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We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator L(u)=Δ u-∇ u,x\,, as a function of the domain, is convex with respect to the Minkowski addition, and we characterize the equality cases in some classes of convex sets. We also prove that the corresponding (positive) eigenfunction is log-concave if the domain is convex.
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Colesanti et al. (2024) studied this question.
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