Minicourse on Calculus of variations done with measures

Abstract: In this short course, I will discuss how Mather-Aubry theory coming from dynamical systems has motivated the development of a new framework for variational analysis using certain measures that represent immersed submanifolds.  

 

In the first lecture, there will be a review of the theory of minimizers of time-independent Lagrangians, which has more a dynamical flavor: it is a description of action-minimizing orbits ('geodesics') in all homology classes of the manifold, of the cases when they are integrable and what happens when the Lagrangian is slightly deformed (KAM theory), of the case of generic Lagrangians, of their induced diffusion regimes, of the way in which they solve the Hamilton-Jacobi equation, and of how this is an approximation to the quantum case.

 

The second lecture will be about the higher-dimensional case. This has a more geometric-measure-theoretical flavor. After motivating with a review of a few traditional results in the theory of minimal surfaces, I will explain the point of view of holonomic measures. I will discuss the full space of distributions that are tangent to the space of holonomic measures, how this can be used to deduce properties of the critical points of the action, and some interesting examples.