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Generic regularity of free boundaries for the obstacle problem

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Abstract

The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in \(\mathbf {R}^{n}\). 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). 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 \(n\leq 4\).

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Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, Raemistrasse 101, 8092, Zürich, Switzerland

    Alessio Figalli & Joaquim Serra

  2. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zürich, Switzerland

    Xavier Ros-Oton

  3. ICREA, Pg. Lluís Companys 23, 08010, Barcelona, Spain

    Xavier Ros-Oton

  4. Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007, Barcelona, Spain

    Xavier Ros-Oton

Authors
  1. Alessio Figalli
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  2. Xavier Ros-Oton
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  3. Joaquim Serra
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Corresponding author

Correspondence to Alessio Figalli.

Additional information

AF and JS have received funding from the European Research Council (ERC) under the Grant Agreement No. 721675. XR was supported by the European Research Council (ERC) under the Grant Agreement No. 801867. JS was supported by Swiss NSF Ambizione Grant PZ00P2 180042. XR and JS were supported by MINECO grant MTM2017-84214-C2-1-P (Spain).

The authors are grateful to the anonymous referee for a careful reading and the useful comments.

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Figalli, A., Ros-Oton, X. & Serra, J. Generic regularity of free boundaries for the obstacle problem. Publ.math.IHES 132, 181–292 (2020). https://doi.org/10.1007/s10240-020-00119-9

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  • DOI: https://doi.org/10.1007/s10240-020-00119-9

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