Embeddings of anisotropic Sobolev spaces
type: Item
instance of: scholarly article
title: Embeddings of anisotropic Sobolev spaces
zbMATH Open document ID: 0628.46028
DOI: 10.1007/BF00279865
author: R. M. Edmunds
author: D. E. Edmunds
published in: Archive for Rational Mechanics and Analysis
publication date: 1986-01-01T00:00:00Z
review text: This paper deals with the approximation and the entropy numbers of the embedding I of an anisotropic Sobolev space \(W^{r,p}(\Omega)\) into an Orlicz space \(L^{\phi}(\Omega)\), where \(\Omega\) is an open subset of \({\mathbb{R}}^ n\). \(W^{r,p}(\Omega)\), \(1\leq p<\infty\), \(r=(r_ 1,...,r_ n)\in {\mathbb{N}}^ n\), is the set of functions u on \(\Omega\) with norm \(\| u\|_{r,p,\Omega}=\| u\|_{p,\Omega}+\sum^{n}_{i=1}\| D_ i^{r_ i}u\|_{p,\Omega}\), \(\| \cdot \|_{p,\Omega}\) being the \(L^ p\)-norm on \(\Omega\), and the closure of \(C_ 0^{\infty}(\Omega)\) in \(W^{r,p}(\Omega)\) is denoted by \(W_ 0^{r,p}(\Omega)\). \(L^{\phi}(\Omega)\) is the linear hull of the functions u on \(\Omega\) such that \(\int^{0}_{\Omega}\phi (| u(x)|)dx<\infty\), with norm \(\| u\|_{\phi,\Omega}=\inf \{\lambda >0;\int_{\Omega}\phi (| u(x)| /\lambda)dx\leq 1\}\), where \(\phi\) is a given Orlicz function, i.e. \(\phi\) : [0,\(\infty)\to R\), continuous, convex and \(\lim_{t\to 0+}\phi (t)/t=0\), \(\lim_{t\to \infty}\phi (t)/t=\infty\). \(L^{\phi}(\Omega)\) and \(W^{r,p}(\Omega)\) are seen to be Banach spaces. The k-th approximation (resp. entropy) number \(a_ k(I)\) (resp. \(e_ k(I))\) of I is defined as \(a_ k(I)=\inf \{\| I-F\|\); F is a bounded linear map of \(W^{r,p}(\Omega)\) into \(L^{\phi}(\Omega)\) with dim F(W\({}^{r,p}(\Omega))<k\}\) (resp. \(e_ k(I)=\inf \{\epsilon >0\); the image by I of the closed unit ball of \(W^{r,p}(\Omega)\) can be covered by \(2^{k-1}\) closed balls of \(L^{\phi}(\Omega)\) with radius \(\epsilon)\). Then it is proved that both \(a_ k(I)\) and \(e_ k(I)\) are estimated at O((log k)\({}^{-1/(\gamma \nu)})\) as \(k\to \infty\) provided that \(\Omega\) is bounded and satisfies a certain \((=\) strong r-) horn condition, \(1<p<\sigma <\infty\), \(p=\sum^{n}_{j=1}1/r\); \(\phi (t)=t^{\sigma}\exp (t^{\nu})\), \(1<\nu <p'=p/(p-1)\), \(\gamma =\max \{\sigma /\nu,p'/(p'-\nu)\}\). As for the embedding I of \(W_ 0^{r,p}(\Omega)\) into \(L^{\phi}(\Omega)\), the…
reviewed by: K.Yoshinaga
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 4021902
zbMATH Keywords: approximation and entropy numbers
zbMATH Keywords: embedding
zbMATH Keywords: anisotropic Sobolev
zbMATH Keywords: Orlicz space
zbMATH Keywords: Orlicz function
zbMATH Keywords: horn condition
MaRDI profile type: Publication
cites work: Q4073917
cites work: Entropy numbers, s-numbers, and eigenvalue problems
cites work: Inequalities between eigenvalues, entropy numbers, and related quantities of compact operators in Banach spaces
cites work: Entropy and Approximation Numbers of Embeddings in Orlicz Spaces
cites work: A formula for the eigenvalues of a compact operator
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full work available at URL: https://doi.org/10.1007/bf00279865
OpenAlex ID: W4232981502
description: scientific article; zbMATH DE number 4021902