PUBLICATIONS MATHÉMATIQUES DE L'IHES

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Generic regularity of free boundaries for the obstacle problem
Publications Mathématiques de l'IHÉS, Volume 132 (2020), pp. 181-292
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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 ∞ 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≤4, the free boundary is generically a C ∞ manifold. This solves a conjecture of Schaeffer (dating back to 1974) on the generic regularity of free boundaries in dimensions n≤4.

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

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