Existence and regularity of solutions to the mixed elliptic–hyperbolic system

Establish existence and regularity properties for solutions of the mixed elliptic–hyperbolic Monge–Ampère transport system coupling an elliptic equation on the left source subdomain with a hyperbolic equation on the right source subdomain.

Background

The paper introduces a coupled transport formulation in which the source domain is partitioned into an elliptic subregion and a hyperbolic subregion. The two mapping components must satisfy Monge–Ampère equations of opposite determinant sign, remain compatible across the interface, map their respective source subdomains onto the target domain, and satisfy a density-coupling condition.

The numerical method demonstrates computed examples, but the paper does not provide an analytical existence or regularity theory for the mixed system. In particular, the authors identify these properties as unknown, making them a direct open problem for the proposed formulation.