Weighted Sobolev spaces for Laplace's equation in \(\mathbb{R}^ n\)
type: Item
instance of: scholarly article
title: Weighted Sobolev spaces for Laplace's equation in \(\mathbb{R}^ n\)
zbMATH Open document ID: 0836.35038
author: V. Girault
author: Chérif Amrouche
author: Jean Giroire
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1995-12-03T00:00:00Z
review text: The purpose of this paper is to solve the problem \(\Delta u = f\) in \(\mathbb{R}^n\). The problem is set in spaces similar to the usual Sobolev spaces \(W^{m,p} (\mathbb{R}^n)\), with weights that control the behaviour of the function at infinity. \(D (\mathbb{R}^n)\) denotes the space of infinitely differentiable functions with compact support and \(D' (\mathbb{R}^n)\) denotes its dual space. Let \(\rho = (1 + |x |^2)^1/2\), \(r = |x |\), \(x \in \mathbb{R}^n\). \[ W^{m, p}_{ \alpha, \beta} (\mathbb{R}^n) = \{u \in D' (\mathbb{R}^n);\;\forall \lambda \in \mathbb{N}^n : 0 \leq |\lambda |\leq k,\;\rho^{\alpha - m + |\lambda |} (\ln (2 + r^2))^{\beta - 1} D^\lambda u \in L^p (\mathbb{R}^n); \] \[ \forall \lambda \in \mathbb{N}^n : k + 1 \leq |\lambda |\leq m,\;\rho^{\alpha - m + |\lambda |} (\ln (2 + r^2))^\beta D^\lambda u \in L^p (\mathbb{R}^n)\}, \] which is a reflexive Banach space with natural norm \(|u |_{W^{m, p}_{\alpha, \beta}} (\mathbb{R}^n)\), and the seminorm: \(|u |_{W^{m,p}_{\alpha, \beta}} (\mathbb{R}^n) = (\sum_{|\lambda |= m} |\rho ^\alpha (\ln (2 + r^2))^\beta D^\lambda u |^p_{L^p (\mathbb{R}^n)} )^{1/p}\). The main results in this paper are: 1) Let \(\alpha\) and \(\beta\) be two real numbers and \(m \geq 1\) an integer not satisfying simultaneously \((n/p) + \alpha \in \{1,2, \dots, m\}\) and \((\beta - 1) p = - 1\). Let \(q' = \inf (q,m - 1)\), where \(q\) is the highest degree of the polynomials contained in \(W^{m, p}_{\alpha, \beta} (\mathbb{R}^n)\). Then the seminorm \(|\cdot |_{W^{m,p}_{\alpha, \beta}} (\mathbb{R}^n)\) defines on \(W^{m,p}_{\alpha, \beta} (\mathbb{R}^n)/P_{q'}\) a norm which is equivalent to the quotient norm. 2) Let \(\ell\) be a nonnegative integer and assume that \(n/p\), does not belong to \(\{1, \dots, \ell\}\), with the conventions that this set is empty when \(\ell = 0\). The following Laplace operators are isomorphisms: \[ \begin{aligned} \Delta : W_\ell^{1,p} (\mathbb{R}^n)/P_{[1 - \ell - n/p]} & \mapsto W_\ell^{-1…
reviewed by: Rudolf Kodnár
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 811456
zbMATH Keywords: weighted Sobolev spaces
zbMATH Keywords: polynomials
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 811456