Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866929244589785088 |
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| author | Lou, Bendong Yuan, Lixia |
| author_facet | Lou, Bendong Yuan, Lixia |
| contents | We study a generalized mean curvature flow involving a positive power of the mean curvature and a driving force. In this paper, we first construct all kinds of radially symmetric translating solutions, and then select one of them to satisfy a prescribed boundary angle in a cylinder. We then consider the flow starting at an initial hypersurface: showing the a priori estimates (especially the uniform-in-time bounds for the mean curvature which guarantee the uniform parabolicity of the corresponding fully nonlinear equation), giving the global existence for the solution of the initial boundary value problem, and proving its convergence to the corresponding translating solution. Our study provides a complete exposition on the influence of the dimension, the power of the mean curvature, the driving force and the boundary angles on the existence and stability of radially symmetric translating solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_14431 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles Lou, Bendong Yuan, Lixia Analysis of PDEs 35K93, 35B45 We study a generalized mean curvature flow involving a positive power of the mean curvature and a driving force. In this paper, we first construct all kinds of radially symmetric translating solutions, and then select one of them to satisfy a prescribed boundary angle in a cylinder. We then consider the flow starting at an initial hypersurface: showing the a priori estimates (especially the uniform-in-time bounds for the mean curvature which guarantee the uniform parabolicity of the corresponding fully nonlinear equation), giving the global existence for the solution of the initial boundary value problem, and proving its convergence to the corresponding translating solution. Our study provides a complete exposition on the influence of the dimension, the power of the mean curvature, the driving force and the boundary angles on the existence and stability of radially symmetric translating solutions. |
| title | Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles |
| topic | Analysis of PDEs 35K93, 35B45 |
| url | https://arxiv.org/abs/2205.14431 |