Power concavity and 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.

The authors withdrew this preprint because its conclusion is now contained as a corollary in arXiv:2207.13449 (see Corollary 1.7)

References in corpus (1)

• To logconcavity and beyond