Rankin, Cale2021-12-142021-12-14http://hdl.handle.net/1885/255920This is a thesis about generated Jacobian equations; our purpose is twofold. First, we aim to provide an introduction to these equations, whilst, at the same time, collating and unifying some results scattered throughout, and hinted at in, the literature. The other goal is to present the author's own results on these equations. These results all concern solutions of the PDE f*(Y(,u,Du))det DY(x,u,Du) = f(x), for a particular family of vector fields Y:Rn x R x Rn -> Rn. Usually this PDE is paired with the second boundary value problem: Y(.,u,Du)(U) = V. for prescribed domains U,V subsets of Rn. We summarize our results as follows, though warn that the word convexity needs to be understood in an appropriately generalised sense. 1. In two dimensions when f/f* > c > 0 and U is convex, solutions are strictly convex. When f/f* is bounded above and V is convex, solutions are continuously differentiable. 2. When f/f* is pinched between two positive constants, V is convex, and the generating function is defined on a convex domain containing U, solutions are strictly convex and in C1,alpha. The same result holds provided U is uniformly convex and the generating function is defined on a domain containing U. 3. If two C1,1(U) solutions intersect, then they are the same solution. In addition C^1,1 solutions of the Dirichlet problem are unique. 4. Under appropriate uniform convexity hypothesis on U,V and smoothness conditions on f,f* all weak solutions lying in a particular domain are globally smooth. 5. Solutions of the parabolic version of the generated Jacobian equation remain uniformly bounded, independent of time. We want to emphasize that the results of point 2 have already appeared in the works of Guillen and Kitagawa under stronger hypothesis. Our contribution is the weaker hypothesis. Furthermore, on point 4, the existence of globally smooth solutions is due to Jiang and Trudinger. Our contribution is showing weak solutions are globally smooth solutions.en-AU202110.25911/8KG0-Q497