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Title: | Generic regularity of free boundaries for the obstacle problem |
Author: | Figalli, Alessio Ros, Xavier Serra Montolí, Joaquim |
Keywords: | Problemes de contorn Equacions en derivades parcials Funcions de variables complexes Distribució (Teoria de la probabilitat) Boundary value problems Partial differential equations Functions of complex variables Distribution (Probability theory) |
Issue Date: | 2-Jul-2020 |
Publisher: | Springer |
Abstract: | The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $\mathbb{R}^n$. 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. Our main result establishes that, generically, the singular set has zero $\mathcal{H}^{n-4}$ measure (in particular, it has codimension 3 inside the free boundary). In particular, for $n \leq 4$, the free boundary is generically a $C^{\infty}$ manifold. This solves a conjecture of Schaeffer (dating back to 1974 ) on the generic regularity of free boundaries in dimensions $n \leq 4$ |
Note: | Versió postprint del document publicat a: https://doi.org/10.1007/s10240-020-00119-9 |
It is part of: | Publications mathématiques de l'IHÉS, 2020, vol. 132, num. 1, p. 181-292 |
URI: | http://hdl.handle.net/2445/194135 |
Related resource: | https://doi.org/10.1007/s10240-020-00119-9 |
ISSN: | 0073-8301 |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
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