Weak solutions of generated Jacobian equations

Abstract

We prove two groups of relationships for weak solutions to generated Jacobian equations under proper assumptions on the generating functions and the domains, which are generalizations for the optimal transportation case and the standard Monge-Ampère case respectively. One group of weak solutions is Aleksandrov solution, Brenier solution and $ C $-viscosity solution. The other group of weak solutions is Trudinger solution and $ L^p $-viscosity solution.

Keywords

• generated jacobian equations
• aleksandrov solutions
• brenier solutions
• $ c $-viscosity solutions
• trudinger solutions
• $ l^p $-viscosity solutions