Morrey–Sobolev Extension Domains

Abstract

We show that every uniform domain of R n with n ≥ 2 is a Morrey-Sobolev W 1, p-extension domain for all p ∈ [1, n), and moreover, that this result is essentially best possible for each p ∈ [1, n) in the sense that, given a simply connected planar domain or a domain of R n with n ≥ 3 that is quasiconformal equivalent to a uniform domain, if it is a W 1, p-extension domain, then it must be uniform.