Morrey quasiconvexity

Summary

A function $f$ on $m \times n$ matrices is quasiconvex in the sense of Morrey if perturbations with zero boundary values never help. That is, the average of $f(\xi + \nabla \varphi)$ is never below $f(\xi)$. Under suitable growth conditions on $f$, it is exactly the condition under which $\int_\Omega f(\nabla u) \, dx$ is weakly lower semicontinuous on Sobolev spaces of vector-valued maps. It lies between polyconvexity and rank-one convexity. It coincides with convexity when $m = 1$ or $n = 1$. It is a different notion from the quasiconvex functions of optimisation, which are defined by convex sublevel sets.