Γ‐convergence of stratified media with measure‐valued limits
Björn Gustafsson, Bernard Heron, Jacqueline Mossino · Asymptotic Analysis · 2000
We consider energy functionals, or Dirichlet forms, [Formula: see text] for a class $\mathcal{G}$ of bounded domains $\varOmega\subset\mathbb{R}^N$ , with $\varepsilon >0$ a fine structure parameter and with symmetric conductivity matrices $A^\varepsilon=(a^\varepsilon_{ij})\in L_\mathrm{loc}^\infty(\mathbb{R})^{N\times N}$ which are functions only of the first coordinate $x_1$ and which are locally uniformly elliptic for each fixed $\varepsilon>0$ . We show that if the functions (of $x_1$ ) $b^\varepsilon_{11}={1}/{a^\varepsilon_{11}}$ , $b^\varepsilon_{1j}={a^\varepsilon_{1j}}/{a^\varepsilon_{11}}\ (j\geq 2)$ , $b^\varepsilon_{ij}= a^\varepsilon_{ij} - {a^\varepsilon_{i1}a^\varepsilon_{1j}}/{a^\varepsilon_{11}} \ (i, j\geq 2)$ converge weakly* as measures towards corresponding limit measures $b_{ij}$ as $\varepsilon\to 0$ , if the $(1,1)$ ‐coefficient $m_{11}^\varepsilon$ of $(A^\varepsilon)^{-1}$ is bounded in $L_\mathrm{loc}^1(\mathbb{R})$ and if none of its weak* cluster measures has atoms in common with $b_{ii}$ , $i\geq 2$ , then the family $J^\varepsilon=\{J_\varOmega^\varepsilon \}_{\varOmega\in \mathcal{G}}$ $\varGamma$ ‐converges in a local sense towards a naturally defined limit family $J=\{J_\varOmega \}_{\varOmega\in \mathcal{G}}$ as $\varepsilon\to 0$ . An alternative way of formulating the conclusion is to say that the energy densities $(A^\varepsilon abla u, abla u)$ $\varGamma$ ‐converge in a distributional sense towards the corresponding limit density. Writing $J_\varOmega^\varepsilon$ in terms of $B^\varepsilon=(b_{ij}^\varepsilon)$ it becomes [Formula: see text] and the definition of $J_\varOmega$ and the limit density $(A abla u, abla u)$ is obtained by properly replacing the $b^\varepsilon_{ij}\in L_\mathrm{loc}^\infty (\mathbb{R})$ by the limit measures $b_{ij}$ and making sense to everything for $u$ in a certain linear subspace of $L_\mathrm{loc}^2(\mathbb{R}^N)$ .