14 problems

Let $\Omega$ be a bounded open set in $\mathbb{R}^3$ with smooth boundary, and let $\gamma$ be a Jordan curve on $\partial\Omega$ separating it into relatively open sets $\Gamma^+$…

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…

Assume $m=1$, $d=2$, and let $U_{\Phi}$ be the set of interior slopes for a simply attractive potential. Let $\sigma$ be the surface tension. Cusp conjecture. The smooth phases wit…

Assume $m=1$, $d=2$, and that $\Phi$ is a general discrete simply attractive potential. Let $\sigma$ denote the surface tension, and let the slopes in the dual of $\mathcal L$ be t…

Let $U$ be a uniformly elliptic potential on the lattice-gradient field, and let the corresponding Funaki–Spohn state have slope-dependent surface tension. The surface tension is s…

Lafaurie–Popinet conjecture. The amplitude of spurious currents is proportional to

Let $\Lambda$ be a probabilistic solution of the radial Stefan problem, and let the physicality condition be … which constrains the jumps of the solution. Uniqueness conjecture. Th…

Let $\mathrm{Bo}$ denote the Bond number, and consider a fluid domain with free surface $\mathcal{F}$ and fundamental sloshing profile $\xi$. The high spot is a point where $\xi$ a…

Surface-tension conjecture. The surface tension remains valid when the constant weights $p$ and $q$ are replaced by the position-dependent weights

Surface-tension smoothness conjecture. One conjectures that

Let exp} denote the exponential interaction, and for each $N$ let $\sigma_N^{\mathsf{exp}}$ be its finite surface tension. Exponential surface-tension conjecture. The limit … exi…

Consider the two-dimensional $q$-state Potts model, with critical inverse temperature $\beta_c(q)$ and surface tension describing the macroscopic phase-separation cost. Potts surfa…

Let $\beta_c$ denote the critical inverse temperature, and consider the variational problem governing the macroscopic phase partition in the Ising model, with its surface-tension f…

The Muskat problem describes the evolution of an interface between fluids, and Theorem 1.1 gives a result for the problem without surface tension. Surface-tension analogue conjectu…