The Neumann problem for Sub-Laplacians on Carnot groups and the extension theorem for Sobolev spaces
type: Item
instance of: scholarly article
title: The Neumann problem for Sub-Laplacians on Carnot groups and the extension theorem for Sobolev spaces
zbMATH Open document ID: 1052.46026
author: D. M. Nhieu
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2004-09-07T00:00:00Z
review text: The main aim of the paper is to prove the existence of a unique variational solution to the Neumann boundary value problem for Sub-Laplacians on Carnot groups. As in the classical approach, it is based on the use of suitable extension and trace theorems for appropriate Sobolev spaces. The central result concerns the construction of a linear bounded extension operator \(\mathcal E:\mathcal L^{k,p}(\Omega)\to\mathcal L^{k,p}(G)\), where \(\mathcal L^{k,p}\) is the Folland-Stein non-isotropic Sobolev space, \(1\leq p\leq\infty\), \(k\geq1\), \(G\) is a Carnot group of step \(r<\infty\), and \(\Omega\subset G\) is an \((\varepsilon,\delta)\) domain with \(\text{rad}(\Omega)>0\). The main outline of the proof follows \textit{P. W. Jones} [Acta Math. 147, 71--88 (1981; Zbl 0489.30017)]. An important role in the proof is played by a theorem on the appropriate approximation of functions from \(\mathcal L^{k,p}(\Omega)\) by functions from \(C^\infty(G)\). The extension theorem is then applied to prove a trace theorem for functions from \(\mathcal L^{1,p}(\Omega)\), a compact embedding of \(\mathcal L^{k,p}(\Omega)\) in \(\mathcal L^{k-1,q}(\Omega)\), an existence theorem for the \(L^2\)-variational Neumann problem on \(G\), and several further corollaries.
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35H20
zbMATH DE Number: 2099209
zbMATH Keywords: Carnot groups
zbMATH Keywords: \((\varepsilon,\delta)\) domain
zbMATH Keywords: extension theorem
zbMATH Keywords: Sobolev space
zbMATH Keywords: Neumann problem
zbMATH Keywords: trace
zbMATH Keywords: compact embedding
reviewed by: Jiří Rákosník
MaRDI profile type: Publication
full work available at URL: s10231-001-8195-0
OpenAlex ID: W2017797845
DOI: 10.1007/S10231-001-8195-0
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description: scientific article; zbMATH DE number 2099209