\(L^ n\) is sharp for the anti-maximum principle
type: Item
instance of: scholarly article
title: \(L^ n\) is sharp for the anti-maximum principle
zbMATH Open document ID: 0885.35016
author: Guido Sweers
published in: Journal of Differential Equations
publication date: 1997-05-28T00:00:00Z
review text: This paper extends the following result of Clément and Peletier that reads as follows for the Dirichlet Laplacian on bounded smooth domains \(\Omega\subset\mathbb{R}^n\): for all \(f>0\) with \(f\in L^p(\Omega)\) and \(p>n\), there is \(\lambda_f>\lambda_1\), where \(\lambda_1\) is the first eigenvalue, such that one finds for \(\lambda\in(\lambda_1, \lambda_f)\) that the solution of \[ -\Delta u= \lambda u+f\quad\text{in }\Omega,\qquad u= 0\quad\text{on }\partial\Omega,\tag{\(*\)} \] satisfies \(u<0\). The author shows that the result above is no longer true for all \(f\in L^p(\Omega)\) with \(p\leq n\). In fact the following proposition is established: Let \(n\geq 2\). There exists \(f\in L^n(\Omega)\) with \(f>0\) such that, for all \(\lambda>\lambda_1\) and \(\lambda\) not an eigenvalue, the solution \(u_\lambda\) of \((*)\) changes sign.
reviewed by: G. A. Philippin
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1000134
zbMATH Keywords: Dirichlet Laplacian
MaRDI profile type: Publication
full work available at URL: jdeq.1996.3211
OpenAlex ID: W2049907854
DOI: 10.1006/JDEQ.1996.3211
Recommended article: Sharp maximal function and \(C_ p\) condition
Recommended article: On the best constant in the \(L^p\) estimate for the sharp maximal function
Recommended article: Extremal functions for the sharp \(L^2\)-Nash inequality
Recommended article: Sharp total variation results for maximal functions
Recommended article: Q5402808
Recommended article: Q4241775
Recommended article: Sharp inequalities for maximal functions associated with general measures
Recommended article: Sharp Estimates for the Nontangential Maximal Function and the Lusin Area Function in Lipschitz Domains
Recommended article: Sharp reversed Hardy-Littlewood-Sobolev inequality on \(\mathbf{R}^{n}\)
Recommended article: A Proof of the Fefferman-Stein-Stromberg Inequality for the Sharp Maximal Functions
description: scientific article; zbMATH DE number 1000134