4 problems

Let $\Omega$ be a planar domain of area $|\Omega|$, let $\Lambda_N^{(2)}(\Omega)$ denote the optimal sum of the first Dirichlet eigenvalues over $N$-partitions of $\Omega$, and let…

Let $\Omega\subset\mathbb R^2$ be a bounded open set, let $H_N(\Omega)$ be the optimal Cheeger value for an $N$-partition of $\Omega$, and suppose … Here $h(H)$ denotes the Cheeger…

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…

Caffarelli–Lin conjecture for Cheeger clusters. The asymptotic constant should satisfy