Hölder continuity of surfaces with bounded mean curvature at corners where Plateau boundaries meet free boundaries

Abstract

Let $P={\rm \Gamma}\cap{\cal S}$ be the point of non-tangential intersection of a closed Jordan arc ${\rm \Gamma} \subset {\mathbb R}^{3}$ and an embedded, regular support surface ${\cal S} \subset {\mathbb R}^{3}$. Let ${\bf x}:B \to {\mathbb R}^{3}$ be a conformally parametrized solution of $|{\rm \Delta}{\bf x}|\le a|\nabla {\bf x}|^{2}$with partially free boundaries $\{{\rm \Gamma},{\cal S}\}$. It is proved, that ${\bf x}$ is Hölder continuous up to $w_{0}\in \partial B$ with ${\bf x}(w_{0})=P$, whenever ${\bf x}$ meets ${\cal S}$ orthogonally along its free trace. This provides a regularity result for stationary minimal surfaces and is applicable also to surfaces of prescribed bounded mean curvature.