A. Figalli | Generic regularity in obstacle problems
Channel: EIMI, PDMI RAS and Chebyshev Laboratory
Date: 2022-08-01

O.A. Ladyzhenskaya centennial conference on PDE’s, St. Petersburg, July 16 – July 22, 2022
Alessio Figalli (ETH Zurich) 
Generic regularity in obstacle problems

The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is smooth outside a set of singular points. Explicit examples show that the singular set could be, in general, as large as the regular set. In a recent paper with RosOton and Serra we show that, generically, the singular set has codimension 3 inside the free boundary, solving a conjecture of Schaeffer in dimension 𝑛 ≤ 4. The aim of this talk is to give an overview of these results.
References
[1] A. Figalli, X. Ros-Oton and J. Serra, Generic regularity of free boundaries
for the obstacle problem, Publ. Math. Inst. Hautes Etudes Sci. 132 (2020),
181–292.