A Priori Log-Concavity Estimates for Dirichlet Eigenfunctions

TL;DR

This work addresses the problem of obtaining quantitative a priori Hessian bounds for the log of the first Dirichlet eigenfunction on convex domains in Riemannian manifolds. It develops a barrier- and maximum-principle–based framework, together with a one-parameter continuity argument, to derive the Hessian bound $\nabla^2 v + (\alpha |\nabla \sqrt{u}|^2 + C v - d) g < 0$ for $v=\log u$, with constants depending on the eigenvalue $\lambda$, curvature bounds, Ricci-derivative, and boundary geometry. The main contributions include explicit closed-form constants in spherical and Euclidean geometries and a robust extension to generalized problems with potentials $V$ and densities $\rho$, as well as near-boundary sharpness insights that connect to known barrier lemmas and conjectures on stronger log-concavity. The results have significance for spectral-gap analyses, provide a framework for a priori estimates that can feed into existence proofs via continuity methods, and offer geometric-analytic tools for conformal and Einstein-geometric settings. Overall, the paper advances quantitative control on the concavity properties of ground states in curved spaces, with potential impact on geometric analysis and spectral theory.

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 $\log u$.

Theorems & Definitions (21)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Remark 1
• Proposition 2.5