Sobolev Versus Homogeneous Sobolev Extension

TL;DR

This paper explores the relationship between Sobolev and homogeneous Sobolev extension domains, establishing conditions under which domains serve as extension domains for these function spaces.

Contribution

It provides new characterizations and equivalences between Sobolev and homogeneous Sobolev extension domains under various parameter conditions.

Findings

Bounded $(L^{1, p}, L^{1, q})$-extension domains are also $(W^{1, p}, W^{1, q})$-extension domains.

Equivalence of Sobolev and homogeneous Sobolev extension domains under certain parameter ranges.

Existence of domains that are Sobolev extension domains but not homogeneous Sobolev extension domains for specific parameters.

Abstract

In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results. 1- Let $1\leq q\leq p\leq \infty$. Then a bounded $(L^{1, p}, L^{1, q})$-extension domain is also a $(W^{1, p}, W^{1, q})$-extension domain. 2- Let $1\leq q\leq p<q^\star\leq \infty$ or $n< q \leq p\leq \infty$. Then a bounded domain is a $(W^{1, p}, W^{1, q})$-extension domain if and only if it is an $(L^{1, p}, L^{1, q})$-extension domain. 3- For $1\leq q<n$ and $q^\star<p\leq \infty$, there exists a bounded domain $\Omega\subset\mathbb{R}^n$ which is a $(W^{1, p}, W^{1, q})$-extension domain but not an $(L^{1, p}, L^{1, q})$-extension domain for $1 \leq q <p\leq n$.