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

Abstract

Let $n\ge 2$ 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 $q\le n-1$, $n>2$, then $|\partial \Omega |=0$. In the case $1\le q<n-1$, we give an example of a Sobolev $(p,q)$-extension domain with $|\partial \Omega |>0$.

arXiv comment44 pages, 3 figures