\(\mathcal A\)-quasiconvexity, lower semicontinuity, and Young measures. (English) Zbl 0940.49014

Fonseca, Irene; Müller, Stefan

SIAM J. Math. Anal. 30, No. 6, 1355-1390 (1999).

The paper is devoted to the study of the sequential lower semicontinuity of integral functionals \[ (u,v)\mapsto\int_\Omega f(x,u(x),v(x)) dx \] with respect to strong convergence (or convergence in measure) in \(u\) and weak convergence in \(v\). Here \(\Omega\subset{\mathbb R}^n\) is a bounded open set and \((u,v):\Omega\to{\mathbb R}^m\times{\mathbb R}^d\). The variable \(v\) is supposed to satisfy (only asymptotically, in some cases) the differential constraint \({\mathcal A}v=0\), where \[ {\mathcal A}v:=\sum_{i=1}^n A^{(i)}{\partial v\over\partial x_i} \] and \(A^{(i)}:{\mathbb R}^d\to {\mathbb R}^l\) are linear maps. This general setting obviously generalizes the case when the functions \(v\) are gradients (in this case \({\mathcal A}\) is the curl operator). And indeed, the classical Morrey quasiconvexity becomes (here \(Q=(0,1)^n\)) \[ f(x,u,v)\leq\int_Q f(x,u,v+\phi(y)) dy \quad\forall Q\text{-periodic }\phi\in C^\infty(Q,{\mathbb R}^d) \text{s.t.} \int_Q w dy=0 \] in this more general setting. Under the Murat constant rank condition \[ \exists r \text{such that \(\text{rank }\Biggl(\sum_{i=1}^n A^{(i)}w\Biggr)=r\) for any \(w\in {\mathbb S}^{n-1}\)} \] basically all results of the classical theory of quasiconvexity are recovered: characterization of \({\mathcal A}\)-quasiconvexity as a necessary and sufficient condition, characterization of Young measures and their duality with \({\mathcal A}\)-quasiconvex functions. The main technical novelty of the paper is the use of projection operators on the kernel of \({\mathcal A}\) based on discrete Fourier multipliers. These projection operators are necessary because, unlike the curl-free case, there is no natural way to generate potentials associated to \(v\).

Reviewer: L.Ambrosio (Pisa)

MSC:

49J45

Methods involving semicontinuity and convergence; relaxation

Keywords:

\({\mathcal A}\)-quasiconvexity; equi-integrability; Young measures; lower semicontinuity; integral functionals; differential constraint