99 problems

Pekar's conjecture. The limit $g_0$ exists and satisfies

One-dimensionality conjecture. The optimal transition profile will be one-dimensional when $\beta\in(0,1)$ and $|a^+-a^-|$ is small.

Weighted-minimizer profile conjecture. The minimizers are conjectured to have the form

Let $(M,\zeta)$ be a closed contact $3$-manifold, where $\zeta$ is a contact distribution, and consider the Chern-Hamilton energy functional on the space of compatible metrics for…

Let $q_{\mathrm{ic}}=\delta_{-\zeta}+\delta_{\zeta}$ for $\zeta>0$, set $\mathfrak{m}=1$ and $\xi_1=0$, and consider the variational problem in which $T$ and these data are fixed.…

Let $B_1$ be the open unit ball in $\mathbb{R}^n$, let $G_\lambda(x)=\lambda^{n+1}G(\lambda x)$ for $\lambda>0$, and define … Consider … with $\gamma>0$. Large-parameter unit-ball…

Choksi's conjecture. The three thresholds coincide:

Let … For a positive function $u$ on $S^1$, let ${\bf F}(u)$ denote the functional whose minimizers are being considered. Extremal classification conjecture. If $u$ is a minimizer…

Let $V$ be the integral operator with kernel $v$, and let $\chi_N^{(\otimes)}$ be the Hartree variational quantity obtained by minimising over normalized functions…

Unduloid-extremizer conjecture. In the case $x=0$ and $y<1$, the maximizers and minimizers correspond to the unduloid immersions.

For $p>0$, let $A_p[c]$ be the average chord-length functional on closed, unit-speed curves of length $2\pi$, and let $p^*$ be the critical value at which the circle ceases to maxi…

For $p>0$, let $A_p[c]$ be the average chord-length functional on closed, unit-speed curves $c$ of length $2\pi$ in $\mathbf{R}^n$, … A maximizer exists and every maximizer is conv…

Distortion ropelength minimizer conjecture. For each nontrivial knot type $K$, there exists a shortest curve $[?]\subset\mathbb R^3$ of distortion thickness $1$ in that knot type.…

Aftalion's conjecture. Asymmetric solutions can have lower energy than the symmetric solutions.

Marcus's conjecture. A minimizer of $\mathcal G$ is probably a symmetric solution satisfying (1.2)–(1.6). The source leaves open whether asymmetric solutions may also exist as mini…

Let … and, for $a,c,d>0$, define … The constrained problem is to minimize $J_{a,c,d}(u)$ over $u\in E_2$ subject to $u(0,0)=1$. Ekeland–Nirenberg's positivity conjecture. For every…

Non-trivial interlacement conjecture. For every sufficiently large particle density, the minimizing point process cannot contain loops only; it must have a non-trivial interlacemen…

Let $w\in H^2_{\operatorname{loc}}(\mathbb{R}^n)$ have constant Laplacian. Constant-Laplacian conjecture. Every such $w$ is a global minimizer of $\mathcal{F}$. The conjecture aris…

Let $S_{3/2,\mathrm{curl}}$ denote the best constant in the Sobolev-curl inequality at $p=3/2$, and let $u$ be a vector field of the form defined in the source by … . This conjectu…

Let $d\geq 2$, let $q>1$, and let $\gamma$ be a density matrix satisfying $0\leq\gamma\leq 1$. Write … for a rank-$N$ projection, and let $\rho_\gamma$ denote its density. The asso…

Let $\mathcal{FS}_\rho$ be the perturbed Faddeev–Skyrme energy on maps $u:\mathbb S^3\to\mathbb S^2$, and let $h:\mathbb S^3\to\mathbb S^2$ be the Hopf map. Two maps are in the sam…

Let the Skyrme model have domain $\mathbb S^3_\rho$ and let $x\mapsto x/\rho$ be the homothety, where $\rho>0$. Two maps are in the same homotopy class when they are homotopic as m…

Nam's radial-measure conjecture. The infimum defining $\beta_2$ is achieved by radially symmetric probability measures $\mu$; equivalently, $\beta_2$ can be calculated using only r…

Let $x=(x_1,\ldots,x_d)$ with $x_i\in{\mathbb R}^n$, signed masses $m(x_i)$ satisfying … and define … A configuration is stationary and stable when its first variation vanishes and…

For a tame knot class $K$, let $\mathcal{K}$ be the $W^{2,2}$-closure of the class of unit-length, arclength-parametrized closed curves in $K$: … A stable elastic knot for $K$ is a…