×

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
PDF BibTeX XML Cite
Full Text: DOI

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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.