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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.16721 |
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| _version_ | 1866916261505531904 |
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| author | Ishige, Kazuhiro Liu, Qing Salani, Paolo |
| author_facet | Ishige, Kazuhiro Liu, Qing Salani, Paolo |
| contents | In this paper, we provide a new PDE proof for the celebrated Borell--Brascamp--Lieb inequality. Our approach reveals a deep connection between the Borell--Brascamp--Lieb inequality and properties of diffusion equations of porous medium type pertaining to the large time asymptotics and preservation of a generalized concavity of the solutions. We also recover the equality condition in the special case of the Prékopa--Leindler inequality by further exploiting known properties of the heat equation including the eventual log-concavity and backward uniqueness of solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16721 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A parabolic PDE-based approach to Borell--Brascamp--Lieb inequality Ishige, Kazuhiro Liu, Qing Salani, Paolo Analysis of PDEs 39B62, 35E10, 35K05, 35D40 In this paper, we provide a new PDE proof for the celebrated Borell--Brascamp--Lieb inequality. Our approach reveals a deep connection between the Borell--Brascamp--Lieb inequality and properties of diffusion equations of porous medium type pertaining to the large time asymptotics and preservation of a generalized concavity of the solutions. We also recover the equality condition in the special case of the Prékopa--Leindler inequality by further exploiting known properties of the heat equation including the eventual log-concavity and backward uniqueness of solutions. |
| title | A parabolic PDE-based approach to Borell--Brascamp--Lieb inequality |
| topic | Analysis of PDEs 39B62, 35E10, 35K05, 35D40 |
| url | https://arxiv.org/abs/2405.16721 |