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 \(\) 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.