The affine maximal equation is the Euler-Lagrange equation of the affine area functional for a locally uniformly convex
graph. For a smooth function 
on an open domain in 
with positive definite Hessian,
put 
and 
.
The equation is
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The corresponding functional is 
for compactly
supported variations. Every quadratic polynomial
with positive definite Hessian
solves the equation, since 
is constant. The affine
Bernstein conjecture asks when complete solutions must be quadratic.
