Extending Sobolev Functions with Partially Vanishing Traces from Locally (epsilon,delta)-Domains and Applications to Mixed Boundary Problems

Abstract

We prove that given any positive integer $k$, for each open set $\Omega$ and any closed subset $D$ of its closure such that $\Omega$ is locally an (epsilon,delta)-domain near points in the boundary of $\Omega$ not contained in $D$ there exists a linear and bounded extension operator $E$ mapping, for each $p\in[1,\infty]$, the space $W^{k,p}_D(\Omega)$ into $W^{k,p}_D({\mathbb{R}}^n)$. Here, with $O$ denoting either $\Omega$ or the entire ambient, the space $W^{k,p}_D(O)$ is defined as the completion in the classical Sobolev space $W^{k,p}(O)$ of compactly supported smooth functions whose supports are disjoint from $D$. In turn, this result is used to develop a functional analytic theory for the class $W^{k,p}_D(\Omega)$ (including intrinsic characterizations, boundary traces and extensions results, interpolation theorems, among other things) which is then employed in the treatment of mixed boundary value problems formulated in locally (epsilon,delta)-domains.