33 problems

Consider partitions of the sphere into three cells, with the objective given by the sum of their first Laplace--Beltrami eigenvalues. Let the $Y$ partition be the partition consist…

Genericity conjecture. Triple intersection points are conjectured to be generic for large $k$ in spectral partitions.

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 $\Gamma$ be a planar domain and consider partitions of $\Gamma$ into $m$ parts minimizing the optimal partition energy. Hexagonal asymptotics conjecture. For large $m$, the opt…

Topology conjecture. Locally minimal $k$-partitions $P_1$ and $P_2$ with

Transition-threshold conjecture. For odd $k$, the threshold $b_k$ is given by the upper bound

Let $\Omega$ be a bounded planar domain, and let the second eigenfunction of the Dirichlet Laplacian have two nodal domains, which form a two-partition of $\Omega$. Energy suboptim…

Let a 5-partition be a partition of the domain into five cells, and consider the middle configuration described in the source, which has four odd critical points. Minimality conjec…

Hexagonal conjecture.

Loose-versus-non-exhaustive partition conjecture. The infimum over non-exhaustive rigid $k$-partitions satisfies

Let $\mathcal{G}$ be a metric graph and let $k\geq1$. A rigid $k$-partition is a partition of $\mathcal{G}$ of the type considered in the paper, and $\mathcal{P}$ is called interna…

Let $\mathcal{G}$ be a finite, compact, connected metric graph. An exhaustive rigid $2$-partition of $\mathcal{G}$ is a partition ${\mathcal P}_2=\{\mathcal{G}_1,\mathcal{G}_2\}$ w…

Consider partitions of the plane into cells and the sum of the first Dirichlet eigenvalues of the cells. Caffarelli--Lin conjecture. The honeycomb partition asymptotically minimize…

Consider partitions of the plane into cells, and for each cell take the fundamental, or first Dirichlet, eigenvalue. Hexagonal tiling conjecture. The hexagonal tiling of the plane…

Let $\Omega$ be a domain, let $k$ be a positive integer, let $\mathfrak L_{k,p}(\Omega)$ be the optimal $p$-norm energy of a $k$-partition, and let $p_{\infty}(\Omega,k)$ denote th…

Let $\Omega$ be a disk, a square, or an equilateral triangle, let $k$ be the number of cells, and let $\mathfrak L_{k,p}(\Omega)$ denote the optimal $p$-norm energy of a $k$-partit…

Let $\Omega$ be a disk, and consider partitions of $\Omega$ into subdomains. The “Mercedes partition” is the partition into three congruent sectors meeting at the center, with each…

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…

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 $(\phi_n)$ be a sequence of Laplace eigenfunctions, and let $\mu(\phi_n)$ denote the number of nodal domains of $\phi_n$. Let $Sq_1$ be the square of area $1$, with first eigen…

Hexagonal conjecture for Pleijel.

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 $\Omega$ be a simply connected domain, let $\mathcal D_k$ be a minimal $k$-partition of $\Omega$, let $N(\mathcal D_k)$ be its associated partition graph, and let…

Let $\Omega$ be a bounded planar domain, let $A(\Omega)$ denote its area, let $\mathfrak L_k(\Omega)$ be the minimal spectral energy among $k$-partitions of $\Omega$, and let…

Minimal-energy equality conjecture. The three inequalities in Proposition 1 are equalities.