Quasiconvexity and its role in the Calculus of Variations

Oxford Mathematician Andre Guerra talks about quasiconvexity and its role in the Calculus of Variations:

"Most physical systems can be described through a Lagrangian, which is a function $f\colon \mathbb{R}^{m\times n}\to \mathbb{R}$ that can be thought of as some kind of 'energy.' For our purposes, the Lagrangian induces a functional $\mathcal F$ by $$\mathcal F[u]\equiv \int_\Omega f(\mbox{D} u(x))\mbox{ d}x,$$ where $u\colon \Omega\subset\mathbb R^n\to \mathbb R^m$ is a vector field. Physical configurations of the system correspond to those $u$ which minimise $\mathcal F$ or, more generally, are its critical points. Familiar examples of Lagrangians are the elastic energy of a rubber band, the electrostatic energy of a battery and the gravitational potential energy of an object. The string in a spider web, for instance, attains a configuration that minimizes its gravitational energy.

The only systematic approach to solve minimisation problems is to use the so-called Direct Method: one takes a sequence of fields $u_j$ such that $$\lim_{j\to \infty}\mathcal{F}[u_j]=\inf \mathcal F$$ and one hopes that $u_j$ converges in some sense to a minimiser $u$ of $\mathcal F$. In the 1950s Morrey recognised that the Lagrangians for which the Direct Method works are those possessing a weak type of convexity, known as quasiconvexity, and which reads as $$f(A)\leq \frac{1}{|\Omega|} \int_\Omega f(A+\mbox{D}u) \mbox{ d}x, \mbox{ for all } A\in \mathbb R^{m\times n} \mbox{ and } \varphi \in C^\infty_c(\Omega,\mathbb R^m).$$

In general, critical points of physical systems are expected to be somewhat smooth and so we are led to consider conditions on $f$ ensuring this smoothness. The most natural condition in this direction is known as rank-one convexity, and all quasiconvex Lagrangians are rank-one convex. It is important to keep in mind that when the Lagrangian acts on scalar fields, so that $m=1$, both quasiconvexity and rank-one convexity reduce to the usual notion of convexity, although this is not so whenever $m>1$.

Although 70 years have passed since Morrey's seminal work, quasiconvexity remains very poorly understood, despite formidable efforts from many mathematicians: there are explicit examples of fourth-order polynomials for which we cannot prove nor disprove quasiconvexity! On the other hand, rank-one convexity is an easier-to-verify condition. One of the most challenging open problems in the Calculus of Variations is to decide whether quasiconvexity and rank-one convexity are different. In 1992, Šverák found, for $m> 2$, a remarkable example of a rank-one convex Lagrangian which is not quasiconvex. In particular, the two-dimensional case remains open and this is known in the literature as Morrey's problem.

Morrey's problem asks whether two very large classes of functions are the same, making it a daunting question to answer. Recently I proved that it is enough to solve this problem in a strictly smaller class of functions, the so-called extremal functions, which should be better behaved than the typical rank-one convex function and thus easier to study. Though the set of extremal functions is smaller than the entire class of rank-one convex functions, there is still no satisfying way of characterizing them. To try to gain some intuition, I am interested in obtaining examples of such functions. I have found several, whose extremality had been conjectured by Šverák already in 1992, on my own. Currently, together with Daniel Faraco from Universidad Autónoma de Madrid, I am trying to produce more examples."