Curvature flow with driving force on fixed boundary points

Abstract

In this paper, we consider the mean curvature flow with driving force on fixed extreme points in the plane. We give a general local existence and uniqueness result of this problem with C 2 initial curve. For a special family of initial curves, we classify the solutions into three categories. Moreover, in each category, the asymptotic behavior is given.

Remarks on the generalized Cauchy-Dirichlet problem for graph mean curvature flow with driving force

“…In our paper, we prove by similar arguments to these in [3,7] that if viscosity solutions to (CD) lose the boundary condition, then the gradient blow-up happens. This gradient blow-up phenomenon is observed in [24], which studies the behavior of curvature flows with driving force with singular initial curves. This fact makes hard to get the existence of viscosity solutions to the ergodic problem (or additive eigenvalue problem):…”

“…After a finite time, the solution u goes down on the boundary as in the proof of Theorem 1.1, and u satisfies the Dirichlet boundary condition in the classical sense. Thus, in general, we may have loss of boundary conditions, and a gradient blow-up phenomenon at the same time (see also [24] for a gradient blow-up phenomenon). The gradient blow-up phenomenon is studied in [3,9] for viscous Hamilton-Jacobi equations with super quadratic Hamiltonians.…”