Examples of optimal Hölder regularity in semilinear equations involving the fractional Laplacian
We discuss the Hölder regularity of solutions to the semilinear equation involving the fractional Laplacian [(¿)^s]u = f(u) in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the nonlocality and the semilinearity of the equation, since it does not happen n...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2025 |
| 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:2117/446489 |
| Acceso en línea: | https://hdl.handle.net/2117/446489 https://dx.doi.org/10.1016/j.na.2025.113755 |
| Access Level: | acceso abierto |
| Palabra clave: | Fractional Laplacian Semilinear equations Hölder regularity Classificació AMS::35 Partial differential equations::35B Qualitative properties of solutions Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type Classificació AMS::35 Partial differential equations::35S Pseudodifferential operators and other generalizations of partial differential operators Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en derivades parcials |
| Sumario: | We discuss the Hölder regularity of solutions to the semilinear equation involving the fractional Laplacian [(¿)^s]u = f(u) in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the nonlocality and the semilinearity of the equation, since it does not happen neither for local semilinear equations, nor for nonlocal linear equations. Namely, for nonlinearities f in C^ß and when 2s + ß < 1, the solution is not always C^(2s + ß - e) for all e > 0. Instead, in general the solution u is at most C^(2s/(1-ß)) |
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