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 <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis u right-parenthesis equals normal upper Delta u minus left-parenthesis x comma nabla u right-parenthesis comma"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>u</mml:mi> <mml:mo> − </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(u)=\Delta u-(x,\nabla u),</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis u right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L(u)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the class of convex bodies.