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Journal of Convex Analysis 23 (2016), No. 3, 893--920
Copyright Heldermann Verlag 2016

Integral Inequalities for Infimal Convolution and Hamilton-Jacobi Equations

Patrick J. Rabier
Dept. of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.


Let $f,g:\Bbb{R}^{N}\rightarrow (-\infty ,\infty ]$ be Borel measurable, bounded below and such that $\inf f+\inf g\geq 0.$ We prove that with $% m_{f,g}:=(\inf f-\inf g)/2,$ the inequality $$ ||(f-m_{f,g})^{-1}||_{\phi }+||(g+m_{f,g})^{-1}||_{\phi }\leq 4||(f\Box g)^{-1}||_{\phi } $$ holds in every Orlicz space $L_{\phi },$ where $f\Box g$ denotes the infimal convolution of $f$ and $g$ and where $||\cdot ||_{\phi }$ is the Luxemburg norm (i.e., the $L^{p}$ norm when $L_{\phi }=L^{p}$). \par Although no genuine reverse inequality can hold in any generality, we also prove that such reverse inequalities do exist in the form $$ ||(f\Box g)^{-1}||_{\phi }\leq 2^{N-1}(||(\check{f}-m_{f,g})^{-1}||_{\phi }+||(\check{ g}+m_{f,g})^{-1}||_{\phi }), $$ where $\check{f}$ and $\check{g}$ are suitable transforms of $f$ and $g$ introduced in the paper and reminiscent of, yet very different from, nondecreasing rearrangement. \par Similar inequalities are proved for other extremal operations and applications are given to the long-time behavior of the solutions of the Hamilton-Jacobi and related equations.

Keywords: Brunn-Minkowski inequality, enclosing ball, Hamilton-Jacobi equations, infimal convolution, Orlicz space, rearrangement.

MSC: 26D15, 46E30, 35F25, 49L25

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