I want to find all minimal surfaces of the form $z = \phi(x) + \psi(y)$ over some region $D$ in the xy-plane.
This should be equivalent to minimizing the functional
$$ J[z] = \iint_D \sqrt{1+z_x^2+z_y^2}\,dxdy. $$
The Euler-Lagrange equation is then
$$ F_z - \frac{\partial}{\partial{x}}F_{z_x} - \frac{\partial}{\partial{x}}F_{z_y} = 0 \Rightarrow $$
$$ \frac{\partial}{\partial{x}} \Bigg(\frac{z_x}{\sqrt{1+z_x^2+z_y^2}}\Bigg) + \frac{\partial}{\partial{x}} \Bigg(\frac{z_y}{\sqrt{1+z_x^2+z_y^2}}\Bigg) = 0 \Rightarrow $$
$$ \frac{z_{xx}(1+z_y^2)}{(1+z_x^2+z_y^2)^{3/2}} + \frac{z_{yy}(1+z_x^2)}{(1+z_x^2+z_y^2)^{3/2}} = 0 \Rightarrow $$
$$ z_{xx}(1+z_y^2) + z_{yy}(1+z_x^2) = 0 \Rightarrow $$
$$ \phi''(x)(1+\psi'(y)^2) + \psi''(y)(1+\phi'(x)^2) = 0. $$
Now I don't really know how to attack this equation and I do not recognize its form. I have tried separation of variables but don't know how to handle the squared terms. I can set either $\phi = 0$ or $\psi=0$ and obtain a solution $$ z = Ax + By +C$$ which represents a plane. But I cannot be confident that this respresents all solutions to the given problem.
My question is thus: What is an efficient approach to solving this equation, and how to be certain when I have found all solutions?
$$\phi''(x)(1+\psi'(y)^2) + \psi''(y)(1+\phi'(x)^2) = 0.$$ $$\frac {\phi''(x)}{(1+\phi'(x)^2)}=-\frac {\psi''(y) }{(1+\psi'(y)^2)} $$ Both functions depends on different variables so $$\frac {\phi''(x)}{(1+\phi'(x)^2)}=-\lambda $$ $$\frac {\psi''(y) }{(1+\psi'(y)^2)}=\lambda$$ Integrate both equations: $$\int \frac {d\phi'(x)}{(1+\phi'(x)^2)}=-\lambda \int dx$$ $$\int \frac {d\psi'(y) }{(1+\psi'(y)^2)}=\lambda\int dy$$ And: $$\int \dfrac {dw}{1+w^2}=\arctan ( w ) +C$$