The Sobolev embedding theorem for a domain with irregular boundary.
type: Item
instance of: scholarly article
title: The Sobolev embedding theorem for a domain with irregular boundary.
zbMATH Open document ID: 1081.46502
author: O. V. Besov
published in: Doklady Mathematics
publication date: 2004-06-15T00:00:00Z
review text: 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.
reviewed by: Walter Farkas
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 2074616
zbMATH Keywords: Sobolev embedding
zbMATH Keywords: John domain
zbMATH Keywords: Sobolev--Poincaré inequality
zbMATH Keywords: weighted spaces
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2074616