A case study for free boundary regularity: the obstacle problem
Venue: Federico Franceschini - Aula Buzano (DISMA Politecnico di Torino)
Free boundaries arise naturally in many nonlinear elliptic (or parabolic) problems: roughly speaking, free boundaries are interfaces at which solutions display some kind of discontinuous behaviour. Such interfaces are “free” in the sense that they are not prescribed and do not solve themselves an equation. The main (open) problem in the field is to determine if free boundaries are in fact smooth hypersurfaces and what is going on if they are not. A model case is the so-called Obstacle Problem.In this talk --- aimed to a general Analysis audience --- I will give an introduction to the Obstacle Problem and present the current understanding of the free boundary regularity in this case. I will explain some recent results (obtained with W. Zatoń) concerning the C\infty regularity of the free boundary outside of a lower-dimensional set. Such results bear nice similarities with the regularity theory of Minimal Surfaces.
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