11 problems

Small-mass rigidity conjecture. There exists a universal constant $q_0>0$ such that, for $q<q_0$, balls are rigid minimizers for Problem $$ .

Let $m_{\operatorname{ball}}$ be the critical mass below which balls minimize the liquid drop energy, let $m_\mathrm{ex}$ be the critical mass up to which minimizers exist, and let…

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…

Consider the interaction energy with confinement $V(x)=|x|^2$ and anisotropic kernel as in the paper. For $obreakalphain[0,1)$, the minimiser is two-dimensional, whereas at…

Consider the generalized antiferromagnetic local/nonlocal interaction functionals in dimension $d$, with parameter $\tau$ measuring the relative strength of the short-range attract…

Higher-order truncation conjecture. Truncations to orders other than quadratic should give corresponding analysis and results.

Consider the discrete and continuous periodic variational models with coupling parameter $J$, and let $J^{\mathrm{dsc}}_c$ and $J_c$ be their respective critical constants, defined…

The discrete energy describes arrangements of positive and negative dislocations, with vertical walls formed by dislocations of the same sign. Taylor-lattice conjecture. The minimi…

Consider the repulsive-attractive potential … and the instability region in the $(a,b)$-parameter plane, where the spherical shell is unstable. An integrable radial stationary stat…

In dimension $N=1$, consider the stationary state … Here $\delta_{x=c}$ denotes the Dirac mass at $c$. Instability conjecture. The stationary state is unstable. Numerical simulatio…

Global-attractor conjecture. Based on numerical experiments, these equilibria are global attractors for solutions of the aggregation model.