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Generating monotone quantities for the heat equation

Jonathan Bennett and Neal Bez

Abstract

The purpose of this article is to expose and further develop a simple yet surprisingly far-reaching framework for generating monotone quantities for positive solutions to linear heat equations in euclidean space. This framework is intimately connected to the existence of a rich variety of algebraic closure properties of families of sub/super-solutions, and more generally solutions of systems of differential inequalities capturing log-convexity properties such as the Li–Yau gradient estimate. Various applications are discussed, including connections with the general Brascamp–Lieb inequality and the Ornstein–Uhlenbeck semigroup.

Funding source: H2020 European Research Council

Award Identifier / Grant number: 307617

Funding statement: The work of the first author was supported by the European Research Council (grant number 307617). The work of the second author was supported by JSPS Kakenhi (grant numbers 26887008 and 16H05995).

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Received: 2015-09-22
Revised: 2017-04-25
Published Online: 2017-06-08
Published in Print: 2019-11-01

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