Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 449-460.

We investigate the asymptotic behaviour, as ε0, of a class of monotone nonlinear Neumann problems, with growth p-1 (p]1,+[), on a bounded multidomain Ω ε N (N2). The multidomain Ω ε is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness h ε in the x N direction, as ε0. 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 ε0 ε p h ε =0. After rescaling the equation, with respect to h ε , 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 x N , coupled with an algebraic system. Moreover, the limit solution is independent of x N 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.

DOI : 10.1051/cocv:2003022
Classification : 35B27, 35J60
Mots clés : homogenization, oscillating boundaries, multidomain, monotone problem
Blanchard, Dominique  ; Gaudiello, Antonio 1

1 Università di Cassino, Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale, via G. Di Biasio 43, 03043 Cassino (FR), Italy;
@article{COCV_2003__9__449_0,
     author = {Blanchard, Dominique and Gaudiello, Antonio},
     title = {Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {449--460},
     publisher = {EDP-Sciences},
     volume = {9},
     year = {2003},
     doi = {10.1051/cocv:2003022},
     mrnumber = {1998710},
     zbl = {1071.35012},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2003022/}
}
TY  - JOUR
AU  - Blanchard, Dominique
AU  - Gaudiello, Antonio
TI  - Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2003
SP  - 449
EP  - 460
VL  - 9
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2003022/
DO  - 10.1051/cocv:2003022
LA  - en
ID  - COCV_2003__9__449_0
ER  - 
%0 Journal Article
%A Blanchard, Dominique
%A Gaudiello, Antonio
%T Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2003
%P 449-460
%V 9
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2003022/
%R 10.1051/cocv:2003022
%G en
%F COCV_2003__9__449_0
Blanchard, Dominique; Gaudiello, Antonio. Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 449-460. doi : 10.1051/cocv:2003022. http://archive.numdam.org/articles/10.1051/cocv:2003022/

[1] G. Allaire, Homogenization and Two-Scale Convergence. SIAM J. Math Anal. 23 (1992) 1482-1518. | MR | Zbl

[2] G. Allaire and M. Amar, Boundary Layer Tails in Periodic Homogenization. ESAIM: COCV 4 (1999) 209-243. | Numdam | MR | Zbl

[3] Y. Amirat and O. Bodart, Boundary Layer Correctors for the Solution of Laplace Equation in a Domain with Oscillating Boundary. J. Anal. Appl. 20 (2001) 929-940. | MR | Zbl

[4] N. Ansini and A. Braides, Homogenization of Oscillating Boundaries and Applications to Thin Films. J. Anal. Math. 83 (2001) 151-183. | MR | Zbl

[5] D. Blanchard, L. Carbone and A. Gaudiello, Homogenization of a Monotone Problem in a Domain with Oscillating Boundary. ESAIM: M2AN 33 (1999) 1057-1070. | Numdam | MR | Zbl

[6] R. Brizzi and J.P. Chalot, Boundary Homogenization and Neumann Boundary Value Problem. Ricerche Mat. 46 (1997) 341-387. | MR | Zbl

[7] G. Buttazzo and R.V. Kohn, Reinforcement by a Thin Layer with Oscillating Thickness. Appl. Math. Optim. 16 (1987) 247-261. | MR

[8] G.A. Chechkin, A. Friedman and A.L. Piatniski, The Boundary Value Problem in a Domain with Very Rapidly Oscillating Boundary. J. Math. Anal. Appl. 231 (1999) 213-234. | MR | Zbl

[9] P.G. Ciarlet and P. Destuynder, A Justification of the Two-Dimensional Linear Plate Model. J. Mécanique 18 (1979) 315-344. | MR | Zbl

[10] D. Cioranescu and J. Saint Jean Paulin, Homogenization in Open Sets with Holes. J. Math. Anal. Appl. 71 (1979) 590-607. | MR | Zbl

[11] A. Corbo Esposito, P. Donato, A. Gaudiello and C. Picard, Homogenization of the p-Laplacian in a Domain with Oscillating Boundary. Comm. Appl. Nonlinear Anal. 4 (1997) 1-23. | MR | Zbl

[12] A. Gaudiello, Asymptotic Behaviour of non-Homogeneous Neumann Problems in Domains with Oscillating Boundary. Ricerche Mat. 43 (1994) 239-292. | MR | Zbl

[13] A. Gaudiello, Homogenization of an Elliptic Transmission Problem. Adv. Math Sci. Appl. 5 (1995) 639-657. | MR | Zbl

[14] A. Gaudiello, B. Gustafsson, C. Lefter and J. Mossino, Asymptotic Analysis for Monotone Quasilinear Problems in Thin Multidomains. Differential Integral Equations 15 (2002) 623-640. | MR | Zbl

[15] A. Gaudiello, R. Hadiji and C. Picard, Homogenization of the Ginzburg-Landau Equation in a Domain with Oscillating Boundary. Commun. Appl. Anal. (to appear). | Zbl

[16] A. Gaudiello, R. Monneau, J. Mossino, F. Murat and A. Sili, On the Junction of Elastic Plates and Beams. C. R. Acad. Sci. Paris Sér. I 335 (2002) 717-722. | MR | Zbl

[17] H. Le Dret, Problèmes variationnels dans les multi-domaines : modélisation des jonctions et applications. Masson, Paris (1991). | MR | Zbl

[18] J.L. Lions, Quelques méthodes de résolution de problèmes aux limites non linéaires. Dunod, Paris (1969). | MR | Zbl

[19] T.A. Mel'Nyk, Homogenization of the Poisson Equations in a Thick Periodic Junction. ZAA J. Anal. Appl. 18 (1999) 953-975. | Zbl

[20] T.A. Mel'Nyk and S.A. Nazarov, Asymptotics of the Neumann Spectral Problem Solution in a Domain of “Thick Comb” Type. J. Math. Sci. 85 (1997) 2326-2346. | Zbl

[21] G. Nguetseng, A General Convergence Result for a Functional Related to the Theory of Homogenization. SIAM J. Math Anal. 20 (1989) 608-623. | MR | Zbl

[22] 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. | Zbl

Cité par Sources :