Andriyanova, Elina

### Description

The optimal transportation problem was formulated by Monge in 1781: given two domains $\Omega, \Omega^* \subset \R^n$ and two mass distributions $f\in L^1(\Omega)$ and $g\in L^1(\Omega^*)$ of the same mass, find the optimal volume-preserving map $T$ between them, where optimality is measured against a cost functional $$ \mathcal C(s)=\int_\Omega f(x) c(x,s(x)). $$ %One views the first set $\Omega$ as being filled with mass, %and $c(x,y)$ as being the cost for transporting per unit mass from $x...[Show more] \in \Omega $ to $ y \in \Omega^*$; %the optimal map $T$ minimizes the total cost of redistributing the mass distribution $f\in L^1(\Omega)$ %to $L^1(\Omega^*)$. Optimal transportation has undergone a rapid and important development since the pioneering work of Brenier, who discovered that when the cost is the distance squared, optimal maps for the problem are gradients of convex functions. Later Caffarelli proved that in the case of convex target domain $\Om^*$, potential functions to optimal transportation problem are weak solutions (in the Aleksandrov sense) to the standard Monge-Amp\`ere equation. Following this result and its subsequent extensions, the theory of optimal transportation has flourished, with generalizations to other cost functions and corresponding Monge-Amp\`ere type equations, applications in many other areas of mathematics such as geometric analysis, functional inequalities, fluid mechanics, dynamical systems, and other more concrete applications such as irrigation and cosmology. In this thesis we are concerned with the global regularity problem for the optimal transportation and the corresponding Monge-Amp\`ere type equation. The manuscript consists from 4 chapters. Chapter 1 is an introduction, where collected some background information on the topic and main results been made so far. In Chapter 2 we consider the global regularity of potential functions in optimal transportation with quadratic cost and provide a global $C^{1,\alpha}$ regularity result, which extends the Caffarelli's regularity theory on a class of nonconvex domains. %Chapter 2 concerns global regularity of potential functions in optimal transportation with quadratic cost. First, there is a proof of global $C^1$ regularity on any Lipschitz domain $\Omega$ under some natural conditions. Then using $C^1$ regularity result we obtain the $C^{1,\alpha}$ regularity of potential functions on a domain $\Omega$ obtained by removing finitely many disjoint convex subsets from a convex domain. In Chapter 3 we deal with degenerate Monge-Amp\`ere type equations with the general cost function $c$. There will be obtained global $C^{2}$ a priori estimates for generalized solutions of the corresponding Dirichlet problem as well as the existence and uniqueness of solutions via the classical continuity method. %Chapter 3 is devoted to degenerate Monge-Amp\`ere type equations which belongs to a wider class of equations than ones considered in Chapter 2. %There will be presented the global $C^{2}$ a priori estimates for the corresponding Dirichlet problem. We also show that these estimates imply existence of the unique solution $u\in C^{1,1}(\overline{\Om})$ by the classical continuity method. Finally, Chapter 4 is devoted to singular Monge-Amp\`{e}re equations. There will be shown a symmetry of smooth convex solutions to singular Monge-Amp\`{e}re equations in the entire space $\mathbb{R}^n$ as well as different applications of this result to Monge-Amp\`ere equations, Hessian equations and special Lagrangian equations. In addition, we will construct an example which clearly shows that solutions to singular Monge-Amp\`ere equations in a ball may not have a decomposition of a smooth function and a smooth convex cone.

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