A Priori Log-Concavity Estimates for Dirichlet Eigenfunctions

TL;DR

This paper derives quantitative log-concavity estimates for the first Dirichlet eigenfunction in convex domains of Riemannian manifolds, enhancing understanding of eigenfunction behavior under geometric constraints.

Contribution

It provides new a priori estimates for the Hessian of the logarithm of the principal eigenfunction in convex Riemannian domains, assuming log-concavity.

Findings

Quantitative bounds on the Hessian of log u

Enhanced understanding of eigenfunction concavity

Applications to geometric analysis

Abstract

In this paper, we establish a priori log-concavity estimates for the first Dirichlet eigenfunction of convex domains of a Riemannian manifold. Specifically, we focus on cases where the principal eigenfunction $u$ is assumed to be log-concave and our primary goal is to obtain quantitative estimates for the Hessian of $\log u$.