  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.
