Mara Mather

Minimax Probabilities for Aubry-mather Problems

In this paper we study minimax Aubry-Mather measures and its main properties. We consider first the discrete time problem and then the continuous time case. In the discrete time problem we establish existence, study some of the main properties using duality theory and present some examples. In the continuous time case, we establish both existence and nonexistence results. First we give some exa...

Viscosity Solutions and Aubry-Mather theory

A stochastic analogue of Aubry–Mather theory*

In this paper, we discuss a stochastic analogue of Aubry–Mather theory in which a deterministic control problem is replaced by a controlled diffusion. We prove the existence of a minimizing measure (Mather measure) and discuss its main properties using viscosity solutions of Hamilton–Jacobi equations. Then we prove regularity estimates on viscosity solutions of the Hamilton–Jacobi equation usin...

Optimal mass transportation and Mather theory

We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is a...

Aubry-Mather Measures in the Nonconvex Setting

The adjoint method, introduced in [L. C. Evans, Arch. Ration. Mech. Anal., 197 (2010), pp. 1053–1088] and [H. V. Tran, Calc. Var. Partial Differential Equations, 41 (2011), pp. 301– 319], is used to construct analogues to the Aubry–Mather measures for nonconvex Hamiltonians. More precisely, a general construction of probability measures, which in the convex setting agree with Mather measures, i...