Global Existence of a Mean Curvature Flow in a Cone

Abstract

We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which propagates toward the opening of the cone with normal velocity depending on its mean curvature. In addition, the contact angle between the hypersurface and the cone boundary depending on its position. First, we construct a family of radially symmetric self-similar solutions. Then we use these solutions to give a priori estimates for the solutions of the initial boundary value problems, and show their global existence.