Lower-semicontinuity of variational integrals and compensated compactness. (English) Zbl 0852.49010

Šverák, Vladimír

Chatterji, S. D. (ed.), Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. II. Basel: Birkhäuser. 1153-1158 (1995).

The paper provides a discussion on two subjects: (I) Lower semicontinuity and quasiconvexity, (II) Quasiconvexity and compensated compactness.
In the first part the relationships between sequential weak lower semicontinuity of integral functionals of the form \(I(u)= \int_\Omega f(Du(x)) dx\) (\(\Omega\) being a bounded open subset of \(\mathbb{R}^n\), \(u: \Omega\to \mathbb{R}^m\), \(Du(x)\) denoting the gradient matrix of \(u\) at \(x\) and \(f\) being a given real function defined on the set \(M^{m\times n}\) of the \(m\times n\) real matrices) and quasiconvexity is described, then comparison results between quasiconvexity, convexity, polyconvexity and rank-one convexity are reported and some open problems are illustrated.
In the second part of the paper it is shown how quasiconvexity is related to compactness properties of sets of (approximate) solutions of the system \(Du\in K\), \(K\) being a closed subset of \(M^{m\times n}\). The approach followed is based on ideas from the theory of compensated compactness. Finally, several examples are discussed.
For the entire collection see [Zbl 0829.00015].

Reviewer: R.De Arcangelis (Napoli)

MSC:

49J45

Methods involving semicontinuity and convergence; relaxation

Keywords:

quasiconvexity; compensated compactness; lower semicontinuity; integral functionals