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

Bendong Lou, Lixia Yuan

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.

Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant 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.
Paper Structure (13 sections, 20 theorems, 209 equations)

This paper contains 13 sections, 20 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Translating Solutions in Case $b\leq 0$
  3. TSs on their maximal existence intervals
  4. TSs in the cylinder
  5. Translating Solutions in Case $b>0$
  6. TSs in case $c\geq b>0$
  7. The case $b>c>0$.
  8. Asymptotical Stability of the TSs: the Case $b\leq 0$
  9. A priori estimates
  10. Stability of the TS
  11. Asymptotical Stability of the TSs: the Case $b>0$
  12. The case $b>0, k>0$
  13. The case $b>0>k$

Key Result

Theorem 1.1

Let $u(r,t)$ be a classical solution of E-BC-cond-IC in the time interval $[0,T)$, then there exist positive numbers $M_0, M_1, M_2, V_*, V^*, H_*, H^*$, independent of $t$ and $T$, and $\tilde{c}(b,k)>0$ such that provided the constants $b,k$ and the initial data $u_0$ satisfy one of the following conditions

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 23 more