12 problems

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$, and let $\mathfrak L_k(\Omega)$ denote the infimum of the maximal first Dirichlet eigenvalue over all $k$-partitions of $\Omega$…

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…

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 $(\phi_n)$ 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…

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

Let $\Gamma$ be the sphere and consider the problem of minimizing the sum of the first Dirichlet eigenvalues over partitions of $\Gamma$ into three parts. A Y-partition is the part…

Hexagonal boundary-length conjecture. The normalized boundary length satisfies