18 problems

Let $\psi:\mathbb{R}^2\to\mathbb{C}$ be a vortex solution of the Ginzburg–Landau equation with degree $n$, and call it radially symmetric when its modulus depends only on the radia…

Let $E”(u)$ be the Hessian of the Ginzburg–Landau energy at a minimizer $u$, and let $\rho(\kappa)^{-1}$ denote its coercivity constant. Assume the general hypotheses of the paper…

Consider the two-dimensional Ginzburg–Landau equation and entire solutions $u$ satisfying $|u|\to 1$ at infinity. For each degree $d\in\mathbb{Z}$, let $v_d$ denote a planar vortex…

BerRyb13 conjecture. For sufficiently small $\mathbb{\varepsilon}$, the degrees of the hole vortices are the same for $\mathbb{C}$-valued and $\mathbb{S}^1$-valued maps.

Uniqueness conjecture. The function $V$ is the only non-constant locally minimizing solution, up to the obvious symmetries of the problem. The paper presents this as an open classi…

Boundary-layer current-conversion conjecture. The normal current $-\sigma\nabla\phi$ turns into the superconducting current $\rho_{out}^2\nabla\chi_{out}$ within a thin one-dimensi…

Global sign conjecture. The same sign properties should hold for every $t\in(0,1]$:

Sign conjecture. The radial profiles satisfy

Let $E(L)$ be the limiting energy defined by … and let $\lambda$ be the function introduced in the source. For $b>0$, define $E^{\rm 1D}(b)=\mathfrak{b}(\alpha_0,b)$, where…

Symmetry-breaking bifurcation conjecture. The bifurcation at $\lambda_*$ does occur, producing a non-symmetric family of solutions with distinct vortices in the two components…

Let $A_+,A_->0$ be fixed and define … For the entire equivariant solution … with $B^2<A_+A_-$, write $f^\infty_\pm(r;B)$ for its component profiles. Monotonicity conjecture. The pr…

Consider the equivariant degree $[1,1]$ vortex in a two-component Ginzburg–Landau system, with component profiles $f_+(r)$ and $f_-(r)$ and parameter $B\in(0,B_0)$. The unstable di…

Radial minimizer conjecture. The radial solution $\tilde u_\varepsilon$ is the unique minimizer of $E_\varepsilon$ on $B_1$. This conjecture asks whether the explicitly identified…

Let $q$ be the exponent appearing in the preceding estimates, let $\alpha$ be the discretization exponent, and let the domain be the two-dimensional torus $\mathbb{T}^2$. An a prio…

Vorlatt's hexagonal-lattice conjecture. The minimum of $W$ over $\mathcal{A}_m$ is achieved for currents corresponding to a configuration of points in a perfect hexagonal lattice,…

Instability conjecture. For sufficiently small $\varepsilon$, the solution $u_\varepsilon$ is unstable, regardless of curvature conditions on $\mathcal M$.

In the higher-dimensional setting, let $J$ be a measure-valued $2$-form in ${\mathbb R}^n$, with $n\geq 4$, and consider the energy regimes $g_\epsilon\gg |\log\epsilon|$. The limi…

Let $\mu_*$ be the measure supported on the circle $\partial D(0,R)$ such that, up to extraction of a subsequence, … as $\varepsilon\to0$, where … is the vorticity measure in regim…