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$C^\infty$ partial
regularity of the singular set in the obstacle problem
Federico Franceschini and Wiktoria Zatoń |
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Vol. 18 (2025), No. 1, 199–264
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Abstract |
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We show that the singular set in the classical obstacle problem can be locally covered by a hypersurface, up to an “exceptional” set , which has Hausdorff dimension at most (countable in the case). Outside this exceptional set, the solution admits a polynomial expansion of arbitrarily large order. We also prove that is extremely unstable with respect to monotone perturbations of the boundary datum. We apply this result to the planar Hele-Shaw flow, showing that the free boundary can have singular points for at most countable many times. |
Keywords
obstacle problem, singular set, higher-regularity
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Mathematical Subject Classification
Primary: 35R35
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Milestones
Received: 19 December 2022
Revised: 18 July 2023
Accepted: 31 August 2023
Published: 15 December 2024
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Authors |
| Federico Franceschini | |
| Department of Mathematics ETH Zürich Zürich Switzerland |
School of Mathematics Institute for Advanced Study Princeton, NJ United Stated |
| Wiktoria Zatoń | |
| Institut für Mathematik Universität Zürich Zürich Switzerland |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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