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The Mixed Problem for the Laplacian in Lipschitz Domains

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A Correction to this article was published on 01 February 2020

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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.

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Change history

  • 01 February 2020

    We provide an example to show that a technical estimate in our earlier work, ���The Mixed Problem for the Laplacian in Lipschitz Domains���, is not correct.

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Correspondence to Russell M. Brown.

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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).

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Ott, K.A., Brown, R.M. The Mixed Problem for the Laplacian in Lipschitz Domains. Potential Anal 38, 1333–1364 (2013). https://doi.org/10.1007/s11118-012-9317-6

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