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 \[ L ( u ) = Δ u − ( x , ∇ u ) , L(u)=Δ u-(x,∇ u), \] 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.