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Minimal surface equation for a graph

The graph of

h

is minimal exactly when

(1 + h_{y}^{2}) h_{xx} - 2 h_{x} h_{y} h_{x y} + (1 + h_{x}^{2}) h_{yy} = 0.

Equivalently, it is the Euler--Lagrange equation

div \frac{\nabla h}{1 + ∣\nabla h ∣ ^{2}} = 0

for the area functional

\int 1 + ∣\nabla h ∣^{2}

.

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Solution