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

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arxiv 2407.21354 v2 pith:DZ32IPWC submitted 2024-07-31 math.AP math.MG

Andrea Colesanti , Elisa Francini , Galyna Livshyts , Paolo Salani This is my paper · ORCID
classification math.APmath.MG
keywords convexdomaineigenfunctioneigenvaluefirstoperatorornstein-uhlenbeckprove
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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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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Impossibility of a nontrivial Brunn--Minkowski inequality for higher Dirichlet eigenvalue

    math.MG 2026-07 accept novelty 7.0 of 10

    For every j≥2 and N≥2, if f∘λ_j is concave on convex bodies then f must be constant.

  2. A Brunn-Minkowski inequality for Schr\"odinger operators with Kato class potentials

    math.AP 2026-03 unverdicted novelty 6.0 of 10

    For Schrödinger operators −div(A∇)+V with convex Kato-class potentials, the first Dirichlet eigenvalue is convex under Minkowski combination of convex domains, and the ground state is log-concave.

  3. The strong log-concavity for first eigenfunction of the Ornstein-Uhlenbeck operator in the class of convex bodies

    math.AP 2025-07 conditional novelty 6.0 of 10

    The first Dirichlet eigenfunction of the Ornstein-Uhlenbeck operator on any bounded convex domain is strongly log-concave, and the equality case in the Brunn-Minkowski inequality for its principal frequency is charact...

  4. log-concavity of eigenfunction and Brunn-Minkowski inequality of eigenvalue for weighted p-Laplace operator

    math.AP 2024-11 conditional novelty 5.0 of 10

    For bounded C^2 convex domains and p>1, the first weighted p-Laplace eigenfunction is log-concave and its first eigenvalue satisfies λ(Ω_t) ≤ (1-t)λ(Ω_0)+tλ(Ω_1).

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