Two-sided boundary points of Sobolev-extension domains on Euclidean spaces

TL;DR

This paper investigates the Hausdorff dimension of two-sided boundary points in Sobolev-extension domains on Euclidean spaces, providing estimates and examples to understand their geometric complexity.

Contribution

It establishes new bounds on the Hausdorff dimension of boundary points in Sobolev-extension domains and offers examples illustrating these bounds.

Findings

Derived upper bounds on the Hausdorff dimension of boundary points.

Constructed examples demonstrating lower bounds on the dimension.

Enhanced understanding of boundary geometry in Sobolev-extension domains.

Abstract

We prove an estimate on the Hausdorff-dimension of the set of two-sided boundary points of general Sobolev-extension domains on Euclidean spaces. We also present examples showing lower bounds on possible dimension estimates of this type.