Past lectures

Abstract

The purpose of the lectures is to analyze several examples of nonlinear PDEs (mostly with strong geometric or physical features) which enjoy a hidden convex structure. This is not only a matter of curiosity. Once the convex structure is unvealed, robust existence and uniqueness results can be unexpectedly obtained for very general data. Of course, as usual, regularity issues are left over as a hard post-process; but, at least, existence, uniqueness and stability results are obtained in a large, global, framework.

We will discuss:

- The Real Monge-Ampere Equation and show how the convex structure is related to Optimal Transport Theory;

- The Euler Equations of Fluid Mechanics (which describe the motion of inviscid, incompressible fluids and provide the most famous example of a geodesic flow in infinite dimension) and their Hydrostatic and Semi-Geostrophic Limits;

- The Multi-Dimensional Hyperbolic Scalar Conservation Laws (a simplified model for multidimensional systems of hyperbolic conservation laws)

- The Born-Infeld System (a non-linear electromagnetic model introduced in 1934, playing an important role in high energy Physics since the 1990's).