Sobolev Versus Homogeneous Sobolev II

TL;DR

This paper investigates the differences between Sobolev and homogeneous Sobolev extension domains, providing specific examples where one type extends functions but the other does not, highlighting nuanced distinctions in extension properties.

Contribution

It constructs explicit examples of domains that are Sobolev extension domains but not homogeneous Sobolev extension domains for certain exponents.

Findings

Existence of domains that are Sobolev but not homogeneous Sobolev extension domains

Clarification of the relationship between different types of extension domains

Identification of specific exponent ranges where these distinctions occur

Abstract

We study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, for a certain range of exponents $p$ and $q$, we construct a $(W^{1, p}, W^{1, q})$-extension domain which is not an $(L^{1, p}, L^{1, q})$-extension domain.