A quantitative linearization of the Monge-Ampère equation around the identity is the Poisson equation, used for a variational proof of partial regularity of optimal transport maps and validation of predictions in matching problems.
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A variational approach to regularity theory in optimal transportation
A quantitative linearization of the Monge-Ampère equation around the identity is the Poisson equation, used for a variational proof of partial regularity of optimal transport maps and validation of predictions in matching problems.