Strong log-concavity for the 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) = Δ u − (x, ∇ u), L (u) = u- (x, u), \ is strongly log-concave if the domain is bounded and convex, which improves the conclusion by Colesanti et al. to appear in Anal. PDE. We also provide a characterization of the equality case of the Brunn-Minkowski inequality for the principal frequency of L (u) L (u) in the class of convex bodies.