End-time regularity theorem for Brakke flows
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866918069350170624 |
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| author | Stuvard, Salvatore Tonegawa, Yoshihiro |
| author_facet | Stuvard, Salvatore Tonegawa, Yoshihiro |
| contents | For a general $k$-dimensional Brakke flow in $\mathbb{R}^n$ locally close to a $k$-dimensional plane in the sense of measure, it is proved that the flow is represented locally as a smooth graph over the plane with estimates on all the derivatives up to the end-time. Moreover, at any point in space-time where the Gaussian density is close to $1$, the flow can be extended smoothly as a mean curvature flow up to that time in a neighborhood: this extends White's local regularity theorem to general Brakke flows. The regularity result is in fact obtained for more general Brakke-like flows, driven by the mean curvature plus an additional forcing term in a dimensionally sharp integrability class or in a Hölder class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_07727 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | End-time regularity theorem for Brakke flows Stuvard, Salvatore Tonegawa, Yoshihiro Analysis of PDEs Differential Geometry 53E10, 35B65, 49Q15 For a general $k$-dimensional Brakke flow in $\mathbb{R}^n$ locally close to a $k$-dimensional plane in the sense of measure, it is proved that the flow is represented locally as a smooth graph over the plane with estimates on all the derivatives up to the end-time. Moreover, at any point in space-time where the Gaussian density is close to $1$, the flow can be extended smoothly as a mean curvature flow up to that time in a neighborhood: this extends White's local regularity theorem to general Brakke flows. The regularity result is in fact obtained for more general Brakke-like flows, driven by the mean curvature plus an additional forcing term in a dimensionally sharp integrability class or in a Hölder class. |
| title | End-time regularity theorem for Brakke flows |
| topic | Analysis of PDEs Differential Geometry 53E10, 35B65, 49Q15 |
| url | https://arxiv.org/abs/2212.07727 |