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Stable solutions to semilinear elliptic equations are smooth up to dimension $9$


Cabré, Xavier; Figalli, Alessio; Ros-Oton, Xavier; Serra, Joaquim (2020). Stable solutions to semilinear elliptic equations are smooth up to dimension $9$. Acta Mathematica, 224(2):187-252.

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 ⩽ 9.
This result, that was only known to be true for n ⩽ 4, is optimal: log(1/|x|2) is a W1,2 singular stable solution for n ⩾ 10.
The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension n ⩽ 9, stable solutions are bounded in terms only of their L1 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 W1,2 in every dimension and they are smooth in dimension n ⩽ 9. This answers to two famous open problems posed by Brezis and Brezis–Vázquez.

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 ⩽ 9.
This result, that was only known to be true for n ⩽ 4, is optimal: log(1/|x|2) is a W1,2 singular stable solution for n ⩾ 10.
The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension n ⩽ 9, stable solutions are bounded in terms only of their L1 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 W1,2 in every dimension and they are smooth in dimension n ⩽ 9. This answers to two famous open problems posed by Brezis and Brezis–Vázquez.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:340 Law
610 Medicine & health
510 Mathematics
Scopus Subject Areas:Physical Sciences > General Mathematics
Uncontrolled Keywords:General Mathematics
Language:English
Date:1 January 2020
Deposited On:12 Oct 2020 15:33
Last Modified:29 Oct 2020 13:47
Publisher:International Press
ISSN:0001-5962
OA Status:Closed
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.4310/acta.2020.v224.n2.a1

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