On the compactness of embeddings of weighted Sobolev spaces on a domain with an irregular boundary. (English. Russian original) Zbl 1081.46503

Besov, O. V.

Dokl. Math. 63, No. 1, 95-100 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 376, No. 6, 727-732 (2001).

Sobolev embeddings \[ W^s_p(G)\subset L_q(G), \quad s\in \mathbb N,\;1\leq p<q<\infty, \] on a domain \(G\) with a regular boundary are compact for \(s-\frac{n}{p}+\frac{n}{q}>0\). Sufficient conditions of this type were established by Kondrashov, whereas the case \(q=p\) was investigated earlier by Rellich. The goal of the paper under review is to formulate in simple geometric terms sufficient conditions for the compactness of the Sobolev embedding of the weighted and non-weighted spaces on domains \(G\) with regular and irregular boundaries.

Reviewer: Walter Farkas (Zürich)

MSC:

46E35

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

Keywords:

Sobolev embedding; compact embedding; irregular boundary