Regularity Results for Potential Functions of the Optimal Transportation Problem on Spheres and Related Hessian Equations

Abstract

In this thesis, results will be presented that pertain to the global regularity of solutions to boundary value problems having the general form F [ Du − A( · , u,Du)] = B( · , u,Du), in Ω−, Tu(Ω −) = Ω, (1) where A, B, Tu are all prescribed; and Ω − along with Ω are bounded in R, smooth and satisfying notions of c-convexity and c∗-convexity relative to one another (see [MTW05] for definitions). In particular, the case where F is a quotient of symmetric functions of the eigenvalues of its argument matrix will be investigated. Ultimately, analogies to the global regularity result presented in [TW06] for the Optimal Transportation Problem to this new fully-nonlinear elliptic boundary value problem will be presented and proven. It will also be shown that the (A3w) condition (first presented in [MTW05]) is also necessary for global regularity in the case of (1). The core part of this research lies in proving various a priori estimates so that a method of continuity argument can be applied to get the existence of globally smooth solutions. The a priori estimates vary from those presented in [TW06], due to the structure of F , introducing some complications that are not present in the Optimal Transportation case. In the final chapter of this thesis, the (A3) condition will be reformulated and analysed on round spheres. The example cost-functions subsequently analysed have already been studied in the Euclidean case within [MTW05] and [TW06]. In this research, a stereographic projection is utilised to reformulate the (A3) condition on round spheres for a general class