Monge Ampere equation

The Monge-Ampere equation refers to \[ \det D^2 u = f(x). \] The right hand side $f$ is always nonnegative. We look for a convex function $u$ which solves the equation.

The equation is uniformly elliptic when restricted to a subset of functions which are uniformly $C^{1,1}$ and strictly convex. Moreover, the operator \[ MA(D^2 u) := \left( \det (D^2 u) \right)^{1/n} = \inf \{ a_{ij} \partial_{ij} u : \det \{a_{ij}\} = 1 \},\] is concave.

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