Hostname: page-component-7bb8b95d7b-fmk2r Total loading time: 0 Render date: 2024-09-20T20:24:47.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem

Published online by Cambridge University Press:  15 September 2003

Dominique Blanchard  and
Antonio Gaudiello
Dominique Blanchard
Affiliation:
Université de Rouen, UMR 6085, 76821 Mont-Saint-Aignan Cedex, France, and Laboratoire d'Analyse Numérique, Université P. et M. Curie, Case Courrier 187, 75252 Paris Cedex 05, France; blanchar@ann.jussieu.fr.
Antonio Gaudiello
Affiliation:
Università degli Studi di Cassino, Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell'Informazione e Matematica Industriale, via G. di Biasio 43, 03043 Cassino (FR), Italy; gaudiell@unina.it.
Get access

Abstract

We investigate the asymptotic behaviour, as ε → 0, of a class of monotone nonlinear Neumann problems, with growth p-1 (p ∈]1, +∞[), on a bounded multidomain $\Omega_\varepsilon\subset \mathbb{R}^N$ (N ≥ 2). The multidomain ΩE is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness hE in the xN direction, as ε → 0. The second one is a “forest" of cylinders distributed with ε-periodicity in the first N - 1 directions on the upper side of the plate. Each cylinder has a small cross section of size ε and fixed height (for the case N=3, see the figure). We identify the limit problem, under the assumption: ${\lim_{\varepsilon\rightarrow 0} {\varepsilon^p\over h_\varepsilon}=0}$. After rescaling the equation, with respect to hE, on the plate, we prove that, in the limit domain corresponding to the “forest" of cylinders, the limit problem identifies with a diffusion operator with respect to xN, coupled with an algebraic system. Moreover, the limit solution is independent of xN in the rescaled plate and meets a Dirichlet transmission condition between the limit domain of the “forest" of cylinders and the upper boundary of the plate.


Keywords

Homogenizationoscillating boundariesmultidomainmonotone problem.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 449 - 460
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allaire, G., Homogenization and Two-Scale Convergence. SIAM J. Math Anal. 23 (1992) 1482-1518. CrossRef
Allaire, G. and Amar, M., Boundary Layer Tails in Periodic Homogenization. ESAIM: COCV 4 (1999) 209-243. CrossRef
Amirat, Y. and Bodart, O., Boundary Layer Correctors for the Solution of Laplace Equation in a Domain with Oscillating Boundary. J. Anal. Appl. 20 (2001) 929-940.
Ansini, N. and Braides, A., Homogenization of Oscillating Boundaries and Applications to Thin Films. J. Anal. Math. 83 (2001) 151-183. CrossRef
Blanchard, D., Carbone, L. and Gaudiello, A., Homogenization of a Monotone Problem in a Domain with Oscillating Boundary. ESAIM: M2AN 33 (1999) 1057-1070. CrossRef
Brizzi, R. and Chalot, J.P., Boundary Homogenization and Neumann Boundary Value Problem. Ricerche Mat. 46 (1997) 341-387.
Buttazzo, G. and Kohn, R.V., Reinforcement by a Thin Layer with Oscillating Thickness. Appl. Math. Optim. 16 (1987) 247-261. CrossRef
Chechkin, G.A., Friedman, A. and Piatniski, A.L., The Boundary Value Problem in a Domain with Very Rapidly Oscillating Boundary. J. Math. Anal. Appl. 231 (1999) 213-234. CrossRef
Ciarlet, P.G. and Destuynder, P., Justification, A of the Two-Dimensional Linear Plate Model. J. Mécanique 18 (1979) 315-344.
Cioranescu, D. and Saint Jean Paulin, J., Homogenization in Open Sets with Holes. J. Math. Anal. Appl. 71 (1979) 590-607. CrossRef
Corbo Esposito, A., Donato, P., Gaudiello, A. and Picard, C., Homogenization of the p-Laplacian in a Domain with Oscillating Boundary. Comm. Appl. Nonlinear Anal. 4 (1997) 1-23.
Gaudiello, A., Asymptotic Behaviour of non-Homogeneous Neumann Problems in Domains with Oscillating Boundary. Ricerche Mat. 43 (1994) 239-292.
Gaudiello, A., Homogenization of an Elliptic Transmission Problem. Adv. Math Sci. Appl. 5 (1995) 639-657.
Gaudiello, A., Gustafsson, B., Lefter, C. and Mossino, J., Asymptotic Analysis for Monotone Quasilinear Problems in Thin Multidomains. Differential Integral Equations 15 (2002) 623-640.
A. Gaudiello, R. Hadiji and C. Picard, Homogenization of the Ginzburg-Landau Equation in a Domain with Oscillating Boundary. Commun. Appl. Anal. (to appear).
Gaudiello, A., Monneau, R., Mossino, J., Murat, F. and Sili, A., On the Junction of Elastic Plates and Beams. C. R. Acad. Sci. Paris Sér. I 335 (2002) 717-722. CrossRef
H. Le Dret, Problèmes variationnels dans les multi-domaines : modélisation des jonctions et applications. Masson, Paris (1991).
J.L. Lions, Quelques méthodes de résolution de problèmes aux limites non linéaires. Dunod, Paris (1969).
T.A. Mel'nyk, Homogenization of the Poisson Equations in a Thick Periodic Junction. ZAA J. Anal. Appl. 18 (1999) 953-975.
Mel'nyk, T.A. and Nazarov, S.A., Asymptotics of the Neumann Spectral Problem Solution in a Domain of ``Thick Comb" Type. J. Math. Sci. 85 (1997) 2326-2346. CrossRef
Nguetseng, G., General Convergence Result, A for a Functional Related to the Theory of Homogenization. SIAM J. Math Anal. 20 (1989) 608-623. CrossRef
L. Tartar, Cours Peccot, Collège de France (March 1977). Partially written in F. Murat, H-Convergence, Séminaire d'analyse fonctionnelle et numérique de l'Université d'Alger (1977-78). English translation in Mathematical Modelling of Composite Materials, edited by A. Cherkaev and R.V. Kohn, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser-Verlag (1997) 21-44.