Stable solutions to the fractional Allen-Cahn equation in the nonlocal perimeter regime
We study stable solutions to the fractional Allen-Cahn equation (-¿)^(s/2)u = u - u^3, |u|<1 in R^3. For every s ¿ (0,1) and dimension n >=2, we establish sharp energy estimates, density estimates, and the convergence of blow-downs to stable nonlocal s-minimal cones. As a consequence, we obtai...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2025 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/446469 |
| Online Access: | https://hdl.handle.net/2117/446469 https://dx.doi.org/10.1353/ajm.2025.a966290 |
| Access Level: | Open access |
| Keyword: | Differential equations, Partial Equacions en derivades parcials Classificació AMS::35 Partial differential equations::35R Miscellaneous topics involving partial differential equations Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials |
| Summary: | We study stable solutions to the fractional Allen-Cahn equation (-¿)^(s/2)u = u - u^3, |u|<1 in R^3. For every s ¿ (0,1) and dimension n >=2, we establish sharp energy estimates, density estimates, and the convergence of blow-downs to stable nonlocal s-minimal cones. As a consequence, we obtain a new classification result: if for some pair (n,s), with n>=3, hyperplanes are the only stable nonlocal s-minimal cones in R^n/{0}, then every stable solution to the fractional Allen-Cahn equation in R^n is 1D, namely, its level sets are parallel hyperplanes. Combining this result with the classification of stable s-minimal cones in R^3\{0} for s = 1 obtained by the authors in a recent paper, we give positive answers to the ``stability conjecture'' in R^3 and to the ``De Giorgi conjecture'' in R^4 for the fractional Allen-Cahn equation when the order s ¿ (0,1) of the operator is sufficiently close to 1. |
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