6 problems

Morrey's conjecture. Every rank-one convex function $\mathbf E:\mathbb R^{2\times2}\rightarrow\mathbb R$ is quasiconvex.

Let $K$ be a finite subset of $cmathbb{R}^{n\times m}$, and let $\operatorname{conv}_\mathrm{rc}K$ denote its rank-one convex hull, namely the set of matrices satisfying…

Let $B_p\colon \mathbb{R}^{2\times 2}\to\mathbb{R}$ be the Burkholder integrand, defined by … where $\lvert A\rvert$ denotes the operator norm. Iwaniec's conjecture. The integrand…

Let $W\colon \operatorname{GL}^+(2)\to\mathbb{R}\cup\{+\infty\}$ be an objective, isotropic and rank-one convex function. For each $c\in\mathbb{R}$, define its sublevel set by … As…

Šverák's extremality conjecture. The integrands $\det^+$ and $\det^-$ are extremal in $\mathbb{R}^{n\times n}$, and each $F_k$ is extremal in…

Rank-one convexity region conjecture. For $n=2,3$, the energy $F\mapsto\frac{\mu}{k}e^{k\|\operatorname{dev}_n\log U\|^2}$ with $k>\frac{3}{16}$ is rank-one convex on a set contain…