Mathematische Annalen

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

The mixed problem in L p 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, f D , has one derivative in L p (D) of the boundary and the Neumann data, f N , is in L p (N). We find a p 0 > 1 so that for p in an interval (1, p 0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L p .

Mathematics Subject Classification (2000)



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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

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