Compact embeddings for weighted Orlicz-Sobolov spaces on \({\mathbb R}^ N\)
type: Item
instance of: scholarly article
title: Compact embeddings for weighted Orlicz-Sobolov spaces on \({\mathbb R}^ N\)
zbMATH Open document ID: 0611.46041
DOI: 10.1007/BF01461724
author: Pierre-A. Vuillermot
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164323
review text: We prove the existence of compact embeddings between certain weighted Orlicz-Sobolev spaces on \({\mathbb{R}}^ N\); we also prove the existence of compact trace embeddings between those spaces on \({\mathbb{R}}^ N\) and certain weighted Orlicz-Sobolev spaces defined on lower k-dimensional hyperplanes \({\mathbb{R}}^ k\). The method of proof is similar in both cases: invoking the Donaldson-Trudinger theory, we first devise an approximation scheme to construct a sequence of compact operators \((I_ n)^{\infty}_{n=1}\) between the definition space and the target space of the concerned embedding; we then control the limit lim \(I_ n\) as \(n\to \infty\) in some appropriate operator topology with suitable asymptotic conditions on the weights at infinity. We finally prove that lim \(I_ n\) indeed coincides with the desired embedding. The basic difficulty of this problem is that those asymptotic conditions on the weights needed to control lim \(I_ n\) also interfere with the growth properties of the convex functions which appear in the definition- and in the target space of the embeddings. Accordingly, the standard conditions traditionally used to describe growth properties of convex functions, such as the \(\Delta _ 2\)- and \(\Delta _ 3\)-conditions, ply an important role in our considerations. We discuss several examples in detail.
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 47B06
zbMATH DE Number: 3987987
zbMATH Keywords: existence of compact embeddings between certain weighted Orlicz-Sobolev spaces
zbMATH Keywords: compact trace embeddings
zbMATH Keywords: Donaldson-Trudinger theory
MaRDI profile type: Publication
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OpenAlex ID: W2016033030
description: scientific article; zbMATH DE number 3987987