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

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 H^N, we have: 1. The first (positive) Dirichlet eigenfunction of the Laplacian on a ball in H^N raised to some power α > 0 is strictly concave. 2. Let L be the heat kernel on H^N. Then L(⋅,y,t) is strictly log-concave in H^N for y H^N and t > 0.