# American Institute of Mathematical Sciences

March  2009, 8(2): 683-688. doi: 10.3934/cpaa.2009.8.683

## Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations

 1 Department of Mathematics, Faculty of Science, University of Toyama, Toyama 930-8555, Japan 2 Department of Mathematics, Faculty of Human Development, University of Toyama, Toyama 930-8555, Japan

Received  March 2008 Revised  July 2008 Published  December 2008

We provide a general framework of inequalities induced by the Aubry-Mather theory of Hamilton-Jacobi equations. This framework deals with a sufficient condition on functions $f\in C^1(\mathbb R^n)$ and $g\in C(\mathbb R^n)$ in order that $f-g$ takes its minimum over $\mathbb R^n$ on the set {$x\in \mathbb R^n |Df(x)=0$}. As an application of this framework, we provide proofs of the arithmetic mean-geometric mean inequality, Hölder's inequality and Hilbert's inequality in a unified way.
Citation: Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683
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