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
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909280362299392 |
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| author | Colesanti, Andrea Francini, Elisa Livshyts, Galyna Salani, Paolo |
| author_facet | Colesanti, Andrea Francini, Elisa Livshyts, Galyna Salani, Paolo |
| contents | We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator $$ L(u)=Δu-\langle\nabla u,x\rangle\,, $$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21354 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction Colesanti, Andrea Francini, Elisa Livshyts, Galyna Salani, Paolo Analysis of PDEs Metric Geometry We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator $$ L(u)=Δu-\langle\nabla u,x\rangle\,, $$ 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. |
| title | The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction |
| topic | Analysis of PDEs Metric Geometry |
| url | https://arxiv.org/abs/2407.21354 |