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Boundary trace embedding theorems for variable exponent Sobolev spaces. (English) Zbl 1136.46025

The author studies boundary trace embedding theorems for the variable exponent Sobolev space \(W^{1,p(\cdot)}(\Omega)\), where \(\Omega\) is an open, bounded or unbounded domain in \(\mathbb R^N\). The proofs of these trace embeddings are based on the usage of the known Sobolev embeddings for the spaces \(W^{1,p(\cdot)}(\Omega)\) and the classical boundary trace embedding \(W^{1,1}(\Omega)\to L^1(\partial\Omega)\). The boundary is assumed to satisfy a local Lipshitz condition. The author obtains several theorems on trace embeddings, depending on assumptions on \(p(x)\) and the fact whether \(\Omega\) is bounded or not. One of the theorems (Theorem 2.1) for a bounded domain states that if \(p\in W^{1,\gamma}\) with \(1\leq p_-\leq p_+<N<\gamma\), then the trace embedding \(W^{1,p(\cdot)}(\Omega)\to L^\frac{(N-1)p(\cdot)}{N-p(\cdot)}(\partial\Omega)\) holds. A series of other statements with different assumptions on the variable exponent are also obtained and various corollaries are derived.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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