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

2025/07/01 by Qin, Lei · 1 citation
#52A20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.00819

Abstract

In this paper, we prove that the first (positive) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator L(u)=Δu-(∇ 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.

Citations

• Geometric properties of solutions to elliptic PDE's in Gauss space and related Brunn-Minkowski type inequalities
• log-concavity of eigenfunction and Brunn-Minkowski inequality of eigenvalue for weighted p-Laplace operator
• The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction
• Stability of the Gaussian Faber-Krahn inequality
• The Cheeger problem in abstract measure spaces
• Kohler-Jobin meets Ehrhard: the sharp lower bound for the Gaussian principal frequency while the Gaussian torsional rigidity is fixed, via rearrangements
• Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations in \R³

Cited by

• Sharp Fundamental Gap Estimate on Convex Domains in Gaussian Spaces

Related

• The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction
• log-concavity of eigenfunction and Brunn-Minkowski inequality of eigenvalue for weighted p-Laplace operator
• A Priori Log-Concavity Estimates for Dirichlet Eigenfunctions
• On the Ornstein-Uhlenbeck operator in convex sets of Banach spaces
• The Faber-Krahn position of convex bodies and Gaussian measure inequalities