The role of antisymmetric functions in nonlocal equations
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- by Serena Dipierro, Giorgio Poggesi, Jack Thompson and Enrico Valdinoci;
- Trans. Amer. Math. Soc. 377 (2024), 1671-1692
- DOI: https://doi.org/10.1090/tran/8984
- Published electronically: January 10, 2024
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Abstract:
We use a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms, in combination with the method of moving planes, to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that non-negative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set $\Omega \subset \mathbb {R}^n$ must be radially symmetric if one of their level surfaces is parallel to the boundary of $\Omega$; in turn, $\Omega$ must be a ball.
Furthermore, we discuss maximum principles and the Harnack inequality for antisymmetric functions in the fractional setting and provide counter-examples to these theorems when only ‘local’ assumptions are imposed on the solutions. The construction of these counter-examples relies on an approximation result that states that ‘all antisymmetric functions are locally antisymmetric and $s$-harmonic up to a small error’.
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Bibliographic Information
- Serena Dipierro
- Affiliation: Department of Mathematics and Statistics, The 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, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia
- MR Author ID: 1217338
- Email: giorgio.poggesi@uwa.edu.au
- Jack Thompson
- Affiliation: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia
- MR Author ID: 1553050
- ORCID: 0000-0002-6299-2688
- Email: jack.thompson@research.uwa.edu.au
- Enrico Valdinoci
- Affiliation: Department of Mathematics and Statistics, The 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): March 21, 2022
- Received by editor(s) in revised form: February 13, 2023
- Published electronically: January 10, 2024
- Additional Notes: The first author was supported by the Australian Research Council DECRA DE180100957 “PDEs, free boundaries and applications”.
The second author was supported by the Australian Research Council (ARC) Discovery Early Career Researcher Award (DECRA) DE230100954 “Partial Differential Equations: geometric aspects and applications”, and is member of INdAM/GNAMPA. The second and fourth authors were supported by the Australian Laureate Fellowship FL190100081 “Minimal surfaces, free boundaries and partial differential equations”.
The third author was supported by an Australian Government Research Training Program Scholarship. - © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 1671-1692
- MSC (2020): Primary 35B50, 35N25, 35B06
- DOI: https://doi.org/10.1090/tran/8984
- MathSciNet review: 4744739