Sobolev Estimates for the Green Potential Associated with the Robin—Laplacian in Lipschitz Domains Satisfying a Uniform Exterior Ball Condition

Part of the International Mathematical Series book series (IMAT, volume 9)


We show that if u = Gλf is the solution operator for the Robin problem for the Laplacian, i.e., Δu = f in Ω, ∂ νu + λu = 0 on ∂Ω (with 0 ≤ λ ≤ ∞), then : Lp(Ω) → W 2,p(Ω) is bounded if 1 < p ≤ 2 and Ω ⊂ ℝn is a bounded Lipschitz domain satisfying a uniform exterior ball condition. This extends the earlier results of V. Adolfsson, B. Dahlberg, S. Fromm, D. Jerison, G. Verchota, and T. Wolff, who have dealt with Dirich-let (λ = ∞) and Neumann (λ = 0) boundary conditions. Our treatment of the end-point case p = 1 works for arbitrary Lipschitz domains and is conceptually different from the proof given by the aforementioned authors.


Besov Space Neumann Problem Convex Domain Lipschitz Domain Robin Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adolfsson, V.: Lp-integrability of the second order derivatives of Green potentials in convex domains. Pacific J. Math. 159, no. 2, 201–225 (1993)MATHMathSciNetGoogle Scholar
  2. 2.
    Adolfsson, V., Jerison, D.: Lp-integrability of the second order derivatives for the Neumann problem in convex domains. Indiana Univ. Math. J. 43, no. 4, 1123–1138 (1994)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brown, R., Mitrea, I.: The Mixed Problem for the Lamé System in a Class of Lipschitz Domains. Preprint (2007)Google Scholar
  4. 4.
    Dahlberg, B.E.J.: Lq-estimates for Green potentials in Lipschitz domains. Math. Scand. 44, no. 1, 149–170 (1979)MATHMathSciNetGoogle Scholar
  5. 5.
    Dahlberg, B., Kenig, C.: Hardy spaces and the Lp—Neumann problem for Laplace's equation in a Lipschitz domain. Ann. Math. 125, 437–465 (1987)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Fromm, S.J.: Potential space estimates for Green potentials in convex domains. Proc. Am. Math. Soc. 119, no. 1, 225–233 (1993)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston, MA (1985)MATHGoogle Scholar
  8. 8.
    Jerison, D., Kenig, C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, no. 1, 161–219 (1995)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kalton, N., Mayboroda, S., Mitrea, M.: Interpolation of Hardy—Sobo lev—Besov— Triebel—Lizorkin spaces and applications to problems in partial differential equations. Contemp. Math. 445, 121–177 (2007)MathSciNetGoogle Scholar
  10. 10.
    Kozlov, V.A., Maz'ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities. Am. Math. Soc., Providence, RI (1997)Google Scholar
  11. 11.
    Kalton, N.J., Mitrea, M.: Stability results on interpolation scales of quasi-Banach spaces and applications. Trans. Am. Math. Soc. 350, no. 10, 3903–3922 (1998)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Mayboroda, S., Mitrea, M.: Layer potentials and boundary value problems for Lapla-cian in Lipschitz domains with data in quasi-Banach Besov spaces. Ann. Mat. Pura Appl. (4) 185, no. 2, 155–187 (2006)MathSciNetGoogle Scholar
  13. 13.
    Mayboroda, S., Mitrea, M.: Sharp estimates for Green potentials on non-smooth domains. Math. Res. Lett. 11, 481–492 (2004)MATHMathSciNetGoogle Scholar
  14. 14.
    Mayboroda, S., Mitrea, M.: The solution of the Chang—Krein—Stein conjecture. In: Proc. Conf. Harmonic Analysis and its Applications (March 24–26, 2007), pp. 61–154. Tokyo Woman's Cristian University, Tokyo (2007)Google Scholar
  15. 15.
    Maz'ya, V.G.: Solvability in W 2 2 of the Dirichlet problem in a region with a smooth irregular boundary (Russian). Vestn. Leningr. Univ. 22, no. 7, 87–95 (1967)Google Scholar
  16. 16.
    Maz'ya, V.G.: The coercivity of the Dirichlet problem in a domain with irregular boundary (Russian). Izv. VUZ, Ser. Mat. no. 4, 64–76 (1973)Google Scholar
  17. 17.
    Maz'ya, V.G., Shaposhnikova, T.O.: Theory of Multipliers in Spaces of Differentiable Functions. Pitman, Boston, MA (1985)MATHGoogle Scholar
  18. 18.
    Mitrea, M.: Dirichlet integrals and Gaffney—Friedrichs inequalities in convex domains. Forum Math . 13, no. 4, 531–567 (2001)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Mitrea, M., Taylor, M., Vasy, A.: Lipschitz domains, domains with corners, and the Hodge Laplacian. Commun. Partial Differ. Equ. 30, no. 10–12, 1445–1462 (2005)MATHMathSciNetGoogle Scholar
  20. 20.
    Mitrea, M., Wright, M.: Layer Potentials and Boundary Value Problems for the Stokes system in Arbitrary Lipschitz Domains. Preprint (2008)Google Scholar
  21. 21.
    Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Operators. de Gruyter, Berlin—New York (1996)MATHGoogle Scholar
  22. 22.
    Rychkov, V.: 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
  23. 23.
    Triebel, H.: The Structure of Functions. Basel, Birkhäuser (2001)MATHGoogle Scholar
  24. 24.
    Verchota, G.: Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.University of VirginiaCharlottesvilleUSA
  2. 2.University of MissouriColumbiaUSA

Personalised recommendations