Regularity of Monge-Ampere type equations arising in optimal transportation

Abstract

In this thesis we study the regularity problem in optimal transportation, by establishing the a priori estimates for a class of Monge-Ampere type equations. This thesis consists of four chapters. Chapter 1 is an introduction. In Chapter 2 we study Monge's mass transfer problem. Chapter 3 concerns the regularity of optimal transportation on manifolds. In the last Chapter we deal with the homogeneous Monge-Ampere equations. The regularity is a main issue in optimal transportation, and has received intensive investigations in the last two decades. The regularity was obtained if the cost function satisfies certain conditions. However the cost function, proposed by Monge in 1781, does not satisfy these conditions. Our strategy is to use an approximation, and establish uniform estimates in this approximation. By tedious computation we are able to prove that the eigenvalues of the Jacobian matrix of the approximation map are locally uniformly bounded. On the other hand by constructing an example we also show that the monotone optimal mapping corresponding to Monge's cost is not Lipschitz continuous in general. This implies the Jacobian matrix of the approximation map itself may fail to be uniformly bounded, even though all needed conditions are satisfied. These are the main results in Chapter 2. In Chapter 3 we study the regularity of optimal transportation on manifolds with the quadratic cost function. For this research it is crucial to verify the MTW condition. This problem has been studied by many researchers in recent years but progress has been very slow. The MTW condition has been verified for manifolds such as spheres, quotients or submersions of spheres and smooth perturbations of spheres. In this thesis we consider the case of Riemannian surfaces. We give an explicit condition on Gauss curvature such that the MTW condition is satisfied. Motivated by Monge's problem, in Chapter 4 we study the homogeneous Monge-Ampere equations. We first prove the existence and uniqueness of generalized solutions to the Dirichlet problem in strip regions under sharp conditions. Then we show that, if boundary functions are locally uniformly convex and are in the class which consists of functions having continuous derivatives up to order k+2 and such that the (k+2)-th partial derivatives are Holder continuous with exponent a, then the convex solution is in the same class. By an example we show that the solution is not twice continuously differentiable in the interior if boundary functions are merely strictly convex and the dimension is no less than three. Similar regularity results have also been obtained for the Dirichlet problem in bounded, strictly convex domains. Show more