Rank-one convexity does not imply quasiconvexity

In this talk, we discuss Sverak's celebrated example of a function which is rank-one convex but not quasiconvex. We consider variational integrals of the type $$\displaystyle\int_{\Omega}f(\nabla u(x))dx$$ defined for sufficiently regular functions $u:\Omega\to\mathbb{R}^m$, where $\Omega$ is a bounded open subset of $\mathbb{R}^n$. The class of quasiconvex functions is indepedent of $\Omega$ and it plays a fundamental role in problems concerning stability of sets of solutions of certain nonlinear systems under weak convergence and the quasiconvexity condition is important in results regarding partial regularity of minimizers of the integral. It was a long standing conjecture whether rank-one convexity implies quasiconvexity. It turns out that for $n\geq 2$ and $m\geq 3$ this implication is not true due the Sverak's example. The case $n\geq 2$, $m=2$ is still open. It is worth mentioning the fact that even the case $m=n=2$ would have far-reaching consequences in calculus of variations