On the behaviour of free H-surfaces near singular points of the support surface

Abstract

We study parametric surfaces of prescribed mean curvature in ℝ³, beeing stationary points of an associated functional and having part of their boundary on a support surface with edges. Asymptotic expansions near boundary points, which are mapped onto an edge, are provided. Among other conclusions, these expansions yield the exact asymptotic form of the considered surface and the continuity of the surface normal up to the singular point. Our results generalize known theorems on minimal surfaces in certain polyhedral boundary configurations. The method is based on work by G. Dziuk. Support surfaces with corner-type singularities can be considered as well.