Global regularity for solutions to Monge-Ampere type equations

I will talk about the global regularity problem in the optimal transportation and corresponding Monge-Ampere type equations. Firstly, there will be given some background of the optimal transport problem and its link to the Monge-Ampere equations. Next I will prove $C^{1,\alpha}$ regularity of potential functions on nonconvex domains. Then there will be obtained global $C^2$ a priori estimates and an existence result for degenerate Monge-Ampere type equations.  Finally, I will show the radial symmetry of smooth convex solutions to Monge-Ampere type equations in $\mathbb{R}^n\backslash\{0\}$ and consider different applications of this result.