Stable solutions to semilinear elliptic equations are smooth up to dimension 9
In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension $n \leqslant 9$. This result, that was only known to be true for $n \leqslant 4,$ is optimal: $\log \left(1 /|x|^{2}\right)$ is a $W^{1,2}$ sin...
| Autores: | , , , |
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| Formato: | artículo |
| Fecha de publicación: | 2020 |
| País: | España |
| Recursos: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/175795 |
| Acesso em linha: | https://hdl.handle.net/2445/175795 |
| Access Level: | acceso abierto |
| Palavra-chave: | Equacions en derivades parcials Equacions diferencials el·líptiques Partial differential equations Elliptic differential equations |
| Resumo: | In this paper we prove the following long-standing conjecture: stable solutions to semi-linear elliptic equations are bounded (and thus smooth) in dimension $n \leqslant 9$. This result, that was only known to be true for $n \leqslant 4,$ is optimal: $\log \left(1 /|x|^{2}\right)$ is a $W^{1,2}$ singular stable solution for $n \geqslant 10$. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension $n \leqslant 9,$ stable solutions are bounded in terms only of their $L^{1}$ norm, independently of the non-linearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces. As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary, we obtain that extremal solutions of Gelfand problems are $W^{1,2}$ in every dimension and they are smooth in dimension $n \leqslant 9$. This answers to two famous open problems posed by Brezis and Brezis-Vázquez. |
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