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

By Kevin Brewster, D. Mitrea, I. Mitrea, M. Mitrea. 2012

51 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/f7fd6236f529cc296152208bcc40839c6252e710

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In Applied Mathematics

1. Antoine Lemenant
2. Laura V. Spinolo
3. Emmanouil D. Milakis
4. Luis Silvestre

In Computational Theory and Mathematics

1. Antoine Lemenant
2. Laura V. Spinolo
3. Emmanouil D. Milakis
4. Luis Silvestre

Citations in the corpus

• On the extension property of Reifenberg-flat domains (2012, background): “Mitrea [1].”