Smooth approximation in weighted Sobolev spaces. (English) Zbl 0886.46035

Kilpeläinen, T.

Commentat. Math. Univ. Carol. 38, No. 1, 29-35 (1997).

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:

46E35

Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

Sobolev space; Poincaré inequality; doubling weight; compact imbedding