Spring 2015 Samuel Eilenberg Lectures

Speaker:  Prof. Luis A. Caffarelli  (University of Texas)  

Title: “Future Directions in Nonlinear Partial Differential Equations”

Abstract: “The first three lectures will consists of a description of free boundary problems. Typical examples of free boundary problems  are a solid liquid interphase, the edge of a flame, the shape of a drop sitting on a surface, a discontinuous change if strategy in a game. Mathematically their structure consists of a domain where some phenomena takes place (cooling of a substance, value of a portfolio, wetting of a region) describe by some variable, like temperature, height of the drop, etc. and the behavior of the phenomena, changes discontinuously when this variable goes to some treshold value (solid to liquid, unburnt to burnt, the edge of the drop, etc.) Mathematically this means that the transition surface is part of the problem itself, and a range of issues arises like existence of solutions, regularity of the solution and of the transition surface etc. We will describe some of these problems, and how to attack these regularity issues. For the rest of the course , one possibility will be to study a family of non-linear parabolic equations of local and non-local nature that for their character are naturally treated by energy method , sort on the style of De Giorgi Nash Moser way. These include local and non-local equations like porous media, non-linear diffusion with memory, but I will be glad to discuss other topics.”