Growth Estimates for the Gradient of an $H$-Surface Near Singular Points of the Boundary Configuration

Abstract

We provide estimates for the gradient growth of surfaces with prescribed mean curvature in ${\mathbb{R}}^3$ near boundary points, which are mapped onto singular points of the boundary configuration. For corners of a Jordan arc, such estimates were provided by G. Dziuk [Analysis 1 (1981), 63–81]. We consider meeting points of a Jordan arc and a support manifold, as appearing in a partially free boundary problem (see G. Dziuk [Manuscr. Math. 35 (1981), 105–123] for the minimal surface case), and edge-type singularities of a support manifold. In subsequent papers, these results shall be used to derive asymptotic expansions of surfaces with prescribed mean curvature near such singular points.