Separable surfaces that are critical points of the Dirichlet energy

In this paper, we study surfaces z=φ (x,y) z = φ ( x , y ) in Euclidean space that satisfy the equation φ ₓₓ+φ yy=Λ /2 φ xx + φ yy = Λ 2 where Λ ∈ R Λ ∈ R is a real constant. We classify these surfaces when they are the zero level sets of an implicit equation of the type $$f(x)+g(y)+h(z)=0$$ f ( x ) + g ( y ) + h ( z ) = 0 , where f , g and h are smooth functions of one variable. If Λ =0 Λ = 0 , we find a large family of surfaces with interesting symmetry properties. However, if Λ =0 Λ ≠ 0 , we show that the surfaces must be either surfaces of revolution or of the type $$z=f(x)+g(y)$$ z = f ( x ) + g ( y ) ; furthermore, explicit parametrizations of these surfaces are obtained.