7 problems

Annular-support conjecture. The support of the minimizer is an annulus when $\bar{b}<b<b_*(4)$. This conjecture concerns the transition of the minimizer's support from a ball to a…

Let $g$ be a power-law interaction kernel with exponent $\alpha$ in dimension $N$, and consider the discrete energies … and the continuum energy … For minimizers…

Superiority conjecture.

Consider a system of particles with arbitrary boundary conditions prescribed on the surfaces of the particles, and let the interaction energy denote the energy beyond the individua…

Consider two spherical colloidal particles in a nematogenic medium, with the nonlinear Euler–Lagrange problem and its linearization around the state $u\equiv 0$ having the same rat…

Let $\rho$ be a probability measure on $\mathbb{R}^2$, and consider the interaction energy … with logarithmic anisotropic potential … where $\theta$ is the angle of ${\bf x}$,…

Crystallization conjecture. As $n\to\infty$, the minimizers of $F_n$ over all possible points $y_1,\ldots,y_n\in\Omega$ arrange themselves in a periodic lattice.