Power concavity and Dirichlet heat flow

Kazuhiro Ishige; Paolo Salani; Asuka Takatsu

Power concavity and Dirichlet heat flow

Analysis of PDEs 2022-07-29 v3

Authors: Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

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.

Cite

@article{arxiv.2105.02574,
  title  = {Power concavity and Dirichlet heat flow},
  author = {Kazuhiro Ishige and Paolo Salani and Asuka Takatsu},
  journal= {arXiv preprint arXiv:2105.02574},
  year   = {2022}
}

Comments

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

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