Optimal Transport and the Monge Ampère equation

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The Monge Ampère equation[1] is an nonlinear second-order elliptic partial differential equation. Given sufficient conditions on an optimal transport problem, the Monge Ampère equation defines a change of variables formula for the pushforward of a transport map. In particular, the Monge Ampère equation applies to a variant of the Monge Problem where the cost function is quadratic.

Deriving the Monge Ampère equation from Monge's Problem

References

  1. ↑ F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 1.
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