On the Dirichlet and Regularity Problems for the Bi-Laplacian in Lipschitz Domains
Irina Mitrea, Marius Mitrea · Birkhäuser Boston eBooks · 2009
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) $$ \begin{array}{*{20}c} {(D_\Delta )_p \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\Delta u = 0} & {{\rm in}\,\Omega ,} \\ \end{array}} \\ {\mathcal{N}u \in L^p (\partial \Omega ),} \\ {u|_{\partial \Omega } = f \in L^p (\partial \Omega ),} \\ \end{array}} \right.} & {(R_\Delta )_p \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\Delta u = 0} & {{\rm in}\,\Omega ,} \\ \end{array}} \\ {\mathcal{N}u \in L^p (\partial \Omega ),} \\ {u|_{\partial \Omega } = f \in L_1^p (\partial \Omega ),} \\ \end{array}} \right.} \\ \end{array} $$ .