Characterisation of zero trace functions in variable exponent Sobolev spaces

It is well known that if u belongs to the Sobolev spaceW 1 , p( Ω ) , where Ω is an open subset of R N and p ∈ ( 1 , ∞ ) , then u ∈ W 0 1 , p( Ω )if u / d belongs to weakL p ( 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‐Hölder 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 ∈ W 0 1 , p( Ω )if and only if u ∈ W 1 , p( Ω )and u / d ∈ L 1 ( Ω ) .