Definition and certain convergence properties of a two-scale method for Monge-Amp\`ere type equations
Abstract
The Monge-Amp\`ere equation arises in the theory of optimal transport. When more complicated cost functions are involved in the optimal transportation problem, which are motivated e.g. from economics, the corresponding equation for the optimal transportation map becomes a Monge-Amp\`ere type equation. Such Monge-Amp\`ere type equations are a topic of current research from the viewpoint of mathematical analysis. From the numerical point of view there is a lot of current research for the Monge-Amp\`ere equation itself and rarely for the more general Monge-Amp\`ere type equation. Introducing the notion of discrete Q-convexity as well as specifically designed barrier functions this purely theoretical paper extends the very recently studied two-scale method approximation of the Monge-Amp\`ere itself NochettoNtogkasZhang2018 to the more general Monge-Amp\`ere type equation as it arises e.g. in PhillippisFigalli2013 in the context of Sobolev regularity.
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