Regularity of stable solutions to reaction-diffusion elliptic equations
The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970s. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). This note describes, for non-expert readers, a recent work in collaboration wi...
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| Tipo de recurso: | capítulo de libro |
| Fecha de publicación: | 2023 |
| 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:2117/399178 |
| Acceso en línea: | https://hdl.handle.net/2117/399178 https://dx.doi.org/10.4171/8ECM/13 |
| Access Level: | acceso abierto |
| Palabra clave: | Differential equations, Partial Semilinear elliptic equations stable solutions extremal solutions regularity a priori estimates Equacions en derivades parcials Classificació AMS::35 Partial differential equations Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals |
| Sumario: | The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970s. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). This note describes, for non-expert readers, a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solu- tions are smooth up to the optimal dimension 9. This solves an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems. We also describe, briefly, a famous analogue question in differential geometry: the regularity of stable minimal surfaces. |
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