On this paper by M. Athanassenas it is claimed that given a function $u(x,y)$ defined on the whole of $\mathbb{R}^2$, it satisfies $$ (1+u_y^2)u_{xx} - 2u_xu_yu_{xy}+(1+u_x^2)u_{yy} = 0 $$ If and only if there exists a function $\phi(x,y)$ such that \begin{align} \phi_{xx} = \frac{1+u_x^2}{\sqrt{1+u_x^2+u_y^2}} \\ \phi_{xy} = \frac{u_xu_y}{\sqrt{1+u_x^2+u_y^2}} \\ \phi_{yy} = \frac{1+u_y^2}{\sqrt{1+u_x^2+u_y^2}} \end{align} I do not see why this would be but I suspect that it has something to do with a Taylor expansion. Any help is appreciated.
EDIT: As per @Bob_Terrell 's suggestion I have fixed the equation presented (the minimal surface equation)
First, you have $1+u_x^2$ and $1+u_y^2$ interchanged in the minimal surface equation, because it is misprinted in the paper you cite. Second, just a suggestion, since in two dimensions a divergence can be interpreted as a curl, the integral $$\int_{(0,0)}^{(x,y)}\frac{-u_y dx+u_x dy}{\sqrt{1+u_x^2+u_y^2}}$$ defines a function (is independent of path). That might be what is intended by $\phi$ but I'm not sure.