MATJ5116 MA1: Convexity, Partial Differential Equations and Game Theory (JSS33) (2 op)

This course aims to explain in a self-contained way the main ideas behind the relation between convexity and partial differential equations (PDE).
We will develop the following:

• Introduction to fully nonlinear elliptic and parabolic PDE. Viscosity solutions, comparison arguments.
• Classical convexity. Convex sets. Convex functions. Regularity results. lipschitz continuity. Second order differentiability, Aleksandroff theorem. Different notions of convexity.
• The convex envelope of a boundary datum inside a domain. Characterization as a solution to the Dirichlet problem for a PDE. C^1 regularity.
• A brief introduction to game theory and PDEs. A game for the convex envelope of a boundary datum.

It is expected that the students become familiar with the concept of viscosity solution to a PDE and understand its relation with convexity and game theory.
These lectures aim to provide a general (and as self-contained as possible) exposition with lectures, exercises and discussion.

The only prerequisites are a solid background in Mathematical Analysis in the Euclidean space (including some notions of measure theory) and some basic notions of Probability (at undergraduate level). A basic knowledge of PDEs is desirable.