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

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
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