Multiscale convergence and reiterated homogenisation.(English)Zbl 0866.35017

The paper deals with homogenisation problems with several (more than one) lengthscales. The model problem is $-\text{div} (A_\varepsilon \nabla u_\varepsilon) =f \quad \text{in } \Omega;\quad u_\varepsilon=0 \quad \text{on } \partial\Omega \tag{*}$ where $$\Omega\subset\mathbb{R}^n$$ is a given open set and $$A_\varepsilon(x) =A(x, x/\varepsilon_1, \ldots, x/\varepsilon_k),$$ with $$A(x,y_1, \dots, y_k)$$ periodic with respect to all variables $$y_i$$ and $$\varepsilon= (\varepsilon_1, \dots, \varepsilon_k)$$ infinitesimal. Assuming that $$\varepsilon_{i+1} =o(\varepsilon_i)$$ for $$1\leq i<k$$, the main result of the paper is the weak $$H^1_0$$ convergence of solutions of (*) to the solution of $-\text{div} (A^* \nabla u)= f\quad \text{in } \Omega; \quad u=0 \quad \text{on } \partial \Omega$ where the homogenized matrix $$A^*$$ can be computed by a repeated (one scale) homogenisation process. Moreover, suitable correctors yield strong $$H^1$$ convergence. The main mathematical tool in the proof is a new concept of “multiscale convergence” which has natural compactness properties and enables a description of the limit problem.
Reviewer: L.Ambrosio (Pavia)

MSC:

 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35J25 Boundary value problems for second-order elliptic equations

Keywords:

multiscale analysis
Full Text:

References:

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