The strong log-concavity for first eigenfunction of the Ornstein-Uhlenbeck operator in the class of convex bodies

TL;DR

This paper proves that the first eigenfunction of the Ornstein-Uhlenbeck operator is strongly log-concave in convex domains, advancing understanding of spectral properties and equality cases in related inequalities.

Contribution

It establishes strong log-concavity of the first eigenfunction for the Ornstein-Uhlenbeck operator in convex domains and characterizes equality cases in the Brunn-Minkowski inequality.

Findings

First eigenfunction is strongly log-concave in convex domains

Improves previous results on eigenvalue properties

Characterizes equality cases in Brunn-Minkowski inequality

Abstract

In this paper, we prove that the first (positive) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator \[ L(u)=\Delta u-(\nabla u,x), \] is strongly log-concave if the domain is bounded and convex, which improves the conclusion in [6]. We also provide a characterization of the equality case of the Brunn-Minkowski inequality for the principal frequency of $L(u)$ in the class of convex bodies.