A note on equi-integrability in dimension reduction problems

Abstract

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’ ${\left(\nabla_\alpha u_\varepsilon\big|\frac{1}{\varepsilon}\nabla_\beta u_\varepsilon\right)}$ (where $\nabla_\beta$ is the gradient in the k-dimensional ‘thin variable’ x _β) bounded in ${L^p(\Omega;\mathbb{R}b^{m\times n})}$ (1 < p <  + ∞) as a sum of a sequence ${\left(\nabla_\alpha v_\varepsilon\big|\frac{1}{\varepsilon}\nabla_\beta v_\varepsilon\right)}$ 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).