Characterization of $F$-concavity preserved by the Dirichlet heat flow

TL;DR

This paper characterizes the types of $F$-concavity preserved by the Dirichlet heat flow in convex domains, identifying the strongest and weakest preserved concavities and conditions for preservation, extending classical results on concavity properties.

Contribution

It introduces hot-concavity as the strongest preserved $F$-concavity, and provides a complete characterization of all $F$-concavities preserved by the Dirichlet heat flow, including necessary and sufficient conditions.

Findings

Hot-concavity is the strongest preserved $F$-concavity.

Log-concavity is the weakest preserved $F$-concavity.

Certain $F$-concavities are not preserved unless they coincide with log-concavity.

Abstract

$F$-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the $F$-concavities preserved by the Dirichlet heat flow in convex domains on ${\mathbb R}^n$, and complete the study of preservation of concavity properties by the Dirichlet heat flow, started by Brascamp and Lieb in 1976 and developed in some recent papers. More precisely: (1) we discover hot-concavity, which is the strongest $F$-concavity preserved by the Dirichlet heat flow; (2) we show that log-concavity is the weakest $F$-concavity preserved by the Dirichlet heat flow; quasi-concavity is also preserved only for $n=1$; (3) we prove that if $F$-concavity does not coincide with log-concavity and it is not stronger than log-concavity and $n\ge 2$, then there exists an $F$-concave initial datum such that the corresponding solution to the Dirichlet heat flow is not even quasi-concave, hence losing any reminiscence of concavity. Furthermore, we find a sufficient and necessary condition for $F$-concavity to be preserved by the Dirichlet heat flow. We also study the preservation of concavity properties by solutions of the Cauchy--Dirichlet problem for linear parabolic equations with variable coefficients and for nonlinear parabolic equations such as semilinear heat equations, the porous medium equation, and the parabolic $p$-Laplace equation.