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Nonlinear Analysis

Generic regularity in obstacle problems

Alessio Figalli, ETH

Location:  Link: Meeting ID: 992 8122 5008
Date & time: Wednesday, 18 November 2020 at 9:30AM - 10:30AM

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 \(C^infty\) 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 \(mathcal H^{n-4}\) measure (in particular, it has codimension 3 inside the free boundary), solving a conjecture of Schaeffer in dimension \(n leq 4\). The aim of this talk is to give an overview of these results.

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