Program

The Program of this School consists of 2 Courses and 11 Lectures.

Schedule 

Courses (3 sessions each)

"Fully Nonlinear Tools" 

by  Xavier Cabré (ICREA and UPC).

We will prove the Alexandroff ABP estimate (also in its improved version) and describe the Calderón-Zygmund technique, as well as the Krylov-Safonov, Evans, and Caffarelli results on Fully nonlinear elliptic equations. We will apply this to obtain C^{2,alpha} regularity for a nonconvex Isaacs operator following a result obained with Caffarelli  in 2003; in a particular simpler case, this had been proved previously by Brezis-Evans in 1979 using an obstacle problem approach. We will conclude by using the ABP technique to prove the optimal isoperimetric inequality in Euclidean space, presenting the author’s ABP theory on Riemannian manifolds, and a very recent further development of it, by Brendle, which shows that the the optimal isoperimetric constant on minimal surfaces is the Euclidean one. 

[Download here the syllabus of the course with more details and references]

"Free boundary regularity in obstacle problems" 

by Alessio Figalli (ETH Zürich, FIM).

The aim of this course is to give a overview of the classical theory for the obstacle problem, and then present some recent developments on the regularity of the free boundary.

[Download here the lecture notes of the course with more details and references]

Lectures (45 + 15 mins each)