Minimal surface equation for a graph
= Minimal surface equation for a graph
This equation characterizes a <minimal surface> presented as a graph.
The graph of $h$ is minimal exactly when
$$
(1+h_y^2)h_{xx}-2h_xh_yh_{xy}+(1+h_x^2)h_{yy}=0.
$$
Equivalently, it is the Euler--Lagrange equation
$$
\operatorname{div}\frac{\nabla h}{\sqrt{1+|\nabla h|^2}}=0
$$
for the area functional $\int\sqrt{1+|\nabla h|^2}$.