Generic Regularity in Obstacle Problems

Explore the classical obstacle problem in this 47-minute lecture by Alessio Figalli from ETH Zürich, presented at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the intricacies of determining the equilibrium state of an elastic membrane constrained above a predetermined obstacle. Examine Caffarelli's classical results on the smoothness of the 'free boundary' - the intersection between the obstacle and membrane - outside a set of singular points. Discover how explicit examples demonstrate that the singular set can be as extensive as the regular set. Gain insights into both classical and recent findings regarding the regularity of the free boundary in this captivating exploration of a beautiful mathematical problem.