Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles

TL;DR

This paper investigates a generalized mean curvature flow with a positive power and driving force, constructing solutions, establishing existence, and analyzing stability under boundary conditions in a cylindrical setting.

Contribution

It provides a comprehensive analysis of radially symmetric translating solutions, their existence, and stability influenced by dimension, curvature power, and boundary angles.

Findings

Constructed all radially symmetric translating solutions.

Proved global existence and convergence of solutions.

Analyzed effects of dimension, curvature power, and boundary angles.

Abstract

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.