12 problems

Let $n\geq 2$ and $0<\alpha<n$, and let $m_{\operatorname{ball}}$, $m_\mathrm{ex}$, and $m_\mathrm{crit}$ denote respectively the critical mass below which balls minimize the liqui…

Let $m_{\operatorname{ball}}$ be the critical mass below which balls minimize the liquid drop energy, let $m_\mathrm{ex}$ be the critical mass up to which minimizers exist, and let…

Choksi–Peletier conjecture. For $V\leq V_*$, the round ball of volume $V$ uniquely minimizes $\mathcal E(\cdot)$ among measurable sets $\Omega\subset\mathbb{R}^3$ with $|\Omega|=V$…

Let $\chi$ be the fissility parameter and let $\chi_3\approx0.428$ be the two-dimensional analogue of the Businaro–Gallone point. A scission point is a transition state on the sepa…

Let $\chi$ be the fissility parameter, let $\chi_*$ be the bifurcation value, and let $\chi_4\approx0.904\pm0.001$. The Ren–Wei branch is the two-dimensional equilibrium branch eme…

Let $\chi$ be the fissility parameter. The Bohr–Wheeler branch is the bifurcation branch of three-dimensional saddle points, and a saddle point is the transition state along the co…

Let $\omega\in((4\pi)^{-1},\pi/6)$ be the fixed volume fraction, and let $c$ and $K$ be the phase-field parameters. An unduloid is a constant-mean-curvature surface of revolution i…

Let $\omega=(4\pi-3\sqrt{3})/18\approx0.4095$ be the fixed volume fraction, and let $c$ and $K$ be the phase-field parameters. A transition state is the saddle point along the mini…

Let $\chi$ be the fissility parameter and let $\chi_*$ be the bifurcation value. The Ren–Wei branch is the equilibrium branch emerging at this bifurcation. Subcritical pitchfork co…

Liquid-drop minimizer conjecture. The energy $E^{\rm G}(m)$ has a minimizer if and only if $m\leqslant m_*$. Moreover, whenever a minimizer exists, it is a ball. The assertion refl…

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…