Journal of Mathematical Sciences

, Volume 172, Issue 1, pp 24–134 | Cite as

Optimal estimates for the inhomogeneous problem for the bi-Laplacian in three-dimensional Lipschitz domains

  • I. MitreaEmail author
  • M. Mitrea
  • M. Wright

We establish the well-posedness of the inhomogeneous Dirichlet problem for Δ2 in arbitrary Lipschitz domains in \( {\mathbb{R}^3} \), with data from Besov–Triebel–Lizorkin spaces, for the optimal range of indices. The main novel contribution is to allow for certain nonlocally convex spaces to be considered, and to establish integral representations for the solution. Bibliography: 57 titles. Illustrations: 1 figure.


Hardy Space Besov Space Lipschitz Domain Singular Integral Operator Biharmonic Function 
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  1. 1.
    T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Operators, de Gruyter, Berlin etc. (1996).Google Scholar
  2. 2.
    V. Adolfsson and J. Pipher, “The inhomogeneous Dirichlet problem for Δ2 in Lipschitz domains,” J. Funct. Anal. 159. No. 1, 137–190 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    I. Mitrea and M. Mitrea, Multiple Layer Potentials for Higher Order Elliptic Boundary Value Problems. Preprint (2010).Google Scholar
  4. 4.
    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,” J. Anal. Math. (2010).Google Scholar
  5. 5.
    S. Mayboroda and M. Mitrea, “The solution of the Chang–Krantz–Stein conjecture,” In: Proc. Conference in Harmonic Analysis and Applications, pp. 1–95, Tokyo, Japan (2007).Google Scholar
  6. 6.
    D.-C. Chang, S. G. Krantz, and E. M. Stein, “H p theory on a smooth domain in \( {\mathbb{R}^N} \) and elliptic boundary value problems,” J. Funct. Anal. 114 No. 2, 286–347 (1993).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    D.-C. Chang, S. G. Krantz, and E. M. Stein, “Hardy spaces and elliptic boundary value problems,” Contemp. Math. 137, 119–131 (1992).MathSciNetGoogle Scholar
  8. 8.
    M. Mitrea and M. Wright, “Layer Potentials and Boundary Value Problems for the Stokes System in Arbitrary Lipschitz Domains,” Astérisque (2010).Google Scholar
  9. 9.
    J. Pipher and G. Verchota, “A maximum principle for biharmonic functions in Lipschitz and C 1 domains,” Comment. Math. Helv. 68, No. 3, 385–414 (1993).zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    J. Pipher and G. C. Verchota, “Maximum principles for the polyharmonic equation on Lipschitz domains,” Potential Anal. 4, No. 6, 615–636 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson, Paris etc. (1967).Google Scholar
  12. 12.
    J. K. Seo, “Regularity for solutions of biharmonic equation on Lipschitz domain,” Bull. Korean Math. Soc. 33, No. 1, 17–28 (1996).zbMATHMathSciNetGoogle Scholar
  13. 13.
    S. Mayboroda and V. Maz’ya, “Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation,” Invent. Math. 175, No. 2, 287–334 (2009).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    D. Jerison and C. Kenig, “The inhomogeneous Dirichlet problem in Lipschitz domains,” J. Funct. Anal. 130, No. 1, 161–219 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,” Commun. Pure Appl. Math. 12, 623–727 (1959).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    D.-C. Chang, G. Dafni, and E. M. Stein, “Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in \( {\mathbb{R}^n} \),” Trans. Am. Math. Soc. 351 1605–1661 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    J. Franke and T. Runst, “Regular elliptic boundary value problems in Besov–Triebel–Lizorkin spaces,” Math. Nachr. 174, 113–149 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    B. Dahlberg, “L q-estimates for Green potentials in Lipschitz domains,” Math. Scand. 44, No. 1, 149–170 (1979).zbMATHMathSciNetGoogle Scholar
  19. 19.
    C. E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, Am. Math. Soc., Providence, RI (1994).zbMATHGoogle Scholar
  20. 20.
    E. Fabes, O. Mendéz, and M. Mitrea, “Potential operators on Besov spaces and the Poisson equation with Dirichlet and Neumann boundary conditions on Lipschitz domains,” J. Funct. Anal. 159, 323–368 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    M. Mitrea and M. Taylor, “Potential theory on Lipschitz domains in Riemannian manifolds: Holder continuous metric tensors,” Commun. Partial Differ. Equ. 25, 1487–1536 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    J. Pipher and G. Verchota, “The Dirichlet problem in L p for the biharmonic equation on Lipschitz domains,” Am. J. Math. 114, No. 5, 923–972 (1992).zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    B. Dahlberg and C. Kenig, “Hardy spaces and the L p–Neumann problem for Laplace’s equation in a Lipschitz domain,” Ann. Math. 125, 437–465 (1987).CrossRefMathSciNetGoogle Scholar
  24. 24.
    G. C. Verchota, “The biharmonic Neumann problem in Lipschitz domains,” Acta Math. 194, No. 2, 217–279 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, “The Dirichlet problem for the biharmonic equation in a Lipschitz domain,” Ann. Inst. Fourier (Grenoble) 36, No. 3, 109–135 (1986).zbMATHMathSciNetGoogle Scholar
  26. 26.
    J. Pipher and G. Verchota, “Area integral estimates for the biharmonic operator in Lipschitz domains,” Trans. Am. Math. Soc. 327, No. 2, 903–917 (1991).zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Z. Shen, “The L p boundary value problems on Lipschitz domains,” Adv. Math. 216, 212–254 (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Z. Shen, “On estimates of biharmonic functions on Lipschitz and convex domains,” J. Geom. Anal. 16, No. 4, 721–734 (2006).zbMATHMathSciNetGoogle Scholar
