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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 and Paolo Salani |
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Vol. 19 (2026), No. 7, 1339–1360
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Abstract |
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We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein–Uhlenbeck operator as a function of the domain, is convex with respect to 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. |
Keywords
Ornstein–Uhlenbeck operator, principal frequency, Gauss
space, Brunn–Minkowski inequality, convex body
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Mathematical Subject Classification
Primary: 35E10
Secondary: 35P15, 52A40
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Milestones
Received: 6 August 2024
Revised: 14 July 2025
Accepted: 13 October 2025
Published: 18 September 2026
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Authors |
| Andrea Colesanti | |
| Dipartimento di Matematica e
Informatica “Ulisse Dini” Università di Firenze Firenze Italy |
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| Elisa Francini | |
| Dipartimento di Matematica e
Informatica “Ulisse Dini” Università di Firenze Firenze Italy |
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| Galyna Livshyts | |
| Georgia Institute of
Technology Atlanta, GA United States |
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| Paolo Salani | |
| Dipartimento di Matematica e
Informatica “Ulisse Dini” Università di Firenze Firenze Italy |
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| © 2026 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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