The Dual Hamilton–Jacobi Equation and the Poincaré Inequality

Rigao He (), Wei Wang, Jianglin Fang and Yuanlin Li
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Rigao He: Department of Mathematics, Jiangxi University of Science and Technology, Ganzhou 341000, China
Wei Wang: School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, China
Jianglin Fang: College of Science, Hunan Institute of Engineering, Xiangtan 411104, China
Yuanlin Li: Department of Mathematics, Jiangxi University of Science and Technology, Ganzhou 341000, China

Mathematics, 2024, vol. 12, issue 24, 1-10

Abstract: Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity shown by L. Gross, and applying the ideas and methods of the work by Bobkov, Gentil and Ledoux, we would like to establish a new connection between the logarithmic Sobolev inequalities and the hypercontractivity of solutions of dual Hamilton–Jacobi equations. In addition, Poincaré inequality is also recovered by the dual Hamilton–Jacobi equations.

Keywords: the logarithmic Sobolev inequality; Hamilton–Jacobi equation; the Prékopa–Leindler inequality; Poincaré inequality; the Brunn–Minkowski inequality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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