19 problems

Consider the three-dimensional six-state Potts model in a box with boundary condition $i$ on the $i$-th face. Assume that the sharp simplex inequality holds, that the surface tensi…

Let $\sigma^t$ be the spin configuration at time $t$, and let $\mathcal C_L$ denote the event that the configuration restricted to the specified box of scale $L$ is constant, eithe…

Consider the homogeneous ferromagnetic zero-temperature stochastic Ising model in dimension two, with spins taking values in $\\{+1,-1\\}$ and with domain walls separating clusters…

Coarsening conjecture. The linear behavior of $j^\ast(\rho)$ in this regime does not result from an effective absence of interactions. Instead, the dynamics exhibits coarsening: th…

Let $\mathbf{w}^0$ be a non-degenerate phase-separating solution of the problem under consideration in a domain $\Omega\subset\mathbb{R}^N$, where $N\geq 3$. Non-degeneracy means t…

Three-dimensional Gamma-limit conjecture. For $n=3$ and $\rho\to0$, if $\zeta\in(\zeta_0,\zeta_1)$, the Gamma-limit of $F'_\rho$ is a quadratic form in the principal curvatures for…

Two-dimensional Gamma-limit conjecture. For $n=2$ and $\rho\to0$, if $\zeta\in(0,\zeta_1)$, the Gamma-limit of $F'_\rho$ is the elastica functional for closed $W^{2,2}$ curves in…

Three-dimensional asymptotic morphology conjecture. For $n=3$, fixed $\gamma>0$, and $\zeta>\zeta_0$,

The phase boundary of an area-constrained random walk has a maximal local roughness, namely the maximal deviation of the interface from the convex hull of the interface. Its scalin…

Gravitational-field variational-shape conjecture. The preceding convergence in probability holds. The supplied context says that the solution of this variational problem is known i…

Let $\Omega\subset\mathbb R^n$ be a bounded domain, let $u:\Omega\to\mathbb R$ denote the relative concentration of two phases with pure-phase values $u^-<u^+$, and consider the Ca…

Let $\omega\ll 1$ and $\gamma\gg 1$, and let $\mathcal{A}_N^{\omega}$ be the class of periodic step functions considered in the generalized Ohta–Kawasaki model. Let $U_N^{\omega}$…

Let $[\gamma_{ij}]$ satisfy the conditions in Theorem and have nullity $1$, with $\gamma>0$ its overall factor, so that $[\gamma_{ij}]/\gamma$ is fixed. Let $\vec u(\cdot;\gamma)$…

Consider the bifurcation diagram for the two-drop primary branch at $L=50$, with Hopf bifurcation point $I$ and pitchfork bifurcation point $3$ as described in the paper. A time-pe…

Let $u$ and $v$ denote the lipid concentrations in the reduced model, let $c_1$ and $c_2$ be its parameters, and define the non-equilibrium quantity … Let $[?]$ denote the paramete…

Disk-minimizer conjecture. If

Let $(u_{i,\beta})$ be a sequence of solutions satisfying the nontriviality condition that its limit profile is not identically zero. Suppose that $x_\beta\in\Gamma_\beta$ and…

Consider the elliptic system … in Euclidean space, together with the higher-dimensional constructions and properties discussed for this system and for the Allen–Cahn equation. High…

Let $P={\mathcal P}_{\beta,q}$ with $\beta<\beta_c$ and $q\geq 1$. For a circuit $\Gamma_0$, let $\operatorname{MLR}(\Gamma_0)$ and $\operatorname{MFL}(\Gamma_0)$ denote its maximu…