A Note on Equi-integrability in Dimension Reduction Problems

A very handy tool in the study of the asymptotic behavior of variational problems defined on Sobolev spaces is Fonseca, Müller and Pedregal’s equi-integrability Lemma [8] (see Theorem 2.1 below; see also earlier work by Acerbi and Fusco [2] and by Kristensen [11]), which allows to substitute a sequence (wj) with (∇wj) bounded in L by a sequence (zj) with (|∇zj | ) equi-integrable, such that the two sequences are equal except on a set of vanishing measure. In this way the asymptotic behavior of integral energies of p-growth involving ∇wj can be computed using ∇zj and thus avoiding to consider concentration effects. This method is very helpful for example in the computation of lower bounds for Γ-limits (see, e.g., [5]). In the framework of dimensional reduction, we encounter sequences of functions (wε) defined on cylindrical sets with some ‘thin dimension’ ε; e.g., in the physical three-dimensional case either thin films defined on some set of the type ω×(0, ε) (see, e.g., [10, 6]), or thin wires defined on εω × (0, 1) (see, e.g., [1, 9]), where ω is some two-dimensional bounded open set. In order to carry on some asymptotic analysis such functions are usually rescaled to an ε-independent reference configuration Ω (see Fig. 1), so that a new sequence (uε) is constructed, satisfying some ‘degenerate’ bounds of the form