ABSTRACT: 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 C∞ outside a set of singular points. Explicit examples show that the singular set could be in general (n−1)-dimensional — that is, as large as the regular set. In a recent paper with Ros-Oton and Serra we show that, generically, the singular set has zero Hn−4 measure (in particular, it 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.
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