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Smooth approximation in weighted Sobolev spaces. (English) Zbl 0886.46035
The following problem is considered: When $$H^{1,p}(\Omega;\mu) =W^{1,p}(\Omega;\mu)$$, where $$H^{1,p}(\Omega;\mu)$$ is the weighted Sobolev space defined as a closure of $$C^\infty$$ functions in the usual norm and $$W^{1,p}(\Omega;\mu)$$ is the weighted Sobolev space defined via distributional derivatives, and $$d\mu =w(x) dx$$, is a $$p$$-admissible weight (that is, $$\mu$$ is doubling and the Poincaré inequality with respect to $$\mu$$ holds for smooth functions). The author proves a weighted counterpart of the well known theorem due to Meyers and Serrin; he shows that a necessary and sufficient condition for the coincidence of these spaces is that $$W^{1,p}(\Omega;\mu)$$ is a Banach space and the Poincaré inequality holds for functions in $$W^{1,p}(\Omega;\mu)$$. The main tool for the proof is another result of the paper, namely, a compactness theorem in $$H^{1,p}(\Omega;\mu)$$.
Reviewer: M.Krbec (Praha)

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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