7 problems

Mumford–Shah conjecture. If $(K,u)$ is a minimizer of $E_\lambda$, then $K$ has the structure described above.

Let $(u,K)$ be a minimizer of the Mumford–Shah functional in dimension $n=3$, and call $K$ its singular set. Three-dimensional Mumford–Shah singular-set conjecture. The singular se…

Let $u$ be the function on the plane defined in polar coordinates by … which is equivalently the imaginary part of $\sqrt{2z/\pi}$ with a cut along the negative real axis. Let…

Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with boundary of class $C^{1,\gamma}$, let $g \in C^{1,\gamma}(\partial\Omega)$, and let $(K,u)$ be a normalized minimizer of…

Uniqueness conjecture. The pair $(K,u)$ is either an elementary global minimizer or a cracktip.

De Giorgi's uniqueness conjecture. The function $\mathrm{Rsq}$ is the unique global minimizer in $\mathbb{R}^2$.

Cremers–Strekalovskiy conjecture. The functionals $\tilde{F}_{\varepsilon,g}$ approximate the Mumford–Shah functional.