4 problems

Let $N$ be the number of phases, let $\mathscr{G}_f$ denote an optimal free network, and let $\mathrm{Pol}_N$ denote the regular polygonal network associated with the $N$-phase con…

Let $u^\epsilon$ be a minimizer of the Allen–Cahn problem, let $\mathscr{I}$ be its diffuse interface, let $\hat{\mathscr{G}}$ be the associated optimal network, and let…

Optimal-network approximation conjecture. One may conjecture that

Let $n\geq 2$ be an integer, let $E\subset \mathbb{R}^n$ have boundary $\Sigma:=\partial E$ of class $C^2$, and let $\Omega\subset \mathbb{R}^n$ be open. For $\lambda>0$, define ……