H�lder continuity of surfaces with bounded mean curvature at corners where Plateau boundaries meet free boundaries

M�ller, Frank
November 2005
Calculus of Variations & Partial Differential Equations;Nov2005, Vol. 24 Issue 3, p283
Academic Journal
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 H�lder 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.


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