A necessary condition for Sobolev extension domains in higher dimensions

TL;DR

This paper establishes a necessary geometric condition for higher-dimensional Sobolev extension domains, generalizing planar results and providing quantitative criteria, with an example of a complex boundary domain.

Contribution

It introduces a new necessary condition for Sobolev extension domains in higher dimensions based on boundary measure properties, extending previous planar and low-regularity results.

Findings

Provides a necessary condition involving boundary measure and distance functions.

Generalizes planar curve conditions to higher dimensions.

Constructs an example of a complex boundary extension domain.

Abstract

We give a necessary condition for a domain to have a bounded extension operator from $L^{1,p}(\Omega)$ to $L^{1,p}(\mathbb R^n)$ for the range $1 < p < 2$. The condition is given in terms of a power of the distance to the boundary of $\Omega$ integrated along the measure theoretic boundary of a set of locally finite perimeter and its extension. This generalizes a characterizing curve condition for planar simply connected domains, and a condition for $W^{1,1}$-extensions. We use the necessary condition to give a quantitative version of the curve condition. We also construct an example of an extension domain that is homeomorphic to a ball and has $n$-dimensional boundary.