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

In this paper, we prove that the first (positive) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator $$L(u)=\mathrm{\Delta }u-(\mathrm{\nabla }u,x),$$ 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.