Existence and uniqueness for the hyperbolic Monge–Ampère transport system

Establish existence and uniqueness of solutions to the hyperbolic Monge–Ampère second boundary value problem, for which fundamental questions remain unresolved.

Background

The paper contrasts the elliptic Monge–Ampère transport problem, for which existence, uniqueness, and regularity theory are available under appropriate assumptions, with the corresponding hyperbolic problem. The hyperbolic formulation is used as one component of the paper’s mixed elliptic–hyperbolic transport model, but its underlying analytical theory is described as substantially less developed.

The unresolved issue concerns whether solutions to the hyperbolic system exist and whether they are uniquely determined by the prescribed source density, target density, and transport boundary condition. These questions are foundational for establishing a rigorous theory of the hyperbolic component used in mixed-type optical design.