A sharp minimum principle for the problem of torsionally rigidity
type: Item
instance of: scholarly article
title: A sharp minimum principle for the problem of torsionally rigidity
zbMATH Open document ID: 0928.35036
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-07-20T00:00:00Z
review text: Let \(\Omega\) be a smooth strictly convex plane domain. Let \(u\) be the solution to the equation \(\Delta u=2\) in \(\Omega\) with \(u=0\) on \(\partial\Omega\). The author proves that the function \(P(x)=| Du| ^2-2u\) attains its minimum value on \(\partial\Omega\). This result was known for \(P^\alpha(x)=| Du| ^2-2\alpha u\) with \(\alpha>1\) [\textit{G. A. Philippin}, J. Math. Anal. Appl. 68, 526-535 (1979; Zbl 0412.73035)]. The proof is carried out by using the analyticity of \(u\) in \(\Omega\). As an application, a lower bound for the minimum of \(| Du| \) in terms of the maximum value of the curvature on \(\partial\Omega\) is obtained.
reviewed by: Giovanni Porru
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35J67
zbMATH DE Number: 1289960
zbMATH Keywords: Saint-Venant equation
zbMATH Keywords: minimum principle
zbMATH Keywords: lower bounds
author: Xinan Ma
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6291
OpenAlex ID: W1980345263
cites work: On isoperimetric gradient bounds for Poisson problems and problems of torsional creep
cites work: Solution of Dirichlet's problem for the equation \(\Delta u = -1\) in a convex region
cites work: Sharp size estimates for capillary free surfaces without gravity.
cites work: On nonlinear elliptic partial differential equations and hölder continuity
cites work: A minimum principle for the problem of torsional creep
cites work: Some Applications of the Maximum Principle in the Problem of Torsional Creep
cites work: Q5533735
cites work: Maximum principles and their applications
DOI: 10.1006/JMAA.1999.6291
description: scientific article; zbMATH DE number 1289960