A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder
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| Format: | Preprint |
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2022
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| _version_ | 1866911777919336448 |
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| author | Gao, Zhenghuan Lou, Bendong Xu, Jinju |
| author_facet | Gao, Zhenghuan Lou, Bendong Xu, Jinju |
| contents | In this paper we consider a mean curvature flow $V=H+A$ in a high dimensional cylinder $Ω\times \R$, where, $A$ is a constant, $Ω$ is a bounded domain in $\R^n$, and, for a hypersurface $y=u(x,t)$ over $Ω$, $V$ and $H$ denote its normal velocity and mean curvature, respectively. Assume the hypersurface contacts the cylinder boundary $\partial Ω\times \R$ with prescribed angle $θ(x)$. Under certain assumptions such as $Ω$ is strictly convex and $\|\cosθ\|_{C^2}$ is small, or $Ω$ is not necessarily convex but $|A|$ is sufficiently large, we derive some {\it uniform-in-time gradient bounds} for the solutions to initial boundary value problems. Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when $I:= A|Ω|+\int_{\partial Ω} \cosθ(x) dσ>0$ (resp. $=0$, $<0$), the solution $u$ converges as $t\to \infty$ to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_16475 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder Gao, Zhenghuan Lou, Bendong Xu, Jinju Differential Geometry 35K93, 35B40, 35B45, 35C07 In this paper we consider a mean curvature flow $V=H+A$ in a high dimensional cylinder $Ω\times \R$, where, $A$ is a constant, $Ω$ is a bounded domain in $\R^n$, and, for a hypersurface $y=u(x,t)$ over $Ω$, $V$ and $H$ denote its normal velocity and mean curvature, respectively. Assume the hypersurface contacts the cylinder boundary $\partial Ω\times \R$ with prescribed angle $θ(x)$. Under certain assumptions such as $Ω$ is strictly convex and $\|\cosθ\|_{C^2}$ is small, or $Ω$ is not necessarily convex but $|A|$ is sufficiently large, we derive some {\it uniform-in-time gradient bounds} for the solutions to initial boundary value problems. Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when $I:= A|Ω|+\int_{\partial Ω} \cosθ(x) dσ>0$ (resp. $=0$, $<0$), the solution $u$ converges as $t\to \infty$ to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed). |
| title | A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder |
| topic | Differential Geometry 35K93, 35B40, 35B45, 35C07 |
| url | https://arxiv.org/abs/2210.16475 |