Boundary value problems for rotationally symmetric mean curvature flows

Abstract.

The motion of surfaces by their mean curvature has been studied by several authors from different points of view. K. A. Brake studied this problem from the geometric measure theory point of view, the parametric problem was studied by G. Huisken [5]. Nonparametric mean curavture flow with boundary conditions was studied in [6] and [7]. Rotationally symmetric mean curvature flows have been treated by G. Dziuk, B. Kawohl [3], but also by S. Altschuler, S. B. Angenent and Y. Giga [2].

In this paper we consider the case in which the initial surface has rotational symmetry and we shall generalize the results in [3] in the sense that we shall give more general boundary conditions which enforce the formation of a singularity in finite time. The proofs rely entirely on parabolic maximum principles.