Characterisation of zero trace functions in variable exponent Sobolev spaces

Abstract

It is well known that if u belongs to the Sobolev space W1,p(Ω), where Ω is an open subset of RN and p∈(1,∞), then u∈W01,p(Ω) if u/d belongs to weak Lp(Ω), where d(x)= dist x,∂Ω. Results of this type are given here for Sobolev spaces with a variable exponent p, under the conditions that Ω is bounded and satisfies a mild regularity condition, and p is a bounded, log-Holder continuous function that is bounded away from 1. The outcome includes theorems that are new even when p is constant. In particular it is shown that u∈W01,p(Ω) if and only if u∈W1,p(Ω) and u/d∈L1(Ω).