Sobolev embeddings with variable exponent

Let Ω be a bounded open subset of $ℝ^{n}$ with Lipschitz boundary and let $p:\overline{Ω} → [1,∞)$ be Lipschitz-continuous. We consider the generalised Lebesgue space $L^{p(x)}(Ω)$ and the corresponding Sobolev space $W^{1,p(x)}(Ω)$, consisting of all $f