## 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
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### References:

 [1] Meyers, Ann. Scuola Norm. Sup. Pisa 17 pp 189– (1963) [2] Tartar, Topics in Nonlinear Analysis (1978) [3] Bernardi, Contributions à l’analyse numérique de problèmes non linéaires (1986) [4] Bensoussan, Stefan Banach Symposium on Probabilities 5 pp 15– (1979) [5] Bensoussan, Asymptotic Analysis for Periodic Structures (1978) [6] Bakhvalov, Homogenization: Averaging Processes in Periodic Media 36 (1989) [7] Amrouche, Czechoslovak Math. J 44 pp 119– (1994) [8] Murat, Progress in nonlinear differential equations and their equations 1 [9] Allaire, Asymptotic Anal 7 pp 81– (1993) [10] DOI: 10.1137/0523084 · Zbl 0770.35005 [11] DOI: 10.1016/0362-546X(92)90015-7 · Zbl 0779.35011 [12] Donate, Alcune osservazioni sulla convergenza debolc di funzioni non uniformemente oscillanti 32 (1983) [13] DOI: 10.1007/978-1-4612-0327-8 [14] DOI: 10.1016/0022-247X(79)90211-7 · Zbl 0427.35073 [15] DOI: 10.1007/BF00938526 · Zbl 0617.49005 [16] Briane, J. Math. Pures Appl 73 pp 47– (1994) [17] Briane, Adv. Math. Sci. Appl 4 pp 357– (1994) [18] Bogovski, Soviet Math. Dokl 20 pp 1094– (1979) [19] Boccardo, Atti del Convengno su Sludio dei problemi dell ’Analisi Funzionale’, Bressanone 79 Sett, 1981 pp 13– (1982) [20] DOI: 10.1070/RM1979v034n05ABEH003898 · Zbl 0445.35096 [21] Spagnolo, Ann. Scuola Norm. Sup. Pisa 22 pp 571– (1968) [22] Sanchez-Palencia, Non-homogeneous media and vibration theory 127 (1980) · Zbl 0432.70002 [23] Papanicolaou, Random fields, Esztergom {Hungary 1979) 27 (1981)} [24] DOI: 10.1137/0521078 · Zbl 0723.73011 [25] Kozlov, Soviet Math. Dokl 19 pp 950– (1978)
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