27 problems

Choksi–Peletier conjecture. There exists a unique threshold $\varepsilon^*>0$ such that, for $\varepsilon\leq\varepsilon^*$, the unit ball is the unique minimizer of $G$ under the…

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…

Choksi's conjecture. The three thresholds coincide:

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…

For a ball of radius $R$, let $\mu_*$ be the minimum value of the isolated-droplet energy per unit volume and let $m_*$ be the largest volume among minimizers, while $m_{**}$ is th…

Let … for measurable sets $\Omega\subset\mathbb{R}^3$ of positive volume, and let $\Omega_*$ be a minimizer of $I$. Two sets are considered equivalent under the symmetries of the p…

Positivity conjecture for $I_a$. $I_a>0$ for all $a\in(0,\frac12)$.

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 $E\subset\mathbb{R}^n$ be a finite-perimeter set with mass $|E|=m$, surface energy … and nonlocal repulsion energy … The binding energy is…

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 $\chi$ denote the fissility parameter and let $\chi_*$ be the bifurcation value. A saddle point is a transition state along a minimum energy path for nuclear fission. Bohr–Whee…

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…

Consider the Gamow liquid drop problem … where $V_0\in(0,+\infty)$ and $P(\Omega)$ denotes the perimeter of $\Omega$. Gamow's threshold conjecture. There is a threshold $V^*>0$ suc…

Let $V_Z(x)=Z/|x|$ with $Z\geq 0$. For each $\varepsilon>0$, let $\mu_\varepsilon>0$ be such that the TFDW energy $e_\varepsilon^Z(M)$ is attained for every $0<M<Z+\mu_\varepsilon$…

Ball-minimizer conjecture. The minimizer of $\widetilde E_\infty$ with fixed mass is given by a ball whenever such a minimizer exists.

Ball-minimizer conjecture. If a minimizer of $\widetilde E_\infty$ with fixed mass exists, then it is given by a ball.

Let E_{[?] denote the energy of a configuration in the liquid drop model at background density $[?]$, and consider minimizers in the dilute limit $[?]\to 0$. A minimizer is under…

Equipartition conjecture. The masses of the components of every generalized minimizer are equal:

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…

Consider the liquid drop energy … for measurable sets $\Omega\subset\mathbb{R}^3$ of volume $|\Omega|=A$, and let … Here $B$ denotes a ball, and $\operatorname{Per}\Omega$ is the D…