Embeddings of anisotropic Sobolev spaces (Q1093137)

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 same holds true without the horn condition.