The Neumann problem in the half-space
type: Item
instance of: scholarly article
title: The Neumann problem in the half-space
zbMATH Open document ID: 1019.35026
DOI: 10.1016/S1631-073X(02)02428-7
author: Chérif Amrouche
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-09-21T00:00:00Z
review text: Consider the Neumann boundary value problem for the Laplace equation in the half space, \(-\Delta=f\) in \(\mathbb R^N_+\), \(\partial u/\partial x_N=g\) on \(\mathbb R^{N-1}\) with \(N\geq 2\). The Laplace equation has been studied by many authors either in the whole space \(\mathbb R^N\) or in an exterior domain. Several authors studied the present problem in the Hilbert case of \(p=2\) and without the critical cases corresponding to the logarithmic factor. The author studies the problem in the half-space giving an \(L^p\)-theory with \(1<p<\infty\). He introduces appropriate weighted Sobolev spaces and obtains the corresponding trace space and proves for them basic existence and uniqueness results. Note that similar results were obtained for the case of the Dirichlet boundary value problem by \textit{C. Amrouche} and \textit{Š. Nečasová} in [Math. Bohem. 126, 265-274 (2001; Zbl 0979.35038)].
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1803918
zbMATH Keywords: Laplace equation
zbMATH Keywords: Neumann problem
zbMATH Keywords: halfspace
zbMATH Keywords: \(L^p\)-theory
reviewed by: Jiří Rákosník
MaRDI profile type: Publication
cites work: Laplace equation in the half-space with a~nonhomogeneous Dirichlet boundary condition
cites work: Espaces de Sobolev avec poids pour l'équation de Laplace dans le demi-espace
cites work: Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace
full work available at URL: s1631-073x(02)02428-7
OpenAlex ID: W1978386908
description: scientific article; zbMATH DE number 1803918