The mixed problem in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> for some two-dimensional Lipschitz domains期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

The mixed problem in Lp for some two-dimensional Lipschitz domains

Authors:

Loredana Lanzani Luca Capogna Russell M. Brown

Affiliation:

(1) Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA;(2) Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA

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

Keywords:

Mathematics Subject Classification (2000) 35J05

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