Separable surfaces that are critical points of the Dirichlet energy

Abstract

In this paper, we study surfaces $z=\varphi(x,y)$ in Euclidean space that satisfy the equation $\varphi_{xx}+\varphi_{yy}=\frac{\Lambda}{2}$ where $\Lambda\in\r$ is a real constant. We classify these surfaces when they are the zero level sets of an implicit equation of the type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are smooth functions of one variable. If $\Lambda=0$, we find a large family of surfaces with interesting symmetry properties. However, if $\Lambda\not=0$, we show that the surfaces must be either surfaces of revolution or of the type $z=f(x)+g(y)$; furthermore, explicit parametrizations of these surfaces are obtained.