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
Article

Abstract

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)

35J05 

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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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