Symmetry and quantitative stability for the parallel surface fractional torsion problem
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- by Giulio Ciraolo, Serena Dipierro, Giorgio Poggesi, Luigi Pollastro and Enrico Valdinoci HTML | PDF
- Trans. Amer. Math. Soc. 376 (2023), 3515-3540 Request permission
Abstract:
We study symmetry and quantitative approximate symmetry for an overdetermined problem involving the fractional torsion problem in a bounded open set $\Omega \subset \mathbb R^n$. More precisely, we prove that if the fractional torsion function has a $C^1$ level surface which is parallel to the boundary $\partial \Omega$ then $\Omega$ is a ball. If instead we assume that the solution is close to a constant on a parallel surface to the boundary, then we quantitatively prove that $\Omega$ is close to a ball. Our results use techniques which are peculiar of the nonlocal case as, for instance, quantitative versions of fractional Hopf boundary point lemma and boundary Harnack estimates for antisymmetric functions. We also provide an application to the study of rural-urban fringes in population settlements.References
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Additional Information
- Giulio Ciraolo
- Affiliation: Departimento di Matematica, Università di Milano, Via Cesare Saldini 50, Milan I-20133, Italy
- MR Author ID: 738335
- Email: giulio.ciraolo@unimi.it
- Serena Dipierro
- Affiliation: Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia
- MR Author ID: 924411
- Email: serena.dipierro@uwa.edu.au
- Giorgio Poggesi
- Affiliation: Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia
- MR Author ID: 1217338
- Email: giorgio.poggesi@uwa.edu.au
- Luigi Pollastro
- Affiliation: Departimento di Matematica, Università di Milano, Via Cesare Saldini 50, Milan I-20133, Italy
- ORCID: 0000-0002-2796-1464
- Email: luigi.pollastro@unimi.it
- Enrico Valdinoci
- Affiliation: Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia
- MR Author ID: 659058
- ORCID: 0000-0001-6222-2272
- Email: enrico.valdinoci@uwa.edu.au
- Received by editor(s): October 7, 2021
- Received by editor(s) in revised form: August 30, 2022, and October 10, 2022
- Published electronically: January 18, 2023
- Additional Notes: The first author and the fourth author had been partially supported by the “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni” (GNAMPA) of the “Istituto Nazionale di Alta Matematica” (INdAM, Italy).
The second author, the third author and the fifth author were members of AustMS. S. Dipierro is supported by the Australian Research Council DECRA DE180100957 “PDEs, free boundaries and applications”. The third author was member of INdAM/GNAMPA. The third author and the fifth author were supported by the Australian Laureate Fellowship FL190100081 “Minimal surfaces, free boundaries and partial differential equations”. - © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 3515-3540
- MSC (2020): Primary 35N25, 35B06, 35R11
- DOI: https://doi.org/10.1090/tran/8837
- MathSciNet review: 4577340