Mixed Boundary Value Problems for Nonlinear Elliptic Systems in $n$-Dimensional Lipschitzian Domains

C. Ebmeyer

Abstract: Let $u: \Omega \to \R^N$ be the solution of the nonlinear elliptic system $$ -\sum_{i=1}^n \partial_iF_i(x,\nabla u)=f(x)+\sum_{i=1}^n\partial_if_i(x), $$ where $\Omega \subset \R^n$ is a bounded domain with a piecewise smooth boundary (e.g., $\Omega$ is a polyhedron). It is assumed that a mixed boundary value condition is given. Global regularity results in Sobolev and in Nikolskii spaces are proven, in particular $[W^{s,2}(\Omega)] ^N$-regularity \ \ $(s < {3 \over 2})$ of $u$.

Keywords: mixed boundary value problems, piecewise smooth boundaries, Nikolskii spaces

Classification (MSC2000): 35J55, 35J65, 35J25

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