, Volume 38, Issue 4, pp 1333-1364

The Mixed Problem for the Laplacian in Lipschitz Domains

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Abstract

We consider the mixed boundary value problem, or Zaremba’s problem, for the Laplacian in a bounded Lipschitz domain Ω in R n , n ≥ 2. We decompose the boundary $ \partial \Omega= D\cup N$ with D and N disjoint. The boundary between D and N is assumed to be a Lipschitz surface in $\partial \Omega$ . We find an exponent q 0 > 1 so that for p between 1 and q 0 we may solve the mixed problem for L p . Thus, if we specify Dirichlet data on D in the Sobolev space W 1,p (D) and Neumann data on N in L p (N), the mixed problem with data f D and f N has a unique solution and the non-tangential maximal function of the gradient lies in $L^p( \partial \Omega)$ . We also obtain results for p = 1 when the data comes from Hardy spaces.

Research supported, in part, by the National Science Foundation.
This work was partially supported by a grant from the Simons Foundation (#195075 to Russell Brown).