6 problems

Let $\Omega$ be an open subset of $\mathbb{R}^2$. A function $u$ on $\Omega$ satisfying … defines a minimal graph $G=\{(x,u(x))\mid x\in\Omega\}$ in $\mathbb{R}^3$. It is supported…

Let $u$ be a solution of an anisotropic minimal hypersurface equation, with $B_r$ denoting the ball of radius $r$ and $abla u$ its gradient. Mooney–Yang's conjecture. If … for som…

Let $\gamma_1$ and $\gamma_2$ be curves on two distinct parallel planes, and let $\Sigma$ be a minimal surface spanning them. Assume that the projection of $\Sigma$ onto the plane…

Let $E$subset$\mathbb R^2$ be a planar nonlocal minimal graph as in the surrounding regularity results, and let the Hölder exponent be the one obtained in Theorem. Optimality conje…

Vertex-bound conjecture. For all $d \geq 3$, the number of vertices of a $d$-regular minimal graph attached to $n$ points is

Existence conjecture. There exist minimal graphs attached to some fixed finite set of points in the plane with arbitrarily large numbers of vertices.