  29. 29.
    V. Maz’ya, S. A. Nazarov, and B. A. Plamenevskii, “On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone” [in Russian]: Mat. Cb. 122, No. 4, 435–457; English transl.: Math. USSR Sb. 50, No. 2, 415–437 (1985)Google Scholar
  30. 30.
    V.Maz’ya and B. A. Plamenevskii, “On the maximum principle for the biharmonic equation in a domain with conical points” [in Russian], Izv. Vyssh. Uchebn. Zaved. Mat. No. 2, 52–59 (1981).Google Scholar
  31. 31.
    G. Verchota, “The Dirichlet problem for the biharmonic equation in C 1 domains,” Indiana Univ. Math. J. 36, No. 4, 867–895 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    G. Verchota, “The Dirichlet problem for the polyharmonic equation in Lipschitz domains,” Indiana Univ. Math. J. 39 No. 3, 671–702 (1990).zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    S. Hofmann, M. Mitrea and M. Taylor, “Geometric and transformational properties of Lipschitz domains, Semmes–Kenig–Toro domains, and other classes of finite perimeter domains,” J. Geom. Anal. 17, No. 4, 593–647, (2007).zbMATHMathSciNetGoogle Scholar
  34. 34.
    R. Coifman and G. Weiss, “Extensions of Hardy spaces and their use in analysis,” Bull. Am. Math. Soc. 83, No. 4, 569–645 (1977).zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    V. Rychkov, “On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains,” J. London Math. Soc. (2) 60, No. 1, 237–257 (1999).CrossRefMathSciNetGoogle Scholar
  36. 36.
    S.-Y. Hsu, Removable Singularity of the Polyharmonic Equation. Preprint (2007).Google Scholar
  37. 37.
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ (1970).zbMATHGoogle Scholar
  38. 38.
    N. Kalton and M. Mitrea, “Stability of Fredholm properties on interpolation scales of quasi-Banach spaces and applications,” Trans. Am. Math. Soc. 350, No. 10, 3837–3901 (1998).CrossRefMathSciNetGoogle Scholar
  39. 39.
    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, No. 5, 1425–1461 (1997).zbMATHMathSciNetGoogle Scholar
  40. 40.
    E. B. Fabes, M. Jodeit Jr., and N. M. Rivière, “Potential techniques for boundary value problems on C 1-domains,” Acta Math. 141, No. 3-4, 165–186 (1978).zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    S. Hofmann, M. Mitrea, and M. Taylor, Singular Integrals and Elliptic Boundary Problems on Regular Semmes–Kenig–Toro Domains. Preprint (2007).Google Scholar
  42. 42.
    E. Fabes and U. Neri, “Dirichlet problem in Lipschitz domains with BMO data,” Proc. Am. Math. Soc. 78, No. 1, 33–39 (1980).zbMATHMathSciNetGoogle Scholar
  43. 43.
    C. Fefferman and E. M. Stein, “H p spaces of several variables,” Acta Math. 129, No. 3-4, 137–193 (1972).zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    H. Triebel, The Structure of Functions, Birkhäuser, Basel (2001).zbMATHGoogle Scholar
  45. 45.
    H. Triebel, “Function spaces on Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers,” Rev. Mat. Complut. 15, No. 2, 475–524 (2002).zbMATHMathSciNetGoogle Scholar
  46. 46.
    N. Kalton, S. Mayboroda, and M. Mitrea, “Interpolation of Hardy-Sobolev-Besov–Triebel–Lizorkin spaces and applications to problems in partial differential equations,” Contemp. Math. 445, 121–177 (2007).MathSciNetGoogle Scholar
  47. 47.
    J. Bergh and J. Löfström, Interpolation Spaces. An introduction, Springer, Berlin etc. (1976).zbMATHGoogle Scholar
  48. 48.
    M. Mitrea and M. Taylor, “Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev–Besov space results and the Poisson problem,” J. Funct. Anal. 176, 1–79 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    D. Mitrea, M. Mitrea, and S. Monniaux, “The Poisson problem for the exterior derivative operator with Dirichlet boundary condition on nonsmooth domains,” Commun. Pure Appl. Anal. 7, No. 6, 1295–1333 (2008).CrossRefMathSciNetGoogle Scholar
  50. 50.
    V. Maz’ya and T. Shaposhnikova, “Higher regularity in the layer potential theory for Lipschitz domains,” Indiana Univ. Math. J. 54, No. 1, 99–142 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Dunod, Paris (1968).zbMATHGoogle Scholar
  52. 52.
    O. Mendez and M. Mitrea “The Banach envelopes of Besov and Triebel–Lizorkin spaces and applications to partial differential equations,” J. Fourier Anal. Appl. 6, 503–531 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    N. Kalton, N. T. Peck ,and J. Roberts, An F-Space Sampler, Cambridge Univ. Press, Cambridge (1984).zbMATHGoogle Scholar
  54. 54.
    W. Sickel and H. Triebel, “Hölder inequalities and sharp embeddings in function spaces of B pq s and F pq s type,” Z. Anal. Anwend. 14, No. 1, 105–140 (1995).zbMATHMathSciNetGoogle Scholar
  55. 55.
    W. Rudin, Functional Analysis, McGraw-Hill, New York etc. (1991).zbMATHGoogle Scholar
  56. 56.
    V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Birkhäuser, Basel (2007).zbMATHGoogle Scholar
  57. 57.
    D.-C. Chang, “The dual of Hardy spaces on a bounded domain in \( {\mathbb{R}^n} \),” Forum Math. 6, No. 1, 65–81 (1994).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.University of MinnesotaMinneapolisUSA
  2. 2.University of MissouriColumbiaUSA
  3. 3.Missouri State UniversitySpringfieldUSA

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