×

A note on quasiconvexity and rank-one convexity for \(2\times 2\) matrices. (English) Zbl 0918.49012

The problem of finding the links between the notions of quasiconvexity and rank-one convexity received a lot of attention in the literature, since the Morrey paper of 1952 [Ch. B. Morrey jun., Pac. J. Math. 2, 25-53 (1952; Zbl 0046.10803)]. The interest in the notion of quasiconvexity consists in the fact that for a function \(f: \mathbb{R}^{m\times N}\to \mathbb{R}\) the lower-semicontinuity property \[ \int_\Omega f(\nabla u) dx\leq \liminf_{n\to+\infty} \int_\Omega f(\nabla u_n)dx,\quad \forall u_n\to u\in w^* W^{1,\infty}(\Omega; \mathbb{R}^m) \] turns out to be equivalent to the quasiconvexity of \(f\). On the other hand, due to the nonlocal nature of quasiconvexity [J. Kristensen, “On the non-locality of quasiconvexity”, preprint (1996)], it can be very difficult to decide whether a given function is quasiconvex. On the contrary, rank-one convexity, which is always implied by quasiconvexity, is a notion much easier to check.
Only recently V. Šverák [Proc. R. Soc. Edinb., Sect. A 120, No. 1/2, 185-189 (1992; Zbl 0777.49015)] proved that the two notions do not coincide, by constructing a counterexample which works for mappings on \(\mathbb{R}^{m\times N}\) with \(N\geq 2\) and \(m\geq 3\). The problem still remains open for \(N= m= 2\). The aim of the present paper is to investigate on this case, by presenting some ideas which could lead to a counterexample even for \(N= m= 2\).
Reviewer: G.Buttazzo (Pisa)

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
PDFBibTeX XMLCite
Full Text: EuDML