The volume of the boundary of a Sobolev $(p,q)$-extension domain

TL;DR

This paper investigates the boundary measure of Sobolev (p,q)-extension domains, proving that under certain conditions the boundary has zero volume, and providing examples where it does not.

Contribution

It establishes conditions under which the boundary of Sobolev (p,q)-extension domains has zero volume and constructs examples with positive boundary measure.

Findings

Boundaries of certain Sobolev (p,q)-extension domains have zero volume.

Existence of Sobolev (p,q)-extension domains with positive boundary volume for specific q.

Capacitory restrictions influence boundary measure in Sobolev extension domains.

Abstract

Let $n\geq 2$ and $1\leq q<p<\fz$. We prove that if $\Omega\subset\mathbb R^n$ is a Sobolev $(p, q)$-extension domain, with additional capacitory restrictions on boundary in the case $q\leq n-1$, $n>2$, then $|\partial\Omega|=0$. In the case $1\leq q<n-1$, we give an example of a Sobolev $(p,q)$-extension domain with $|\partial\Omega|>0$.