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The mixed problem in L p for some two-dimensional Lipschitz domains

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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 an 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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Authors and Affiliations

  1. Department of Mathematics, University of Arkansas, Fayetteville, AR, 72701, USA

    Loredana Lanzani & Luca Capogna

  2. Department of Mathematics, University of Kentucky, Lexington, KY, 40506, USA

    Russell M. Brown

Authors
  1. Loredana Lanzani
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  2. Luca Capogna
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  3. Russell M. Brown
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Correspondence to Russell M. Brown.

Additional information

L. Lanzani, L. Capogna and R. M. Brown were supported, in part, by the U.S. National Science Foundation.

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Lanzani, L., Capogna, L. & Brown, R.M. The mixed problem in L p for some two-dimensional Lipschitz domains. Math. Ann. 342, 91–124 (2008). https://doi.org/10.1007/s00208-008-0223-6

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  • DOI: https://doi.org/10.1007/s00208-008-0223-6

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