2 problems

Sharp four-dimensional PES threshold. The free energy $\mathcal F_{\mathrm{PES}}$ should be bounded below on $\mathcal A_M(\Omega)$ for $M<M_*$, should lose compactness at $M=M_*$,…

Let $N=4$ and let $\Omega$ be $\mathbb{R}^{4}$, $\mathbb{T}^{4}$, or a bounded $C^{2}$ Neumann domain. Let $\mathcal{A}_M(\Omega)$ denote the admissible class of densities of mass…