Regularity of the solution to a real Monge--Ampère equation on the boundary of a simplex
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916561733812224 |
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| author | Andreasson, Rolf Hultgren, Jakob Jonsson, Mattias Mazzon, Enrica McCleerey, Nicholas |
| author_facet | Andreasson, Rolf Hultgren, Jakob Jonsson, Mattias Mazzon, Enrica McCleerey, Nicholas |
| contents | Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge--Ampère operator on the boundary of a simplex. This can be formulated in terms of optimal transport, and we consider, more generally, the problem of optimal transport between symmetric probability measures on the boundary of a simplex and of the dual simplex. For suitably regular measures, we obtain regularity properties of the transport map, and of its convex potential. To do so, we exploit boundary regularity results for optimal transport maps by Caffarelli, together with the symmetries of the simplex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01620 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity of the solution to a real Monge--Ampère equation on the boundary of a simplex Andreasson, Rolf Hultgren, Jakob Jonsson, Mattias Mazzon, Enrica McCleerey, Nicholas Analysis of PDEs Differential Geometry 35J96, 53A15 (primary), 14J32, 14J33, 14T90, 32Q25 (secondary) Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge--Ampère operator on the boundary of a simplex. This can be formulated in terms of optimal transport, and we consider, more generally, the problem of optimal transport between symmetric probability measures on the boundary of a simplex and of the dual simplex. For suitably regular measures, we obtain regularity properties of the transport map, and of its convex potential. To do so, we exploit boundary regularity results for optimal transport maps by Caffarelli, together with the symmetries of the simplex. |
| title | Regularity of the solution to a real Monge--Ampère equation on the boundary of a simplex |
| topic | Analysis of PDEs Differential Geometry 35J96, 53A15 (primary), 14J32, 14J33, 14T90, 32Q25 (secondary) |
| url | https://arxiv.org/abs/2403.01620 |