Several versions of the compensated compactness principle

The convergence of the product of a solenoidal vector wε and a gradient ∇uε in L1(Ω) (where Ω is a region in Rd) is investigated in the case when the factors converge weakly in the spaces Lγ(Ω)d and Lα(Ω)d, respectively, with 1/γ+1/α>1, which means that the main assumption of the classical div-curl lemma fails. Nevertheless, the same convergence (in the sense of distributions in Ω) as in the framework of the div-curl lemma, survives under certain additional assumptions. The new versions of the compensated compactness principle proved in the paper can be used in homogenization and in the theory of G-convergence of monotone operators with non-standard coercivity and growth properties, for instance, some degenerate operators. Bibliography: 20 titles.