Calculus of Variations and Geometric Measure Theory

A. Braides - M. Briane

Homogenization of non-linear variational problems with thin low-conducting layers

created by braidesa on 10 May 2005
modified on 02 Feb 2008


Published Paper

Inserted: 10 may 2005
Last Updated: 2 feb 2008

Journal: Appl. Math. Optim.
Volume: 55
Pages: 1-29
Year: 2007


This paper deals with the homogenization of a sequence of non-linear conductivity energies in a bounded open set $\Omega$ of $*R*^d$, for $d\geq 3$. The energy density is of the same order as $a_ \varepsilon({x\over \varepsilon})\,
^p$, where $ \varepsilon\to 0$, $a_ \varepsilon$ is periodic, $u$ is vector-valued function in $W^{1,p}(\Omega;*R*^m)$ and $p>1$. The conductivity $a_ \varepsilon$ is equal to $1$ in the ``hard" phases composed by $N\geq 2$ two by two disjoint-closure periodic sets while $a_ \varepsilon$ tends uniformly to $0$ in the ``soft" phases composed by periodic thin layers which separate the hard phases. We prove that the limit energy, according to $\Gamma$-convergence, is a multi-phase functional equal to the sum of the homogenized energies (of order $1$) induced by the hard phases plus an interaction energy (of order $0$) due to the soft phases. The number of limit phases is less than or equal to $N$ and is obtained by evaluating the $\Gamma$-limit of the rescaled energy of density $ \varepsilon^{-p}\,a_ \varepsilon(y)\,
^p$ in the torus. Therefore, the homogenization result is achieved by a double $\Ga$-convergence procedure since the cell problem depends on $ \varepsilon$.

Keywords: Homogenization, double porosity, multi-phase limits