Remarks on the generalized Cauchy-Dirichlet problem for graph mean curvature flow with driving force
Remarks on the generalized Cauchy-Dirichlet problem for graph mean curvature flow with driving force
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DOI：
10.1007/s42985-020-00066-4
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发表时间：
2021-05
期刊：
影响因子：
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通讯作者：
Hiroyoshi Mitake;Longjie Zhang
中科院分区：
文献类型：
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作者：
Hiroyoshi Mitake;Longjie Zhang
本文研究了图强迫平均曲率流方程在粘性解意义下的广义Cauchy-Dirichlet问题。众所周知，Cauchy-Dirichlet问题的粘性解一般不满足逐点边界条件。我们证明了如果粘性解失去Dirichlet边界条件，则解满足奇异Neumann边界条件。这一事实给研究大时间行为带来了困难。本文证明了当区域为N维球时，广义Cauchy-Dirichlet问题的粘性解收敛于适当的行波型解。
We study a generalized Cauchy-Dirichlet problem for graph forced mean curvature flow equations in the sense of viscosity solutions. It is well-known that viscosity solutions of Cauchy-Dirichlet problems may not satisfy the boundary condition pointwise in general. We prove that if viscosity solutions lose the Dirichlet boundary condition, then the solution satisfies the singular Neumann boundary condition. This fact causes a difficulty to study the large-time behavior. In this paper, we prove that the viscosity solution to a generalized Cauchy-Dirichlet problem converges to the appropriate traveling wave type solution when the domain is aN-dimensional ball.