Potential Analysis

, Volume 38, Issue 4, pp 1333-1364

First online:

The Mixed Problem for the Laplacian in Lipschitz Domains

  • Katharine A. OttAffiliated withDepartment of Mathematics, University of Kentucky
  • , Russell M. BrownAffiliated withDepartment of Mathematics, University of Kentucky Email author 

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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.


Mixed boundary value problem Laplacian Non-smooth domain

Mathematics Subject Classification (2010)