I will describe recent results with Alessio Figalli on 1D symmetry of stable critical points of the Ginzburg-Landau functional associated to the half-Laplacian. As we will discuss, an interesting feature of these ...

Given an arbitrary convex cone of Rn, we find a geometric class of homogeneous weights for which balls centered at the origin and intersected with the cone are minimizers of the weighted isoperimetric problem in the convex ...

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 ...