7 problems

Let $\varOmega$ be a convex body in $\mathbb{R}^m$, and for $\alpha>0$ let its $r^{\alpha-m}$-center be a point where the extreme value of the potential … is attained, with the rel…

Let $n\geq 2$ and $0<\alpha<n$, and let $m_{\operatorname{ball}}$, $m_\mathrm{ex}$, and $m_\mathrm{crit}$ denote respectively the critical mass below which balls minimize the liqui…

Lorentz-Morrey Riesz-potential conjecture. If $p_1$ and $q_1$ satisfy

Burchard–Guo stability conjecture. There is a positive constant $C_\lambda$, depending on $n$, such that

Let $d\geqslant 1$ and $-2\leqslant s<d$. Consider the Jellium model in $d$ dimensions with Riesz interaction and define its lowest energy in a background region $\Omega_N$ of volu…

Let $\rho\colon\mathbb{R}^n\to[0,1]$ be a measurable function with compact support. For $0\leq\alpha<n$, define its Riesz potential by … where $d_\omega x=\frac{1}{\omega_n}\,dx$ a…

Let $\varOmega$ be a convex body in $\mathbb{R}^m$. For $\alpha>0$, define … and call a point where this potential attains its relevant extreme value an $r^{\alpha-m}$-center. Uniq…