The Sobolev embedding theorem for a domain with irregular boundary.
The Sobolev embedding \[ W^s_p(G)\subset L_q(G), \quad s\in \mathbb N,\;1< p < q <\infty, \] on a domain \(G\in \mathbb R^n\) with the cone condition was proved by S.~L.~Sobolev for \(s-\frac{n}{p}+\frac{n}{q}\geq 0\). The last relation is also a necessary condition for the embedding. Sobolev's result was transferred later on to domains of more general form (domains with the John condition, and domains with the flexible cone condition, etc.). For a domain \(G\) with an irregular boundary, the embedding holds only under stronger conditions that relate the numbers \(s\), \(p\), and \(q\) and depend on the geometric properties of the boundary\(\partial G\). The goal of the paper under review is to investigate the validity of the Sobolev embedding with \(s\in\mathbb N\) for domains \(G\) from a certain class, in particular, for the so-called \(\sigma\)-John domains. For \(s=1\), the method of the author makes it possible to extend the class of domains for which the Sobolev--Poincaré inequality is valid. The results are generalized to the case of weighted spaces.
scientific article; zbMATH DE number 2074616