Generated Jacobian equations are Monge–Ampère type equations which
contain optimal transport as a special case. Therefore, the optimal transport
case has its own special structure, which is not necessarily true for more
general generated Jacobian equations. Hence, the theory for optimal transport
cannot be directly applied to generated Jacobian equations. In this paper, we
point out the difficulties that prevent applying the proof of the local Hölder
regularity of solutions of optimal transport problem from Loeper (2009)
directly to generated Jacobian equations. We then discuss how to handle these
difficulties and prove local Hölder regularity in the generated Jacobian equation
case.
