Separable surfaces obeying the PDE φ_xx + φ_yy = Λ/2 are revolution surfaces or z = f(x) + g(y) when Λ is nonzero.
L\'opez, Circles-foliated stationary surfaces of the Dirichlet energy, preprint
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Separable surfaces that are critical points of the Dirichlet energy
Separable surfaces obeying the PDE φ_xx + φ_yy = Λ/2 are revolution surfaces or z = f(x) + g(y) when Λ is nonzero.