• Acta Math. (IF 2.458) Pub Date : 2020-06-23
Xavier Cabré; Alessio Figalli; Xavier Ros-Oton; Joaquim Serra

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 (1 / {\lvert x \rvert}^2)\$ is a \$W^{1,2}\$ singular stable solution for \$n \geqslant 10\$. The proof of this conjecture is a consequence of a new

更新日期：2020-07-20
• Acta Math. (IF 2.458) Pub Date : 2020-06-23
Jinxin Xue

In this paper, we show that there is a Cantor set of initial conditions in the planar \$4\$‑body problem such that all four bodies escape to infinity in a finite time, avoiding collisions. This proves the Painlevé conjecture for the \$4\$‑body case, and thus settles the last open case of the conjecture.

更新日期：2020-07-20
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