Neumann comparison theorems in elliptic PDEs
type: Item
instance of: scholarly article
title: Neumann comparison theorems in elliptic PDEs
zbMATH Open document ID: 1336.35125
author: Jeffrey J. Langford
published in: Potential Analysis
publication date: 2016-01-27T00:00:00Z
review text: In this paper, the author exemplary compares the solution of two PDEs, namely the Poisson PDE and the symmetrized Poisson PDE with Neumann boundary conditions with the aim to adopt Talenti's Theorem, since it cannot be applied directly. In order to do that, he starts with the background of the spherical and hemispherical setting. Next, he takes a look at the required rearrangements -- more precisely the \((k,n)\)-spherical and \((k,n)\)-hemispherical rearrangement to prove his Neumann comparison theorem for the Poisson PDE and further elliptic PDEs. After that, he also presents comparison results for the Euclidean space. In addition, the author states that ``This new technique of using projections to obtain original comparison results is one of the major contributions of our paper'' (see page 418). Finally, the paper is closed with some open problems and conjectures in the unit disk, where different examples concerning heat flow are discussed.
reviewed by: Sarah Eberle
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 6536138
zbMATH Keywords: elliptic PDEs
zbMATH Keywords: comparison results
zbMATH Keywords: symmetrization
zbMATH Keywords: oscillation inequalities
zbMATH Keywords: Poisson PDE
zbMATH Keywords: symmetrized Poisson PDE
zbMATH Keywords: Neumann boundary conditions
zbMATH Keywords: heat flow
MaRDI profile type: Publication
full work available at URL: s11118-015-9479-0
OpenAlex ID: W1975575909
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DOI: 10.1007/S11118-015-9479-0
description: scientific article; zbMATH DE number 6536138