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Sobolev embeddings with variable exponent. (English) Zbl 0974.46040

Let Ω be a bounded open subset of n with Lipschitz boundary and let p:Ω ¯[1,) be Lipschitz-continuous. The authors consider the generalised Lebesgue space L p(x) (Ω) and the corresponding Sobolev space W 1,p(x) (Ω), consisting of all fL p(x) (Ω) with first-order distributional derivatives in L p(x) (Ω).

It is shown that if 1p(x)<n for all xΩ, then there is a constant c>0 such that for all fW 1,p(x) (Ω),

f M,Ω cf 1,p,Ω ·

Here · M,Ω is the norm on an appropriate space of Orlicz-Musielak type and · 1,p,Ω is the norm on W 1,p(x) (Ω). The inequality reduces to the usual Sobolev inequality if sup Ω p<n. Corresponding results are proved for the case in which p(x)>n for all xΩ.


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