Calculus of Variations and Geometric Measure Theory

A. Braides - C. I. Zeppieri

A note on equi-integrability in dimension reduction problems

created by braidesa on 20 Oct 2005
modified on 11 Apr 2007


Published Paper

Inserted: 20 oct 2005
Last Updated: 11 apr 2007

Journal: Calc. Var. Partial Differential Equations
Volume: 29
Pages: 231-238
Year: 2007


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