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  • Stable solutions to semilinear elliptic equations are smooth up to dimension $9$
    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
  • Non-collision singularities in a planar 4-body problem
    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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