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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| Acceso en línea: | https://arxiv.org/abs/2109.06380 |
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| _version_ | 1866916497560961024 |
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| author | Mori, Ryunosuke Tomimatsu, Eita Tonegawa, Yoshihiro |
| author_facet | Mori, Ryunosuke Tomimatsu, Eita Tonegawa, Yoshihiro |
| contents | Suppose that a family of $k$-dimensional surfaces in $\mathbb R^n$ evolves by the motion law of $v=h+u^\perp$ in the sense of Brakke's formulation of velocity, where $v$ is the normal velocity vector, $h$ is the generalized mean curvature vector and $u^\perp$ is the normal projection of a given vector field $u$ in a dimensionally sharp integrability class. When the flow is locally close to a time-independent $k$-dimensional plane in a weak sense of measure in space-time, it is represented as a graph of a $C^{1,α}$ function over the plane. On the other hand, it is not known if the graph satisfies the PDE of $v=h+u^\perp$ pointwise in general. For this problem, when $k=n-1$ and under the additional assumption that the distributional time derivative of the graph is a signed Radon measure, it is proved that the graph satisfies the PDE pointwise. An application to a short-time existence theorem for a surface evolution problem is given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_06380 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Brakke's formulation of velocity and the second order regularity property Mori, Ryunosuke Tomimatsu, Eita Tonegawa, Yoshihiro Analysis of PDEs Differential Geometry 53E10 Suppose that a family of $k$-dimensional surfaces in $\mathbb R^n$ evolves by the motion law of $v=h+u^\perp$ in the sense of Brakke's formulation of velocity, where $v$ is the normal velocity vector, $h$ is the generalized mean curvature vector and $u^\perp$ is the normal projection of a given vector field $u$ in a dimensionally sharp integrability class. When the flow is locally close to a time-independent $k$-dimensional plane in a weak sense of measure in space-time, it is represented as a graph of a $C^{1,α}$ function over the plane. On the other hand, it is not known if the graph satisfies the PDE of $v=h+u^\perp$ pointwise in general. For this problem, when $k=n-1$ and under the additional assumption that the distributional time derivative of the graph is a signed Radon measure, it is proved that the graph satisfies the PDE pointwise. An application to a short-time existence theorem for a surface evolution problem is given. |
| title | Brakke's formulation of velocity and the second order regularity property |
| topic | Analysis of PDEs Differential Geometry 53E10 |
| url | https://arxiv.org/abs/2109.06380 |