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arXiv · 2105.02574

Power concavity and Dirichlet heat flow

Kazuhiro Ishige·Paolo Salani·Asuka Takatsu
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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.

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Kazuhiro Ishige, Paolo Salani, Asuka Takatsu. 2021-05-06. Power concavity and Dirichlet heat flow. https://arxiv.org/abs/2105.02574

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