On stable surfaces of prescribed mean curvature with partially free boundaries

M�ller, Frank
November 2005
Calculus of Variations & Partial Differential Equations;Nov2005, Vol. 24 Issue 3, p289
Academic Journal
We study surfaces of prescribed bounded mean curvature H in a partially free boundary configuration $$\langle\Gamma,{\cal S}\rangle$$. If $$\Gamma$$ is projectable onto and {\cal S} is a cylinder surface over the x1, x2-plane, we show that also the spanned H-surface $${\bf x}$$ is projectable onto this plane. Besides certain conditions on $$\langle\Gamma,{\cal S}\rangle$$ and H, we have to suppose that $${\bf x}$$ is stationary and freely stable w.r.t. the generalized area functional $$A_{{\bf Q}}({\bf x})$$, and the vector field $${\bf Q}={\bf Q}({\bf x})$$ is assumed to be tangential on $${\cal S}$$. Consequences are uniqueness of freely stable H-surfaces and solvability of a mixed boundary problem for the nonparametric prescribed mean curvature equation.


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