Boundary trace embedding theorems for variable exponent Sobolev spaces ✩

We study boundary trace embedding theorems for variable exponent Sobolev space W1,p(·)(Ω). Let Ω be an open (bounded or unbounded) domain in RN satisfying strong local Lipschitz condition. Under the hypotheses that p ∈ L∞(Ω), 1 inf p(x) sup p(x) < N, |∇p| ∈ Lγ (·)(Ω), where γ ∈ L∞(Ω) and infγ (x) > N, we prove that there is a continuous boundary trace embedding W1,p(·)(Ω)→Lq(·)(∂Ω) provided q(·), a measurable function on ∂Ω, satisfies condition p(x) q(x) (N−1)p(x) N−p(x) for x ∈ ∂Ω. © 2007 Elsevier Inc. All rights reserved