A note on equi-integrability in dimension reduction problems

In the framework of the asymptotic analysis of thin structures, we prove that, up to an extraction, it is possible to decompose a sequence of 'scaled gradients' (∇αuε|1\ ε∇βuε) (where ∇ β is the gradient in the k-dimensional 'thin variable' x β) bounded in Lp(ΩRm×n)(1 < p < + ∞) as a sum of a sequence (∇αv ε|1\ε∇βvε) whose p-th power is equi-integrable on Ω and a 'rest' that converges to zero in measure. In particular, for k = 1 we recover a well-known result for thin films by Bocea and Fonseca (ESAIM: COCV 7:443-470; 2002). © Springer-Verlag 2007.