On stable surfaces of prescribed mean curvature with partially free boundaries

Abstract

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.