Synthetic MTW conditions and their equivalence under mild regularity assumption on the cost function
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866915679554240512 |
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| author | Jeong, Seonghyeon |
| author_facet | Jeong, Seonghyeon |
| contents | Loeper's condition in \cite{Loe09} and the quantitatively quasi-convex condition (QQconv) from \cite{GK15} are synthetic expressions of the analytic MTW condition from \cite{TW} since they only require $C^2$ differentiability of the cost function $c$. When the cost function $c$ is $C^4$, it is known that the two synthetic MTW conditions are equivalent to the analytic MTW condition. However, when the cost function has regularity weaker than $C^4$, it is not known that if the two synthetic MTW conditions are equivalent. In this paper, we show the equivalence of the synthetic MTW conditions when the cost function has only $C^2$ regularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2010_14471 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Synthetic MTW conditions and their equivalence under mild regularity assumption on the cost function Jeong, Seonghyeon Analysis of PDEs Optimization and Control 35J96, 26B25, 49Q22, 49J52 Loeper's condition in \cite{Loe09} and the quantitatively quasi-convex condition (QQconv) from \cite{GK15} are synthetic expressions of the analytic MTW condition from \cite{TW} since they only require $C^2$ differentiability of the cost function $c$. When the cost function $c$ is $C^4$, it is known that the two synthetic MTW conditions are equivalent to the analytic MTW condition. However, when the cost function has regularity weaker than $C^4$, it is not known that if the two synthetic MTW conditions are equivalent. In this paper, we show the equivalence of the synthetic MTW conditions when the cost function has only $C^2$ regularity. |
| title | Synthetic MTW conditions and their equivalence under mild regularity assumption on the cost function |
| topic | Analysis of PDEs Optimization and Control 35J96, 26B25, 49Q22, 49J52 |
| url | https://arxiv.org/abs/2010.14471 |