3 problems

Annular-support conjecture. The support of the minimizer is an annulus when $\bar{b}<b<b_*(4)$. This conjecture concerns the transition of the minimizer's support from a ball to a…

Let $\mu=e^{v(r)}\,dx$ be a radially symmetric measure on $\mathbb{R}^d$, where $v$ is a convex smooth potential and $d\geq 2$. The balls about the origin are $B_R=\{x:|x|\leq R\}$…

Let $\rho$ be a smooth radial density on $\mathbb{R}^d$, so that $\mu=\rho(r)\,dx$, and suppose that $\log \rho(r)$ is smooth and convex. An isoperimetric region is a set minimizin…