On uniqueness of weak solutions to the second boundary value problem for generated prescribed Jacobian equations

TL;DR

This paper establishes the uniqueness of weak solutions to generated prescribed Jacobian equations by showing that Aleksandrov solutions share the same gradients at common differentiability points, extending to optimal transportation and general cases.

Contribution

It provides a new proof of uniqueness with weaker regularity assumptions for solutions to generated prescribed Jacobian equations, including the optimal transportation case.

Findings

Aleksandrov solutions have identical gradients at common differentiability points.

Uniqueness up to a constant is confirmed for optimal transportation solutions.

A novel proof technique reduces regularity requirements for key theorems.

Abstract

We prove that two Aleksandrov solutions of a generated prescribed Jacobian equation have the same gradients at points where they are both differentiable. For the optimal transportation case where two solutions can be translated to agree at a point without changing the $g$-subdifferential at that point, we recover the uniqueness up to a constant of solutions. For the general case, our result is a new proof with less regularity assumptions of a key theorem recently used to prove the uniqueness of solutions.