlet problem for Δ2 in Lipschitzdom ains, J. Funct. Anal. 159 (1998), no. 1, 137–190.

    Google Scholar 

  12. P. Auscher and A. Rosén, Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II, ArXiv e-prints (2011).

    Google Scholar 

  13. P. Auscher and P. Tchamitchian, Square root problem for divergence operators and related topics, Astérisque (1998), no. 249, viii+172.

    Google Scholar 

  14. P. Auscher, On estimates for square roots of second order elliptic operators on , Publ. Mat. 48 (2004), no. 1, 159–186.

    Google Scholar 

  15. , On necessary and sufficient conditions for -estimates of Riesz transforms associated to elliptic operators on and related estimates, Mem. Amer. Math. Soc. 186 (2007), no. 871, xviii+75.

    Google Scholar 

  16. S. Blunck and P.C. Kunstmann, Weak type estimates for Riesztr ansforms, Math. Z. 247 (2004), no. 1, 137–148.

    Google Scholar 

  17. A. Calder´on, E.B. Fabes, and Y. Sagher, Nestor M. Riviére: June 10, 1940– January 3, 1978, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. vii– xvii (1 plate).

    Google Scholar 

  18. J. Cohen and J. Gosselin, The Dirichlet problem for the biharmonic equation in a 1 domain in the plane, Indiana Univ. Math. J. 32 (1983), no. 5, 635– 685.

    Google Scholar 

  19. , Adjoint boundary value problems for the biharmonic equation on 1 domains in the plane, Ark. Mat. 23 (1985), no. 2, 217–240.

    Google Scholar 

  20. B.E.J. Dahlberg, On the Poisson integral for Lipschitzand 1-domains, Studia Math. 66 (1979), no. 1, 13–24.

    Google Scholar 

  21. , Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitzdom ain, Studia Math. 67 (1980), no. 3, 297–314.

    Google Scholar 

  22. B.E.J. Dahlberg, D.S. Jerison, and C.E. Kenig, Area integral estimates for elliptic differential operators with nonsmooth coefficients, Ark.Mat. 22 (1984), no. 1, 97–108.

    Google Scholar 

  23. B.E. J. Dahlberg and C.E. Kenig, estimates for the three-dimensional systems of elastostatics on Lipschitzdom ains, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 621–634.

    Google Scholar 

  24. B.E.J. 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.

    Google Scholar 

  25. B.E.J. Dahlberg, C.E. Kenig, and G.C. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitzdom ain, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 3, 109–135.

    Google Scholar 

  26. G. Dal Maso and U. Mosco, Wiener criteria and energy decay for relaxed Dirichlet problems, Arch. Rational Mech. Anal. 95 (1986), no. 4, 345–387.

    Google Scholar 

  27. M. DindoÅ¡, S. Petermichl, and J. Pipher, The Dirichlet problem for second order elliptic operators and a -adapted square function, J. Funct. Anal. 249 (2007), no. 2, 372–392.

    Google Scholar 

  28. M. DindoÅ¡ andD.J. Rule, Elliptic equations in the plane satisfying a Carleson measure condition, Rev. Mat. Iberoam. 26 (2010), no. 3, 1013–1034.

    Google Scholar 

  29. L.C. Evans and R.F. Gariepy, Wiener’s criterion for the heat equation, Arch. Rational Mech. Anal. 78 (1982), no. 4, 293–314.

    Google Scholar 

  30. E.B. Fabes, N. Garofalo, and E. Lanconelli, Wiener’s criterion for divergence form parabolic operators with 1-Dini continuous coefficients, Duke Math. J. 59 (1989), no. 1, 191–232.

    Google Scholar 

  31. E.B. Fabes, D.S. Jerison, and C.E. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, vi, 151– 182.

    Google Scholar 

  32. E.B. Fabes, M. Jodeit, Jr., and J.E. Lewis, Double layer potentials for domains with corners and edges, Indiana Univ. Math. J. 26 (1977), no. 1, 95– 114.

    Google Scholar 

  33. E.B. Fabes, M. Jodeit, Jr., and N.M. Riviére, Potential techniques for boundary value problems on 1-domains, ActaMath. 141 (1978), no. 3-4, 165–186.

    Google Scholar 

  34. S. Hofmann, C.E. Kenig, S. Mayboroda, and J. Pipher, Square function/nontangential maximal function estimates and the Dirichlet problem for nonsymmetric elliptic operators, ArXiv e-prints (2012).

    Google Scholar 

  35. D.S. Jerison and C.E. Kenig, The Dirichlet problem in nonsmooth domains, Ann. of Math. (2) 113 (1981), no. 2, 367–382.

