Delayed parabolic regularity for curve shortening flow
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908541582835712 |
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| author | Sobnack, Arjun Topping, Peter M. |
| author_facet | Sobnack, Arjun Topping, Peter M. |
| contents | Given two curves bounding a region of area $A$ that evolve under curve shortening flow, we propose the principle that the regularity of one should be controllable in terms of the regularity of the other, starting from time $A/π$. We prove several results of this form and demonstrate that no estimate can hold before that time. As an example application, we construct solutions to graphical curve shortening flow starting with initial data that is merely an $L^1$ function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04049 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Delayed parabolic regularity for curve shortening flow Sobnack, Arjun Topping, Peter M. Analysis of PDEs Differential Geometry 35B45, 35B65, 35K55, 35K20, 53E10 Given two curves bounding a region of area $A$ that evolve under curve shortening flow, we propose the principle that the regularity of one should be controllable in terms of the regularity of the other, starting from time $A/π$. We prove several results of this form and demonstrate that no estimate can hold before that time. As an example application, we construct solutions to graphical curve shortening flow starting with initial data that is merely an $L^1$ function. |
| title | Delayed parabolic regularity for curve shortening flow |
| topic | Analysis of PDEs Differential Geometry 35B45, 35B65, 35K55, 35K20, 53E10 |
| url | https://arxiv.org/abs/2408.04049 |