Separable surfaces that are critical points of the Dirichlet energy

Abstract

In this paper, we study surfaces $z=φ(x,y)$ in Euclidean space that satisfy the equation $φ_{xx}+φ_{yy}=\fracΛ{2}$ where $Λ\in\r$ 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 $Λ=0$, we find a large family of surfaces with interesting symmetry properties. However, if $Λ\not=0$, 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.