Morrey’s Problem and a Conjecture of Šverák

Quasiconvexity and rank-one convexity play a fundamental role in the vectorial Calculus of Variations; Morrey’s problem is to decide whether these notions coincide. In this talk we will see that to solve this problem it is sufficient to check whether all extremal rank-one convex functions are quasiconvex. We will also identify a class of such extremal functions, thus proving a conjecture made by Šverák in 1992.