6 problems

Let $d4b\mu=e^W\mathcal{L}^N$ be a measure on $d4b\mathbb{R}^N$, where $W$ is smooth, convex, and radial. A set is isoperimetric for a given weighted volume if it minimizes weighte…

Let $\mathbf{R}^3$ carry the radial density $r^p$. A standard double bubble is the usual two-region double bubble bounded by spherical surfaces meeting along a singular circle at…

Log-convex density conjecture. Balls centered at the origin provide isoperimetric regions of every prescribed weighted volume. Chambers subsequently proved the assertion for densit…

Higher-dimensional double-bubble classification conjecture. A perimeter-minimizing double bubble is either

Higher-dimensional behavior conjecture. Some of the behavior on $\mathbb{R}^1$ will recur in higher dimensions.

Let $(M^n,g,v,m,0)$ be a closed weighted Einstein manifold with $P_\phi^m=\lambda g>0$ and scale $\kappa>0$. Let $I_1,I_2\colon C^\infty(M)\to\mathbb{R}$ be the functionals defined…