A function being a minimal surface is a global condition, since is can be written as an integral over its domain of definition:
$$\int\int_\Delta \sqrt{1+\big(\frac{\partial f}{\partial x}\big)^2+\big(\frac{\partial f}{\partial y}\big)^2}dxdy$$
However, Lagrange's minimal surfaces equation is only "local":
$$(1+q^2)r-2pqs+(1+p^2)t=0$$
(Using Monge's notations).
Why does that happen? Why does this seemingly global condition translate into a local condition?