Morrey quasiconvexity

Remarks

If the inequality holds for one bounded open set $D$ with $\abs{\partial D} = 0$, it holds for all of them; often $D$ is the unit cube.

Example: every convex $f$ is quasiconvex. Since $\varphi = 0$ on $\partial D$, the average of $\nabla \varphi$ over $D$ is $0$, and Jensen's inequality gives the inequality.

Non-example: $f(\xi) = -\abs{\xi}^2$ is not quasiconvex. For $\varphi \ne 0$, the cross term $2 \int_D \xi : \nabla \varphi \, dx$ vanishes, so the average of $f(\xi + \nabla \varphi)$ is $-\abs{\xi}^2 - \frac{1}{\abs{D}} \int_D \abs{\nabla \varphi}^2 \, dx < f(\xi)$.

Every quasiconvex function is rank-one convex. For $m = 1$ or $n = 1$, quasiconvexity is ordinary convexity. For $m \ge 3$ and $n \ge 2$, Šverák gave a rank-one convex function that is not quasiconvex; the case $m = 2$ is open.