    Google Scholar 

  36. , The Neumann problem on Lipschitzdom ains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203–207.

    Google Scholar 

  37. C.E. Kenig, Progress on two problems posed by Riviére, Harmonic analysis and partial differential equations (Boca Raton, FL, 1988), Contemp. Math., vol. 107, Amer. Math. Soc., Providence, RI, 1990, pp. 101–107.

    Google Scholar 

  38. , Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1994.

    Google Scholar 

  39. C.E. Kenig, H. Koch, J. Pipher, and T. Toro, A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, Adv. Math. 153 (2000), no. 2, 231–298.

    Google Scholar 

  40. V.A. Kozlov, V.G. Maz’ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical Surveys and Monographs, vol. 85, American Mathematical Society, Providence, RI, 2001.

    Google Scholar 

  41. C.E. Kenig and J. Pipher, The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math. 113 (1993), no. 3, 447–509.

    Google Scholar 

  42. C.E. Kenig and D.J. Rule, The regularity and Neumann problem for nonsymmetric elliptic operators, Trans. Amer. Math. Soc. 361 (2009), no. 1, 125– 160.

    Google Scholar 

  43. J. Kilty and Z. Shen, A bilinear estimate for biharmonic functions in Lipschitz domains, Math. Ann. 349 (2011), no. 2, 367–394.

    Google Scholar 

  44. , The regularity problem on Lipschitzdom ains, Trans. Amer.Math. Soc. 363 (2011), no. 3, 1241–1264.

    Google Scholar 

  45. D.A. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J. 111 (2002), no. 1, 1–49.

    Google Scholar 

  46. H. Lebesgue, Sur des cas d’impossibilité du probléme de Dirichlet ordinaire, C.R. des Séances de la Société Mathématique de France 17 (1913).

    Google Scholar 

  47. G. Luo and V.G. Maz’ya, Wiener type regularity of a boundary point for the 3D Lamé system, Potential Anal. 32 (2010), no. 2, 133–151.

    Google Scholar 

  48. W. Littman, G. Stampacchia, and H.F.Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 43–77.

    Google Scholar 

  49. S. Mayboroda, The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients, Adv. Math. 225 (2010), no. 4, 1786–1819.

    Google Scholar 

  50. V.G. Maz’ya, On the behavior near the boundary of solutions of the Dirichletproblem for the biharmonic operator, Dokl. Akad. Nauk SSSR 18 (1977), 15– 19. English translation: Soviet Math. Dokl. 18 (1977), no. 4, 1152–1155.

    Google Scholar 

  51. , Behaviour of solutions to the Dirichlet problem for the biharmonic operator at a boundary point, Equadiff IV (Proc. Czechoslovak Conf. Differential Equations and their Applications, Prague, 1977), Lecture Notes in Math., vol. 703, Springer, Berlin, 1979, pp. 250–262.

    Google Scholar 

  52. , Unsolved problems connected with the Wiener criterion, The Legacy of Norbert Wiener: A Centennial Symposium (Cambridge, MA, 1994), Proc. Sympos. Pure Math., vol. 60, Amer. Math. Soc., Providence, RI, 1997, pp. 199–208.

    Google Scholar 

  53. , On the Wiener type regularity of a boundary point for the polyharmonic operator, Appl. Anal. 71 (1999), no. 1-4, 149–165.

    Google Scholar 

  54. , On Wiener’s type regularity of a boundary point for higher order elliptic equations, Nonlinear analysis, function spaces and applications, Vol. 6 (Prague, 1998), Acad. Sci. Czech Repub., Prague, 1999, pp. 119–155.

    Google Scholar 

  55. , The Wiener test for higher order elliptic equations, Duke Math. J. 115 (2002), no. 3, 479–512.

    Google Scholar 

  56. V.G.Maz’ya and T. Donchev, Regularity in the sense of Wiener of a boundary point for a polyharmonic operator, C. R. Acad. Bulgare Sci. 36 (1983), no. 2, 177–179.

    Google Scholar 

  57. V.V. Meleshko, Selected topics in the theory of two-dimensional biharmonic problem, Appl. Mech. Rev. 56 (2003), no. 1, 33–85.

    Google Scholar 

  58. C. Miranda, Formule di maggiorazione e teorema di esistenza per le funzioni biarmoniche de due variabili, Giorn. Mat. Battaglini (4) 2(78) (1948), 97–118.

