## Mathematical Sciences Faculty Publications

Article

9-1-2008

#### Publication Title

Mathematische Annalen

#### 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, 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 the interval (1,p_0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L^p.

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#### DOI

10.1007/s00208-008-0223-6

Preprint

#### Publisher Statement

This is a pre-print of an article published in Mathematische Annalen. The final authenticated version is available online at: https://doi.org/10.1007/s00208-008-0223-6.