Regularity and uniqueness results for generated Jacobian equations

TL;DR

This thesis explores generated Jacobian equations, providing foundational overview, collating existing results, and presenting new findings on solution regularity, convexity, and uniqueness, with implications for global regularity and parabolic cases.

Contribution

It offers new regularity and uniqueness results for solutions of generated Jacobian equations, especially in higher dimensions, and consolidates scattered literature into a comprehensive overview.

Findings

Proved strict convexity and $C^1$ differentiability in 2D.

Extended regularity results to higher dimensions with additional hypotheses.

Established uniqueness results for the second boundary value problem.

Abstract

This is a PhD thesis about generated Jacobian equations; our purpose is twofold. First, we provide an introduction to these equations, whilst, at the same time, collating some results scattered throughout the literature. The other goal is to present the author's own results on these equations. These results all concern solutions of generated Jacobian equations, usually paired with the second boundary value problem. We prove strict convexity and $C^1$ differentiability results under optimal hypothesis in two dimensions, and the same results in higher dimensions with some additional hypothesis. We also consider uniqueness results for the second boundary value problem, and the application of the uniqueness results to global regularity. We conclude with notes on the parabolic generated Jacobian equation. The arXiv version contains minor updates to the ANU open research repository version which is available from the listed DOI.