Uniform gradient bounds and convergence of mean curvature flows in a cylinder

Abstract

We consider a mean curvature flow V=H+A in a cylinder Ω×R, where, Ω is a bounded domain in Rn, A is a constant driving force, V and H are the normal velocity and the mean curvature respectively of a moving hypersurface, which contacts the cylinder boundary with prescribed angle θ(x). Under certain conditions such as Ω is convex and ‖cos⁡θ‖C2 is small, or Ω is non-convex and |A| is large, we derive the uniform gradient bounds for bounded and unbounded solutions (which is crucial in the study of the asymptotic behavior of the solutions). Then we present a trichotomy result on the convergence of the solutions as well as its criterion: when A|Ω|+∫∂Ωcos⁡θ(x)dσ>0 (resp. =0, <0), the solution converges as t→∞ to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed).