7 problems

Let $\Omega\subset\mathbb{R}^n$ be a domain of unit area, and let $l_m^1(\Omega)$ denote the normalized sum of the first Dirichlet eigenvalues of an $l^1$-minimal $m$-partition, ……

For each $k\in\mathbb N$, let $\mathcal D_k$ be a minimal $k$-partition, let $N(\mathcal D_k)$ be its associated adjacency or boundary network, and let $X^{odd}(N(\mathcal D_k))$ d…

Let $\Omega$ be a bounded planar domain of area $A(\Omega)$, let $\mathfrak L_k(\Omega)$ denote the minimal energy of a $k$-partition of $\Omega$, and let $Hexa_1$ be the regular h…

Let $\mathcal D_k$ be a minimal $k$-partition in the asymptotic regime of the hexagonal conjecture. A cell is a hexagonal cell or a pentagonal cell according to its polygonal combi…

Let $\Omega$ be a bounded planar domain, and let $\mathfrak L_k(\Omega)$ be the energy of a minimal $k$-partition of $\Omega$. Let $\hexagon$ be the regular hexagon of area $1$, an…

Let $\Omega$ be a regular bounded open set in $\mathbb R^2$ or $\mathbb S^2$, let $\lambda_j(\Omega)$ be the $j$th Dirichlet eigenvalue of the Laplacian on $\Omega$, and let…

Let $\mathfrak L_k(\mathbb S^2)$ be the infimum of the maximal ground-state energy over spectral minimal $k$-partitions of the sphere, and let $\mathfrak L_{k,1}(\mathbb S^2)$ be t…