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

TL;DR

This paper proves that for certain Sobolev extension domains, the boundary's volume is zero under specific conditions relating p and q, extending understanding of boundary properties in Sobolev spaces.

Contribution

It establishes a new result linking the parameters p and q to the boundary measure of Sobolev extension domains, specifically when the boundary volume is zero.

Findings

Boundary volume of Sobolev (p,q)-extension domains is zero under given conditions.

The result applies when 1 ≤ q < p < qn/(n-q).

Advances the understanding of boundary regularity in Sobolev spaces.

Abstract

We show that the volume of the boundary of a bounded Sobolev $(p,q)$-extension domain is zero when $1\leq q <p< \frac{qn}{(n-q)}.$