    Google Scholar 

  59. , Teorema del massimo modulo e teorema di esistenza e di unicità per il problema di Dirichlet relativo alle equazioni ellittiche in due variabili, Ann. Mat. Pura Appl. (4) 46 (1958), 265–311.

    Google Scholar 

  60. I. Mitrea, Mapping properties of layer potentials associated with higher-order elliptic operators in Lipschitzdom ains, Topics in operator theory. Volume 2. Systems and mathematical physics, Oper. Theory Adv. Appl., vol. 203, Birkhäuser Verlag, Basel, 2010, pp. 363–407.

    Google Scholar 

  61. S. Mayboroda and V.G. Maz’ya, Boundedness of the Hessian of a biharmonic function in a convex domain, Comm. Partial Differential Equations 33 (2008), no. 7-9, 1439–1454.

    Google Scholar 

  62. , Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation, Invent. Math. 175 (2009), no. 2, 287–334.

    Google Scholar 

  63. , Pointwise estimates for the polyharmonic Green function in general domains, Analysis, partial differential equations and applications, Oper. Theory Adv. Appl., vol. 193, Birkhäuser Verlag, Basel, 2009, pp. 143–158.

    Google Scholar 

  64. I. Mitrea and M. Mitrea, On the Dirichlet and regularity problems for the bi- Laplacian in Lipschitzdom ains, Integral methods in science and engineering. Vol. 1, Birkhäuser Boston Inc., Boston, MA, 2010, pp. 245–254.

    Google Scholar 

  65. D. Mitrea and I. Mitrea, On the regularity of Green functions in Lipschitz domains, Comm. Partial Differential Equations 36 (2011), no. 2, 304–327.

    Google Scholar 

  66. V.G. Maz’ya, M. Mitrea, and T. Shaposhnikova, The Dirichlet problem in Lipschitzdom ains for higher order elliptic systems with rough coefficients, J. Anal. Math. 110 (2010), 167–239.

    Google Scholar 

  67. I. Mitrea, M. Mitrea, and M. Wright, Optimal estimates for the inhomogeneous problem for the bi-laplacian in three-dimensional Lipschitzdom ains, J. Math. Sci. 172 (2011), no. 1, 24–134.

    Google Scholar 

  68. V.G. Maz’ya and S.A. Nazarov, The apex of a cone can be irregular in Wiener’s sense for a fourth-order elliptic equation, Mat. Zametki 39 (1986), no. 1, 24–28, 156. English translation: Math. Notes 39 (1986), no. 1-2, 14–16.

    Google Scholar 

  69. V.G. Maz’ya, S.A. Nazarov, and B.A. PlamenevskiÄ­, Singularities of solutions of the Dirichlet problem in the exterior of a thin cone, Mat. Sb. (N.S.) 122 (164) (1983), no. 4, 435–457. English translation: Math. USSR-Sb. 50 (1985), no. 2, 415–437.

    Google Scholar 

  70. V.G.Maz’ya and B.A. PlamenevskiÄ­, On the maximum principle for the biharmonic equation in a domain with conical points, Izv. Vyssh. Uchebn. Zaved. Mat. (1981), no. 2, 52–59. English translation: Soviet Math. (Iz. VUZ) 25 (1981), no. 2, 61–70.

    Google Scholar 

  71. V.G. Maz’ya and J. Rossmann, On the Agmon-Miranda maximum principle for solutions of elliptic equations in polyhedral and polygonal domains, Ann. Global Anal. Geom. 9 (1991), no. 3, 253–303.

    Google Scholar 

  72. , On the Agmon-Miranda maximum principle for solutions of strongly elliptic equations in domains of with conical points, Ann. Global Anal. Geom. 10 (1992), no. 2, 125–150.

    Google Scholar 

  73. J. Mal and W.P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, vol. 51, American Mathematical Society, Providence, RI, 1997.

    Google Scholar 

  74. A. Nadai, Theory of Flow and Fracture of Solids, vol. II, McGraw-Hill, 1963.

    Google Scholar 

  75. J. Neˇcas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, éditeurs, Paris, 1967.

    Google Scholar 

  76. H. Poincaré, Sur les Equations aux Dérivées Partielles de la Physique Mathématique, Amer. J. Math. 12 (1890), no. 3, 211–294.

    Google Scholar 

  77. J. Pipher and G.C. Verchota, Area integral estimates for the biharmonic operator in Lipschitzdom ains, Trans. Amer. Math. Soc. 327 (1991), no. 2, 903– 917.

    Google Scholar 

  78. , The Dirichlet problem in for the biharmonic equation on Lipschitz domains, Amer. J. Math. 114 (1992), no. 5, 923–972.

