8 problems

Let $\Omega\subset\mathbb{R}^2$ be a domain, and consider the analogous optimization problems obtained by replacing $\lambda(\Omega,\beta)$ with the first Dirichlet Laplacian eigen…

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…

A Dirichlet partition of a sphere divides it into subdomains, with the partition energy determined by the corresponding Dirichlet eigenvalues. The Y-partition is the three-partitio…

For every $k\in\mathbb{N}$, a $k$-cell is a connected region obtained as the union of $k$ unit-area regular hexagons from the hexagonal tiling of $\mathbb{R}^2$. Let $D$ be a $k$-c…

Let a partition of $\Bbb{R}^3$ into cells of equal volume be evaluated by its total perimeter. Kelvin's conjecture. Truncated octahedra may give an optimal partition for this probl…

Let $\Gamma$ be one of the surfaces considered in the source, and let $\mathcal{E}_p$ denote the partition energy with $p=\infty$, namely the maximum of the first eigenvalues of th…

Let $\Gamma$ be a smooth surface, and let an optimal partition of $\Gamma$ have boundaries meeting at junctions. The geodesic curvature of a boundary is its intrinsic curvature as…

Let $\Gamma$ be the sphere and consider the optimal partition problem with $m=4$ under the $p=\infty$ energy, which minimizes the largest first Dirichlet eigenvalue among the four…