Mathematische Annalen

, Volume 342, Issue 1, pp 91–124 | Cite as

The mixed problem in Lp for some two-dimensional Lipschitz domains

  • Loredana Lanzani
  • Luca Capogna
  • Russell M. Brown


We consider the mixed problem,
$$\left\{ \begin{array}{ll} \Delta u = 0 \quad & {\rm in }\, \Omega\\ \frac{\partial u }{\partial \nu} = f_N \quad & {\rm on }\, {\rm N} \\ u = f_D \quad & {\rm on}\,D \end{array} \right.$$
in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, fD, has one derivative in Lp(D) of the boundary and the Neumann data, fN, is in Lp(N). We find a p0 > 1 so that for p in an interval (1, p0), we may find a unique solution to the mixed problem and the gradient of the solution lies in Lp.

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Azzam, A., Kreyszig, E.: On solutions of elliptic equations satisfying mixed boundary conditions. SIAM J. Math. Anal. 13(2), 254–262 (1982)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Bell, S.R.: The Cauchy Transform, Potential Theory, and Conformal Mapping. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)Google Scholar
  3. 3.
    Bergh, J., Löfström, J.: Interpolation Spaces. Springer, Heidelberg (1976)MATHGoogle Scholar
  4. 4.
    Brown, R.M.: The mixed problem for Laplace’s equation in a class of Lipschitz domains. Commun. Partial Diff. Eqns. 19, 1217–1233 (1994)CrossRefMATHGoogle Scholar
  5. 5.
    Brown, R.M.: The Neumann problem on Lipschitz domains in Hardy spaces of order less than one. Pac. J. Math. 171(2), 389–407 (1995)MATHGoogle Scholar
  6. 6.
    Chang, D.C., Krantz, S.G., Stein, E.M.: H p theory on a smooth domain in R n and elliptic boundary value problems. J. Funct. Anal. 114(2), 286–347 (1993)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Coifman, R.R., McIntosh, A., Meyer, Y.: L’intégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes lipschitziennes. Ann. Math. 116, 361–387 (1982)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1976)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Dahlberg, B.E.J.: Estimates of harmonic measure. Arch. Rational Mech. Anal. 65(3), 275–288 (1977)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Dahlberg, B.E.J., Kenig, C.E.: Hardy spaces and the Neumann problem in L p for Laplace’s equation in Lipschitz domains. Ann. Math. 125, 437–466 (1987)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Escauriaza, L.: Personal communication, 2000Google Scholar
  12. 12.
    Fabes, E.B.: Layer potential methods for boundary value problems on Lipschitz domains. In: Potential Theory—Surveys and Problems, vol 1344 of Lecture notes in math., pp. 55–80. Springer, Heidelberg (1988)Google Scholar
  13. 13.
    Fabes, E.B., Mendez, O., Mitrea, M.: Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159(2), 323–368 (1998)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, vol 105 of Annals of Mathematics Studies (1983). Princeton University Press, PrincetonGoogle Scholar
  15. 15.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)MATHGoogle Scholar
  16. 16.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, London (1985)MATHGoogle Scholar
  17. 17.
    Jerison, D.S., Kenig, C.E.: Hardy spaces, A , and singular integrals on chord-arc domains. Math. Scand. 50(2), 221–247 (1982)MathSciNetMATHGoogle Scholar
  18. 18.
    Jerison, D.S., Kenig, C.E.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc. 4, 203–207 (1982)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Kenig, C.E.: Weighted H p spaces on Lipschitz domains. Am. J. Math. 102(1), 129–163 (1980)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Kenig, C.E.: Harmonic analysis techniques for second order elliptic boundary value problems. Published for the Conference Board of the Mathematical Sciences, Washington (1994)MATHGoogle Scholar
  21. 21.
    Kenig, C.E., Ni, W.M.: On the elliptic equation Lu  −  k  +  K exp[2u]  =  0. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12(2), 191–224 (1985)MathSciNetMATHGoogle Scholar
  22. 22.
    Lagnese, J.: Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Diff. Equat. 50(2), 163–182 (1983)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Lieberman, G.M.: Mixed boundary value problems for elliptic and parabolic differential equations of second order. J. Math. Anal. Appl. 113(2), 422–440 (1986)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Maz’ya, V.G., Rossman, J.: L p estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains. Preprint, 2006Google Scholar
  25. 25.
    Maz’ya, V.G., Rossman, J.: Mixed boundary value problems for the Navier-Stokes system in polyhedral domains. Preprint, 2006Google Scholar
  26. 26.
    Maz’ya, V.G., Rossman, J.: Pointwise estimates for Green’s kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone. Preprint, 2006Google Scholar
  27. 27.
    Mitrea, I., Mitrea, M.: The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in nonsmooth domains. Trans. Amer. Math. Soc. (to appear)Google Scholar
  28. 28.
    Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice-Hall Inc., Englewood Cliffs (1967)Google Scholar
  29. 29.
    Savaré, G.: Regularity and perturbation results for mixed second order elliptic problems. Commun. Partial Diff. Eqns. 22, 869–899 (1997)CrossRefMATHGoogle Scholar
  30. 30.
    Shen, Z.: Weighted estimates in L 2 for Laplace’s equation on Lipschitz domains. Trans. Am. Math. Soc. 357, 2843–2870 (2005)CrossRefMATHGoogle Scholar
  31. 31.
    Stein, E.M.: Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals. Princeton, New Jersey (1993)MATHGoogle Scholar
  32. 32.
    Strömberg, J.O., Torchinsky, A.: Weighted Hardy Spaces. Springer, Berlin (1989)MATHGoogle Scholar
  33. 33.
    Sykes, J.D.: L p regularity of solutions of the mixed boundary value problem for Laplace’s equation on a Lipschitz graph domain. PhD thesis, University of Kentucky (1999)Google Scholar
  34. 34.
    Sykes, J.D., Brown, R.M.: The mixed boundary problem in L p and Hardy spaces for Laplace’s equation on a Lipschitz domain. In: Harmonic Analysis and Boundary Value Problems (Fayetteville, AR, 2000), vol 277 of Contemp. Math., pp. 1–18. Amer. Math. Soc., Providence (2001)Google Scholar
  35. 35.
    Verchota, G.C.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation on Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Loredana Lanzani
    • 1
  • Luca Capogna
    • 1
  • Russell M. Brown
    • 2
  1. 1.Department of MathematicsUniversity of ArkansasFayettevilleUSA
  2. 2.Department of MathematicsUniversity of KentuckyLexingtonUSA

Personalised recommendations