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 depend- ing 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 exis-tence.

Keywords

Mean curvature flow / quasilinear parabolic equation / free boundary problem / self-similar solution