8 problems

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

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…

De Giorgi's finite-traces conjecture. For every $x\in\Omega$, the set $E(x)$ has finite cardinality $\alpha(x)\leq N+1$.

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

Pólya's conjecture. The optimal profile in three dimensions is a prolate spheroid with sharp ends of angle $120$ degrees.

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

Let $I$ be the energy functional on the domain $4\Omega_\delta4$, with boundary conditions 44 and periodicity in $x_1,x_2$. For $4\delta>04$, let … be the two smooth minimizers des…

Let $(u,K)$ be a reduced global minimizer in $\mathbb R^3$, meaning a Mumford–Shah global minimizer whose singular set $K$ is reduced, and suppose that $K$ is contained in a plane…