A universal Hölder estimate up to dimension 4 for stable solutions to half-Laplacian semilinear equations
We study stable solutions to the equation , posed in a bounded domain of . For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions . This result, which was known only for , follows from a new interior Hölder estimate that is completely independent of the nonlin...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/388320 |
| Acceso en línea: | https://hdl.handle.net/2117/388320 https://dx.doi.org/10.1016/j.jde.2022.02.001 |
| Access Level: | acceso abierto |
| Palabra clave: | Differential equations, Partial Half-Laplacian Stable solutions Extremal solution Interior estimates Dirichlet problem Equacions en derivades parcials Classificació AMS::35 Partial differential equations::35B Qualitative properties of solutions Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials |
| Sumario: | We study stable solutions to the equation , posed in a bounded domain of . For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions . This result, which was known only for , follows from a new interior Hölder estimate that is completely independent of the nonlinearity f. A main ingredient in our proof is a new geometric form of the stability condition. It is still unknown for other fractions of the Laplacian and, surprisingly, it requires convexity of the nonlinearity. From it, we deduce higher order Sobolev estimates that allow us to extend the techniques developed by Cabré, Figalli, Ros-Oton, and Serra for the Laplacian. In this way we obtain, besides the Hölder bound for , a universal estimate in all dimensions. Our bound is expected to hold for , but this has been settled only in the radial case or when . For other fractions of the Laplacian, the expected optimal dimension for boundedness of stable solutions has been reached only when , even in the radial case. |
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