dc.contributor.author
Caponi, M.
dc.contributor.author
Carbotti, A.
dc.contributor.author
Maione, Alberto
dc.date.accessioned
2026-01-19T12:00:05Z
dc.date.available
2026-01-19T12:00:05Z
dc.date.issued
2025-10-10
dc.identifier.uri
http://hdl.handle.net/2072/489124
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
Open access funding provided by Università del Salento within the CRUI-CARE Agreement. The authors are members of GNAMPA of the Istituto Nazionale di Alta Matematica (INdAM). M.C. has been funded by the European Union - NextGenerationEU under the Italian Ministry of University and Research (MUR) National Centre for HPC, Big Data and Quantum Computing (CN_00000013 - CUP: E13C22001000006). M.C. acknowledges also the support of the MUR - PRIN 2022 project “Variational Analysis of Complex Systems in Materials Science, Physics and Biology” (N. 2022HKBF5C), funded by European Union NextGenerationEU, and of the INdAM - GNAMPA 2025 Project “DISCOVERIES - Difetti e Interfacce in Sistemi Continui: un’Ottica Variazionale in Elasticità con Risultati Innovativi ed Efficaci Sviluppi” (Grant Code: CUP_E5324001950001). A.C. acknowledges the support of the MUR - PRIN 2022 project “Elliptic and parabolic problems, heat kernel estimates and spectral theory” (N. 20223L2NWK); the INdAM - GNAMPA 2024 Project “Ottimizzazione e disuguaglianze funzionali per problemi geometricospettrali locali e nonlocali” (Grant Code: CUP_E53C23001670001). A.M. is supported by the Spanish State Research Agency, through the Severo Ochoa and Maria de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M), by MCIN/AEI/10.13039/501100011033 (PID2021-123903NB-I00), by Generalitat de Catalunya (2021-SGR-00087). M.C. and A.M. acknowledge the support of the INdAM - GNAMPA2024 Project “Pairing and div-curl lemma: extensions to weakly differentiable vector fields and nonlocal differentiation” (Grant Code: CUP_E53C23001670001). A.C. and A.M. acknowledge the support of the INdAM -GNAMPA2025 Project “Metodi variazionali per problemi dipendenti da operatori frazionari isotropi e anisotropi” (Grant Code: CUP_E5324001950001). This research was supported by the Centre de Recerca Matemàtica of Barcelona (CRM), under the International Programme for Research in Groups (IP4RG).
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dc.format.extent
36 p.
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dc.relation.ispartof
Calculus of Variations and Partial Differential Equations
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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
Non-local linear operators
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dc.title
H-compactness for nonlocal linear operators in fractional divergence form
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dc.type
info:eu-repo/semantics/article
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dc.description.version
info:eu-repo/semantics/publishedVersion
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dc.identifier.doi
10.1007/s00526-025-03139-7
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dc.rights.accessLevel
info:eu-repo/semantics/openAccess
