  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.

Analysis of PDEs