The Mixed Problem for the Laplacian in Lipschitz Domains
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- Ott, K.A. & Brown, R.M. Potential Anal (2013) 38: 1333. doi:10.1007/s11118-012-9317-6
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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 Rn, 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 q0 > 1 so that for p between 1 and q0 we may solve the mixed problem for Lp. Thus, if we specify Dirichlet data on D in the Sobolev space W1,p(D) and Neumann data on N in Lp (N), the mixed problem with data fD and fN 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.