
Journal of Convex Analysis 23 (2016), No. 3, 893920 Copyright Heldermann Verlag 2016 Integral Inequalities for Infimal Convolution and HamiltonJacobi Equations Patrick J. Rabier Dept. of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. rabier@imap.pitt.edu [Abstractpdf] 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 $$ (fm_{f,g})^{1}_{\phi }+(g+m_{f,g})^{1}_{\phi }\leq 4(f\Box g)^{1}_{\phi } $$ 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 $$ (f\Box g)^{1}_{\phi }\leq 2^{N1}((\check{f}m_{f,g})^{1}_{\phi }+(\check{ g}+m_{f,g})^{1}_{\phi }), $$ 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 longtime behavior of the solutions of the HamiltonJacobi and related equations. Keywords: BrunnMinkowski inequality, enclosing ball, HamiltonJacobi equations, infimal convolution, Orlicz space, rearrangement. MSC: 26D15, 46E30, 35F25, 49L25 [ Fulltextpdf (249 KB)] for subscribers only. 