## 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:

 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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### References:

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