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http://hdl.handle.net/2445/175795
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DC Field | Value | Language |
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dc.contributor.author | Cabré, Xavier | - |
dc.contributor.author | Figalli, Alessio | - |
dc.contributor.author | Ros, Xavier | - |
dc.contributor.author | Serra Montolí, Joaquim | - |
dc.date.accessioned | 2021-03-26T08:36:03Z | - |
dc.date.available | 2021-03-26T08:36:03Z | - |
dc.date.issued | 2020-09-01 | - |
dc.identifier.issn | 0001-5962 | - |
dc.identifier.uri | http://hdl.handle.net/2445/175795 | - |
dc.description.abstract | 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. | ca |
dc.format.extent | 66 p. | - |
dc.format.mimetype | application/pdf | - |
dc.language.iso | eng | ca |
dc.publisher | International Press | ca |
dc.relation.isformatof | Versió postprint del document publicat a: https://doi.org/10.4310/ACTA.2020.v224.n2.a1 | - |
dc.relation.ispartof | Acta Mathematica, 2020, vol. 224, num. 2, p. 187-252 | - |
dc.relation.uri | https://doi.org/10.4310/ACTA.2020.v224.n2.a1 | - |
dc.rights | (c) Institut Mittag-Leffler , 2020 | - |
dc.source | Articles publicats en revistes (Matemàtiques i Informàtica) | - |
dc.subject.classification | Equacions en derivades parcials | - |
dc.subject.classification | Equacions diferencials el·líptiques | - |
dc.subject.other | Partial differential equations | - |
dc.subject.other | Elliptic differential equations | - |
dc.title | Stable solutions to semilinear elliptic equations are smooth up to dimension 9 | ca |
dc.type | info:eu-repo/semantics/article | ca |
dc.identifier.idgrec | 708550 | - |
dc.date.updated | 2021-03-26T08:32:53Z | - |
dc.relation.projectID | info:eu-repo/grantAgreement/EC/H2020/801867/EU//EllipticPDE | - |
dc.rights.accessRights | info:eu-repo/semantics/openAccess | ca |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
Files in This Item:
File | Description | Size | Format | |
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708550.pdf | 682.57 kB | Adobe PDF | View/Open |
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