Conditions for existence of single valued optimal transport maps on convex boundaries with nontwisted cost
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917948835233792 |
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| author | Jeong, Seonghyeon Kitagawa, Jun |
| author_facet | Jeong, Seonghyeon Kitagawa, Jun |
| contents | We prove that if $Ω\subset \mathbb{R}^{n+1}$ is a (not necessarily strictly) convex, $C^1$ domain, and $μ$ and $\barμ$ are probability measures absolutely continuous with respect to surface measure on $\partial Ω$, with densities bounded away from zero and infinity, whose $2$-Monge-Kantorovich distance is sufficiently small, then there exists a continuous Monge solution to the optimal transport problem with cost function given by the quadratic distance on the ambient space $\mathbb{R}^{n+1}$. This result is also shown to be sharp, via a counterexample when $Ω$ is uniformly convex but not $C^1$. Additionally, if $Ω$ is $C^{1, α}$ regular for some $α$, then the Monge solution is shown to be Hölder continuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06826 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Conditions for existence of single valued optimal transport maps on convex boundaries with nontwisted cost Jeong, Seonghyeon Kitagawa, Jun Analysis of PDEs 35J96, 49Q22 We prove that if $Ω\subset \mathbb{R}^{n+1}$ is a (not necessarily strictly) convex, $C^1$ domain, and $μ$ and $\barμ$ are probability measures absolutely continuous with respect to surface measure on $\partial Ω$, with densities bounded away from zero and infinity, whose $2$-Monge-Kantorovich distance is sufficiently small, then there exists a continuous Monge solution to the optimal transport problem with cost function given by the quadratic distance on the ambient space $\mathbb{R}^{n+1}$. This result is also shown to be sharp, via a counterexample when $Ω$ is uniformly convex but not $C^1$. Additionally, if $Ω$ is $C^{1, α}$ regular for some $α$, then the Monge solution is shown to be Hölder continuous. |
| title | Conditions for existence of single valued optimal transport maps on convex boundaries with nontwisted cost |
| topic | Analysis of PDEs 35J96, 49Q22 |
| url | https://arxiv.org/abs/2308.06826 |