$L^2$ normal velocity implies strong solution for graphical Brakke flows
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918159457452032 |
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| author | Motegi, Kotaro |
| author_facet | Motegi, Kotaro |
| contents | We prove that if a one-parameter family of varifolds has an $L^2$ normal velocity $v$ in the sense of Brakke, and if the family is represented as the graph of a continuous function $f$ with continuous spatial derivative $\nabla f$, then $f$ has weak derivatives $\partial_t f, \nabla^2 f \in L^2$, and $v$ coincides with the usual normal velocity of the graph. Moreover, by combining this result with parabolic regularity theory, we show that graphical Brakke flows with forcing term in $L^{p,q}$ and $C^{0,α}$ are strong and classical solutions to the forced mean curvature flow equation, respectively. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_11377 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L^2$ normal velocity implies strong solution for graphical Brakke flows Motegi, Kotaro Analysis of PDEs Differential Geometry 53E10 (Primary) 35B65 (Secondary) We prove that if a one-parameter family of varifolds has an $L^2$ normal velocity $v$ in the sense of Brakke, and if the family is represented as the graph of a continuous function $f$ with continuous spatial derivative $\nabla f$, then $f$ has weak derivatives $\partial_t f, \nabla^2 f \in L^2$, and $v$ coincides with the usual normal velocity of the graph. Moreover, by combining this result with parabolic regularity theory, we show that graphical Brakke flows with forcing term in $L^{p,q}$ and $C^{0,α}$ are strong and classical solutions to the forced mean curvature flow equation, respectively. |
| title | $L^2$ normal velocity implies strong solution for graphical Brakke flows |
| topic | Analysis of PDEs Differential Geometry 53E10 (Primary) 35B65 (Secondary) |
| url | https://arxiv.org/abs/2510.11377 |