On the Dirichlet and Regularity Problems for the Bi-Laplacian in Lipschitz Domains

I. Mitrea () and M. Mitrea ()
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I. Mitrea: Worcester Polytechnic Institute
M. Mitrea: University of Missouri

Chapter 24 in Integral Methods in Science and Engineering, Volume 1, 2010, pp 245-254 from Springer

Abstract: Abstract Recall that a Lipschitz domain is a domain whose boundary is locally given by graphs of Lipschitz functions. The formulation of, respectively, the Dirichlet and regularity problems for the Laplacian in a Lipschitz domain $$\Omega \subset \mathbb{R}^n $$ is (24.1) $$ \begin{array}{*{20}c} {(D_\Delta )_p \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\Delta u = 0} & {{\rm in}\,\Omega ,} \\ \end{array}} \\ {\mathcal{N}u \in L^p (\partial \Omega ),} \\ {u|_{\partial \Omega } = f \in L^p (\partial \Omega ),} \\ \end{array}} \right.} & {(R_\Delta )_p \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\Delta u = 0} & {{\rm in}\,\Omega ,} \\ \end{array}} \\ {\mathcal{N}u \in L^p (\partial \Omega ),} \\ {u|_{\partial \Omega } = f \in L_1^p (\partial \Omega ),} \\ \end{array}} \right.} \\ \end{array} $$ .

Date: 2010
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DOI: 10.1007/978-0-8176-4899-2_24

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