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Sobolev embeddings with variable exponent. (English) Zbl 0974.46040
Let $\Omega$ be a bounded open subset of $\Bbb R^n$ with Lipschitz boundary and let $p:\overline{\Omega}\to [1,\infty)$ be Lipschitz-continuous. The authors consider the generalised Lebesgue space $L^{p(x)}(\Omega)$ and the corresponding Sobolev space $W^{1,p(x)}(\Omega)$, consisting of all $f\in L^{p(x)}(\Omega)$ with first-order distributional derivatives in $L^{p(x)}(\Omega)$. It is shown that if $1\le p(x)<n$ for all $x\in\Omega$, then there is a constant $c>0$ such that for all $f\in W^{1,p(x)}(\Omega)$, $$ \|f\|_{M,\Omega}\le c \|f\|_{1,p,\Omega}. $$ Here $\|\cdot \|_{M,\Omega}$ is the norm on an appropriate space of Orlicz-Musielak type and $\|\cdot \|_{1,p,\Omega}$ is the norm on $W^{1,p(x)}(\Omega)$. The inequality reduces to the usual Sobolev inequality if $\sup _{\Omega}p<n$. Corresponding results are proved for the case in which $p(x)>n$ for all $x\in\Omega$.

46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10Inequalities involving derivatives, differential and integral operators
Full Text: EuDML