A Priori Log-Concavity Estimates for Dirichlet Eigenfunctions

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 $\log u$.