Strengthened Sobolev Spaces for Domains with Irregular Boundary

E. G. D'yakonov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 213-229

Résumé

The properties of strengthened Sobolev spaces $G^{1,m}\equiv G^{1,m}(\Omega ;S)$, $m\geq 1/2$, are studied. These spaces are constructed on the basis of the classical space $W_2^1(\Omega )\equiv H^1(\Omega )$ for a bounded plane domain $\Omega$ whose boundary $\Gamma$ is not, in general, Lipschitzian; $S\subset \bar\Omega\equiv\Omega\cup \Gamma$; and $S=\bar S$ consists of finitely many smooth arcs. Special attention is given to situations when either a singular point of the boundary (the definition is given below) belongs to $S$ or two arcs from $S$ are tangent at their common endpoint, whereby the interior angle between them is zero. Characteristics of traces on $S$ and $\Gamma$ are obtained that make it possible to prove not only an extension theorem but also theorems on approximation of elements from $G^{1,1}$ and their traces by smooth functions.