Power concavity and Dirichlet heat flow

TL;DR

This paper investigates the preservation of power concavity under the Dirichlet heat flow in convex domains, establishing that log-concavity is uniquely preserved among all power concavities.

Contribution

It proves that log-concavity is the weakest power concavity preserved by the Dirichlet heat flow, completing the characterization of preserved concavities.

Findings

Log-concavity is the weakest preserved power concavity.

Negative power concavity initial data can lose all concavity immediately.

Log-concavity is the only power concavity preserved by the Dirichlet heat flow.

Abstract

We show that log-concavity is the weakest power concavity preserved by the Dirichlet heat flow in $N$-dimensional convex domains, where $N\ge 2$ (indeed, we prove that starting with a negative power concave initial datum may result in losing immediately any reminiscence of concavity). Jointly with what we already know, i.e. that log-concavity is the strongest power concavity preserved by the Dirichlet heat flow, we see that log-concavity is indeed the only power concavity preserved by the Dirichlet heat flow.