*Published Paper*

**Inserted:** 20 oct 2005

**Last Updated:** 11 apr 2007

**Journal:** Calc. Var. Partial Differential Equations

**Volume:** 29

**Pages:** 231-238

**Year:** 2007

**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'$x_\beta$) bounded in $L^p(\Omega;*R*^{m\times n})$ ($1<p<+\infty$) as a sum of a sequence $\left(\nabla_\alpha v_\varepsilon\big

\frac{1}{\varepsilon}\nabla_\beta v_\varepsilon\right)$ whose $p$-th power equi-integrable on $\Omega$ and a `rest' that converges
to zero in measure. In particular, for $k=1$ we recover a well-known result for thin films.

**Keywords:**
equi-integrability, thin structures

**Download:**