Publication: Stable solutions to semilinear elliptic equations are smooth up to dimension $9$
Stable solutions to semilinear elliptic equations are smooth up to dimension $9$
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Cabré, X., Figalli, A., Ros-Oton, X., & Serra, J. (2020). Stable solutions to semilinear elliptic equations are smooth up to dimension $9$. Acta Mathematica, 224(2), 187–252. https://doi.org/10.4310/acta.2020.v224.n2.a1
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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 establis
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- Cabré, Xavier
- Figalli, Alessio
- Ros-Oton, Xavier
- Serra, Joaquim
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Cabré, X., Figalli, A., Ros-Oton, X., & Serra, J. (2020). Stable solutions to semilinear elliptic equations are smooth up to dimension $9$. Acta Mathematica, 224(2), 187–252. https://doi.org/10.4310/acta.2020.v224.n2.a1