    Google Scholar 

  79. , A maximum principle for biharmonic functions in Lipschitzand 1 domains, Comment. Math. Helv. 68 (1993), no. 3, 385–414.

    Google Scholar 

  80. , Dilation invariant estimates and the boundary GËšarding inequality for higher order elliptic operators, Ann. of Math. (2) 142 (1995), no. 1, 1–38.

    Google Scholar 

  81. , Maximum principles for the polyharmonic equation on Lipschitzdomains, Potential Anal. 4 (1995), no. 6, 615–636.

    Google Scholar 

  82. D.J. Rule, Non-symmetric elliptic operators on bounded Lipschitzdom ains in the plane, Electron. J. Differential Equations (2007), No. 144, 8.

    Google Scholar 

  83. Z. Shen, The Dirichlet problem for elliptic systems on Lipschitzdom ains, Math. Res. Lett. 13 (2006), no. 1, 143–159.

    Google Scholar 

  84. , Necessary and sufficient conditions for the solvability of the Dirichlet problem on Lipschitzdom ains, Math. Ann. 336 (2006), no. 3, 697– 725.

    Google Scholar 

  85. , On estimates of biharmonic functions on Lipschitzand convex domains, J. G eom. Anal. 16 (2006), no. 4, 721–734.

    Google Scholar 

  86. , The boundary value problems on Lipschitzdom ains, Adv. Math. 216 (2007), no. 1, 212–254.

    Google Scholar 

  87. , A relationship between the Dirichlet and regularity problems for elliptic equations, Math. Res. Lett. 14 (2007), no. 2, 205–213.

    Google Scholar 

  88. R. Selvaggi and I. Sisto, An existence theorem for the Dirichlet problem with respect to the operator Δ2 in certain domains of class 1, Boll. Un. Mat. Ital. B (5) 18 (1981), no. 2, 473–483.

    Google Scholar 

  89. B.V. Å¡ul’ce, A priori estimates in uniform norms for strongly elliptic systems, Sibirsk. Mat. Ž. 16 (1975), 384–394, 422, English translation: Siberian Math. J. 16 (1975), no. 2, 297–305.

    Google Scholar 

  90. G. Sweers, A survey on boundary conditions for the biharmonic, Complex Var. Elliptic Equ. 54 (2009), no. 2, 79–93.

    Google Scholar 

  91. N.S. Trudinger and X.J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124 (2002), no. 2, 369– 410.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  94. , Potentials for the Dirichlet problem in Lipschitzdom ains, Potential theory – ICPT 94 (Kouty, 1994), de Gruyter, Berlin, 1996, pp. 167–187.

    Google Scholar 

  95. , The biharmonic Neumann problem in Lipschitzdom ains, ActaMath. 194 (2005), no. 2, 217–279.

    Google Scholar 

  96. , Boundary coerciveness and the Neumann problem for 4th order linear partial differential operators, Around the research of Vladimir Maz’ya. II, Int. Math. Ser. (N. Y.), vol. 12, Springer, New York, 2010, pp. 365–378.

    Google Scholar 

  97. N. Wiener, The Dirichlet problem, J. Math. Phys. 3 (1924), 127–146.

    Google Scholar 

  98. S.C. Zaremba, Sur le principe du minimum, Bull. Acad. Sci. Cracovie (1909), 197–264.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA

    Ariel Barton & Svitlana Mayboroda

Authors
  1. Ariel Barton
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Svitlana Mayboroda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ariel Barton .

Editor information

Editors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de Sevilla, Sevilla, Spain

    Manuel Cepedello Boiso

  2. Department of Mathematics, The Royal Institute of Technology, Stockholm, Sweden

    Håkan Hedenmalm

  3. Department of Mathematics & Computer Sci, Free University Amsterdam, Amsterdam, The Netherlands

    Marinus A. Kaashoek

  4. Departamento de Análisis Matemático, Universidad de Sevilla, Sevilla, Spain

    Alfonso Montes Rodríguez

  5. Department of Mathematics, Brown University, Providence, Rhode Island, USA

    Sergei Treil

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Basel

About this paper

Cite this paper

Barton, A., Mayboroda, S. (2014). Boundary-value Problems for Higher-order Elliptic Equations in Non-smooth Domains. In: Cepedello Boiso, M., Hedenmalm, H., Kaashoek, M., Montes Rodríguez, A., Treil, S. (eds) Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, vol 236. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0648-0_4

Download citation

Publish with us

Policies and ethics

Search

Navigation