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

Abstract

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 ${\mathbf{H}}^N$ we have: $\bullet$ The first Dirichlet eigenfunction on a ball in ${\mathbf{H}}^N$ is strictly positive power concave; $\bullet$ Let $\Gamma$ be the heat kernel on ${\mathbf{H}}^N$. Then $\Gamma (\cdot ,y,t)$ is strictly log-concave on ${\mathbf{H}}^N$ for $y\in {\mathbf{H}}^N$ and $t>0$.

24 pages. Comments are welcome!

References in corpus (1)

• Parabolic Minkowski convolutions of viscosity solutions to fully nonlinear equations

Cited by in corpus (1)

• The fundamental gap of horoconvex domains in ${\mathbb{H}}^n$