6 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…

Honeycomb conjecture. The hexagonal lattice is the globally minimal solution among tessellations of the plane with equal cell size.

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…

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 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…