3 problems

Critical-volume conjecture. For $m\leq m_{*}$, the ball of volume $m$ uniquely minimizes $\mathcal{E}(\cdot)$ among measurable sets $F\subset\mathbb{R}^n$ with $|F|=m$, and for…

Let $n=2$, let $\alpha\in(0,2)$, and define $f(m)=P(B[m])+\mathcal{V}_\alpha(B[m])$, where $B[m]$ is a ball of volume $m$. Set $m_0=0$, and for $k\geq1$ let $m_k$ be the critical m…

For $n=3$ and $\alpha=2$, consider the liquid-drop energy $P(E)+\mathcal{V}_\alpha(E)$ among sets $E\subset\mathbb{R}^n$ of prescribed volume. A minimizer is a set attaining the in…