Generic regularity of free boundaries for the obstacle problem

Figalli, Alessio; Ros-Oton, Xavier; Serra, Joaquim

doi:10.1007/s10240-020-00119-9

Article

Generic regularity of free boundaries for the obstacle problem

Figalli, Alessio ; Ros-Oton, Xavier ; Serra, Joaquim

Publications Mathématiques de l'IHÉS, Tome 132 (2020), pp. 181-292

Résumé

The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $𝐑^{n}$ . 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 $ℋ^{n - 4}$ measure (in particular, it has codimension 3 inside the free boundary). Thus, 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 $n \leq 4$ .

Reçu le : 2019-11-29
Accepté le : 2020-06-23
Première publication : 2020-07-02
Publié le : 2020-12-28

MR Zbl

DOI : 10.1007/s10240-020-00119-9

@article{PMIHES_2020__132__181_0,
     author = {Figalli, Alessio and Ros-Oton, Xavier and Serra, Joaquim},
     title = {Generic regularity of free boundaries for the obstacle problem},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {181--292},
     year = {2020},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {132},
     doi = {10.1007/s10240-020-00119-9},
     mrnumber = {4179834},
     zbl = {1456.35234},
     language = {en},
     url = {https://www.numdam.org/articles/10.1007/s10240-020-00119-9/}
}

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Figalli, Alessio; Ros-Oton, Xavier; Serra, Joaquim. Generic regularity of free boundaries for the obstacle problem. Publications Mathématiques de l'IHÉS, Tome 132 (2020), pp. 181-292. doi: 10.1007/s10240-020-00119-9

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