Zero volume boundary for extension domains from Sobolev to $BV$

TL;DR

This paper proves that the boundary of certain extension domains in Sobolev and BV spaces has zero volume under fatness conditions, especially in the planar case, advancing understanding of boundary regularity in these function spaces.

Contribution

It establishes that the boundary of $(W^{1,p}, BV)$-extension domains has zero volume under fatness assumptions, including the planar case, which was previously unknown.

Findings

Boundary of $(W^{1,p}, BV)$-extension domains has volume zero.

Planar $(W^{1,p}, BV)$-extension domains have boundary of volume zero.

Fatness condition ensures boundary regularity in extension domains.

Abstract

In this note, we prove that the boundary of a $(W^{1, p}, BV)$-extension domain is of volume zero under the assumption that the domain $\boz$ is $1$-fat at almost every $x\in\partial\boz$. Especially, the boundary of any planar $(W^{1, p}, BV)$-extension domain is of volume zero.