Regularity for stationary surfaces of constant mean curvature with free boundaries (Q1086832)

The authors prove regularity for the free boundary of the partitioning problem for a convex body in \({\mathbb{R}}^ 3\) with boundary T. The body K is assumed to be partitioned into two parts \(K_ 1\) and \(K_ 2\) with given relative volumes by a \(C^ 1\)-surface S with finite stationary, for example minimal, area. It is shown that a solution S of this problem is a surface of constant mean curvature which meets the supporting surface \(T=\partial K\) orthogonally in a weak sense and that the trace is smooth. The most important step in the regularity proof is to prove the continuity of the free boundary.