Integral inequalities for infimal convolution and Hamilton-Jacobi equations

Abstract

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 $L_{\phi }=L^{p}$). 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. 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.