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arXiv · 2407.21354

The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction

Andrea Colesanti·Elisa Francini·Galyna Livshyts·Paolo Salani
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Abstract

We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator $$ L(u)=\Delta u-\langle\nabla u,x\rangle\,, $$ 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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Andrea Colesanti, Elisa Francini, Galyna Livshyts, Paolo Salani. 2024-07-31. The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction. https://arxiv.org/abs/2407.21354

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