On Poincaré-type integral inequalities
type: Item
instance of: scholarly article
title: On Poincaré-type integral inequalities
zbMATH Open document ID: 0599.26028
DOI: 10.1016/0022-247X(86)90069-7
author: B. G. Pachpatte
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: Let \(Q=[0,\sigma]^ n\). The author used very simple tools to prove for functions \(f,g\in C^ 1(Q)\) vanishing on \(\partial Q\) the estimates \[ \int_{Q}| f(x)g(x)| dx\leq \frac{\sigma^ 2}{8n}\int_{Q}[| \nabla f(x)|^ 2+| \nabla g(x)|^ 2]dx, \] \[ \int_{Q}[| f(x)| | \nabla g(x)| +| g(x)| | \nabla f(x)|]dx\leq \frac{\sigma}{2\sqrt{n}}\int_{Q}[| \nabla f(x)|^ 2+| \nabla g(x)|^ 2]dx. \] Here \(| \nabla f|^ 2=\sum^{n}_{i=1}| \partial f/\partial x_ i|^ 2\). The choice of \(g=f\) yields the Friedrichs-Poincaré inequality with a constant better then the ones usually mentioned in the literature.
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 3966535
zbMATH Keywords: multidimensional Poincaré-type integral inequalities
zbMATH Keywords: Friedrichs- Poincaré inequality
reviewed by: Jiří Rákosník
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3966535