Rank-One Convexity vs Quasiconvexity

Problem's Description

A function $f\colon \mathbb R^{n\times N}\to\mathbb R$ is called rank-one convex if

$$f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y),$$ for all $\lambda\in[0,1]$ and $x,y\in\mathbb R^{n\times N}$ such that $\mathrm{rank}(x-y)=1.$ A locally bounded and Borel measurable function $f\colon\mathbb R^{n\times N}\to\mathbb R$ is calles quasiconvex if for any $\xi\in\mathbb R^{n\times N},$ any $\varphi\in C_0^1([0,1]^n,\mathbb R^N)$ there holds: $$\int_{[0,1]^n} f(\xi+\nabla\varphi(x))dx\geq f(\xi).$$ It is know that quasiconvexity implies rank-one convexity. Sverak proved in 1992 that rank-one convexity does not imply quasicomnvexity in the case $n\geq 2$ and $N\geq 3.$ The problem whether those two are the same in the case $N=2$ and $n\geq 2$ is open (this is probably the central question of the calculus of variations and materials science).

References

1. Article Rank-one convexity does not imply quasi convexity

   • Vladimir Sverak

   Proceedings of the Royal Society of Edinburgh: Section A Mathematicsfulltext

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