Heat equation and the sharp Young's inequality

2012/04/10 by Giuseppe Toscani, Giuseppe, Toscani
Mathematics · Physics and Astronomy · #82C40 #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Biology Tumor Growth #Mathematical Physics (math-ph) #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.1204.2086

openalex publication_date 2012/04/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We show that the sharp Young's inequality for convolutions first obtained by Bechner and Brascamp-Lieb can be derived from the monotone in time evolution of a Lyapunov functional of the convolution of two solutions to the heat equation, with different diffusion coefficients, first introduced by Bennett and Bez. Our proof is based on a suitable adaptation of an old idea of Stam and Blachman, used to obtain Shannon's entropy power inequality.

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