dc.contributor.author
Ros-Oton, Xavier
dc.contributor.author
Weidner, M.
dc.date.accessioned
2026-01-19T15:04:45Z
dc.date.available
2026-01-19T15:04:45Z
dc.date.issued
2025-10-29
dc.identifier.uri
http://hdl.handle.net/2072/489131
dc.description.abstract
In this article, we study for the first time the regularity of the free boundary in the one-phase free boundary problem driven by a general nonlocal operator. Our main results establish that the free boundary is C-1,C-alpha near regular points, and that the set of regular free boundary points is open and dense. Moreover, in 2D we classify all blow-up limits and prove that the free boundary is C-1,C-alpha everywhere. The main technical tool of our proof is an improvement of flatness scheme, which we establish in the general framework of viscosity solutions, and which is of independent interest. All of these results were only known for the fractional Laplacian, and are completely new for general nonlocal operators. In contrast to previous works on the fractional Laplacian, our method of proof is purely nonlocal in nature.
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dc.description.sponsorship
The authors were supported by the European Research Council under the Grant Agreements No. 801867 (EllipticPDE) and No. 101123223 (SSNSD), and by the AEI project PID2024-156429NB-I00 (Spain). Moreover, X. Ros-Oton was supported by the grant RED2024-153842-T(Spain) funded by AEI/10.13039/501100011033, by the grant AEI-DFG project PCI2024-155066-2(Spain-Germany), by AGAUR Grant 2021 SGR 00087 (Catalunya), and by the Spanish State Research Agency through the Maria de Maeztu Program for Centers and Units of Excellence in R&D(CEX2020-001084-M). Moreover, M. Weidner was also supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC-2047/1 - 390685813 and through the CRC 1720.
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dc.format.extent
57 p.
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dc.publisher
EMS Press
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dc.relation.ispartof
Journal of the European Mathematical Society
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dc.rights
Attribution 4.0 International
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dc.rights.uri
http://creativecommons.org/licenses/by/4.0/
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dc.source
RECERCAT (Dipòsit de la Recerca de Catalunya)
dc.subject.other
nonlocal
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dc.subject.other
regularity
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dc.subject.other
one-phase problem
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dc.subject.other
free boundary
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dc.subject.other
flatness
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dc.title
Improvement of flatness for nonlocal free boundary problems
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dc.type
info:eu-repo/semantics/article
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dc.description.version
info:eu-repo/semantics/acceptedVersion
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dc.identifier.doi
10.4171/jems/1724
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dc.rights.accessLevel
info:eu-repo/semantics/openAccess
