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We prove that given any positive integer $k$, for each open set $\Omega$ andany 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 theentire ambient, the space $W^{k,p}_D(O)$ is defined as the completion in theclassical Sobolev space $W^{k,p}(O)$ of compactly supported smooth functionswhose supports are disjoint from $D$. In turn, this result is used to develop afunctional analytic theory for the class $W^{k,p}_D(\Omega)$ (includingintrinsic characterizations, boundary traces and extensions results,interpolation theorems, among other things) which is then employed in thetreatment of mixed boundary value problems formulated in locally(epsilon,delta)-domains.

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