Definition and certain convergence properties of a two-scale method for Monge-Amp\`ere type equations

TL;DR

This paper extends a two-scale numerical method for the Monge-Ampère equation to more general Monge-Ampère type equations, introducing discrete convexity and barrier functions to analyze convergence properties.

Contribution

It introduces a novel extension of the two-scale method to Monge-Ampère type equations, incorporating discrete convexity and barrier functions for convergence analysis.

Findings

Defined discrete Q-convexity for Monge-Ampère type equations.

Proved convergence properties of the two-scale method for these equations.

Extended the method's applicability to equations arising in Sobolev regularity contexts.

Abstract

The Monge-Amp\`{e}re 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\`{e}re type equation. Such Monge-Amp\`{e}re 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\`{e}re equation itself and rarely for the more general Monge-Amp\`{e}re 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\`{e}re itself \cite{NochettoNtogkasZhang2018} to the more general Monge-Amp\`{e}re type equation as it arises e.g. in \cite{PhillippisFigalli2013} in the context of Sobolev regularity.