Abstract

Let f,g:RN→(−∞,∞]f,g:\Bbb{R}^{N}\rightarrow (-\infty,\infty ] be Borel measurable, bounded below and such that inf⁡f+inf⁡g≥0.\inf f+\inf g\geq 0. We prove that with mf,g:=(inf⁡f−inf⁡g)/2,m_{f,g}:=(\inf f-\inf g)/2, the inequality ∣∣(f−mf,g)−1∣∣ϕ+∣∣(g+mf,g)−1∣∣ϕ≤4∣∣(f□g)−1∣∣ϕ||(f-m_{f,g})^{-1}||_{\phi }+||(g+m_{f,g})^{-1}||_{\phi }\leq 4||(f\Box g)^{-1}||_{\phi } holds in every Orlicz space Lϕ,L_{\phi }, where f□gf\Box g denotes the infimal convolution of ff and gg and where ∣∣⋅∣∣ϕ||\cdot ||_{\phi } is the Luxemburg norm (i.e., the LpL^{p} norm when Lϕ=LpL_{\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□g)−1∣∣ϕ≤2N−1(∣∣(fˇ−mf,g)−1∣∣ϕ+∣∣(gˇ+mf,g)−1∣∣ϕ),||(f\Box g)^{-1}||_{\phi }\leq 2^{N-1}(||(\check{f}-m_{f,g})^{-1}||_{\phi }+||(\check{ g}+m_{f,g})^{-1}||_{\phi }), where fˇ\check{f} and gˇ\check{g} are suitable transforms of ff and gg 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.

Author information

Contact details are reproduced from the original publication and may be historical.

Patrick J. Rabier

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

rabier@imap.pitt.edu

Keywords

  • Brunn-Minkowski inequality
  • enclosing ball
  • Hamilton-Jacobi equations
  • infimal convolution
  • Orlicz space
  • rearrangement

Mathematics Subject Classification

26D15, 46E30, 35F25, 49L25

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Suggested citation

P. J. Rabier. “Integral Inequalities for Infimal Convolution and Hamilton-Jacobi Equations.” Journal of Convex Analysis 23 (2016), No. 3, 893–920. https://doi.org/10.68381/jca23034

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Published by Heldermann Verlag, 2016. Rights now held by Banach Press LLC.