A geometric approach to apriori estimates for optimal transport maps
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912056996790272 |
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| author | Brendle, Simon Léger, Flavien McCann, Robert J. Rankin, Cale |
| author_facet | Brendle, Simon Léger, Flavien McCann, Robert J. Rankin, Cale |
| contents | A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma--Trudinger--Wang condition is the Pogorelov second derivative bound. This translates to an apriori interior $C^1$ estimate for smooth optimal maps. Here we give a new derivation of this estimate which relies in part on Kim, McCann and Warren's observation that the graph of an optimal map becomes a volume maximizing spacelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_10208 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A geometric approach to apriori estimates for optimal transport maps Brendle, Simon Léger, Flavien McCann, Robert J. Rankin, Cale Differential Geometry Analysis of PDEs Optimization and Control A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma--Trudinger--Wang condition is the Pogorelov second derivative bound. This translates to an apriori interior $C^1$ estimate for smooth optimal maps. Here we give a new derivation of this estimate which relies in part on Kim, McCann and Warren's observation that the graph of an optimal map becomes a volume maximizing spacelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost. |
| title | A geometric approach to apriori estimates for optimal transport maps |
| topic | Differential Geometry Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2311.10208 |