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arXiv 2002.101412020-02-24math.AP

Power concavity for elliptic and parabolic boundary value problems on rotationally symmetric domains

Kazuhiro Ishige, Paolo Salani, Asuka Takatsu

备注：: 24 pages. Comments are welcome!

英文摘要：

We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space ${\bf H}^N$ we have: $\bullet$ The first Dirichlet eigenfunction on a ball in ${\bf H}^N$ is strictly positive power concave; $\bullet$ Let $Γ$ be the heat kernel on ${\bf H}^N$. Then $Γ(\cdot,y,t)$ is strictly log-concave on ${\bf H}^N$ for $y\in {\bf H}^N$ and $t>0$.

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