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

Loredana Lanzani, Luca Capogna, Russell M. Brown

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18 Scopus citations


We consider the mixed problem, {Δ u = 0 in Ω ∂u = f N on N u = fD on D 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 .

Original languageEnglish
Pages (from-to)91-124
Number of pages34
JournalMathematische Annalen
Issue number1
StatePublished - Sep 2008

Bibliographical note

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

ASJC Scopus subject areas

  • General Mathematics


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