Rank-one convexity and lower semicontinuity of functionals

Question

While studying the direct method in the Calculus of Variations, the main two steps in determining the existence of a minimizer of an energy functional are determining compactness of a minimizing sequence and then verifying lower semicontinuity of the functional. For the second step, one usually needs convexity of the function one integrates over. However, this condition can be weakened if I understood correctly. I know for example that quasiconvexity is enough for LSC. Is rank-one convexity also enough?

For reference, the definition of rank-one convexity I have is:

A function $f: \mathbb{R}^{N \times n} \longrightarrow \mathbb{R} \cup \{\infty\}$ is called rank-one convex iff $$f(\lambda \xi_1 + (1-\lambda)\xi_2) \leq \lambda f(\xi_1) + (1-\lambda) f(\xi_2)$$ for all $\lambda \in (0,1)$ and all $\xi_1, \xi_2 \in \mathbb{R}^{N \times n}$ with $rk(\xi_1 - \xi_2) \leq 1$.

Answer 1

Morrey showed that quasiconvexity is equivalent to lower semicontinuity, i.e. it is necessary and sufficient, in the '50s.

Quasiconvexity implies rank-one convexity. This was also known to Morrey and is not hard to check.

Sverak showed in '92 that there exist functions which are rank-one convex but not quasiconvex. In particular, this means rank-one convexity does not imply lower semicontinuity.

There is a huge literature on this topic; here are some themes covered:

• What conditions do imply quasiconvexity but are easier to check (quasiconvexity is obnoxious to check)?
• Does rank-one convexity, or other conditions, imply quasiconvexity under some further structural assumptions on $f$?
• If one minimizes over some restricted class of functions instead, what are the necessary and sufficient conditions for lower semicontinuity? For example, what if $\nabla u$ is restricted to some small number of possible values, or has some special symmetries?
• Can one quantify the manner in which lower semicontinuity fails?