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...
| Authors: | , , , |
|---|---|
| Format: | article |
| Publication Date: | 2020 |
| Country: | España |
| Institution: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repository: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2445/175795 |
| Online Access: | https://hdl.handle.net/2445/175795 |
| Access Level: | Open access |
| Keyword: | Equacions en derivades parcials Equacions diferencials el·líptiques Partial differential equations Elliptic differential equations |
| Summary: | 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. |
|---|