On Monge-Ampere type equations arising in optimal transportation problems

In this dissertation we study Monge-Ampere type equations arising in optimal transportation problems. We introduce notions of weak solutions, and prove the stability of solutions, the comparison principle and the analogous maximum principle of Aleksandrov-Bakelman-Pucci. We also establish a quantitative estimate of Aleksandrov type for c-convex functions which generalizes the well known estimate of Aleksandrov proved for convex functions. These results are in turn used to give a positive answer for the solvability and uniqueness of the Dirichlet problems for any continuous boundary condition and for finite Borel measures provided the domains satisfy a so called c-strictly convex condition which we have introduced.