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

• [1] Università di Cassino, Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale, via G. Di Biasio 43, 03043 Cassino (FR), Italy;
• Volume: 9, page 449-460
• ISSN: 1292-8119

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Abstract

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We investigate the asymptotic behaviour, as $\epsilon \to 0$, of a class of monotone nonlinear Neumann problems, with growth $p-1$ ($p\in \right]1,+\infty \left[$), on a bounded multidomain ${\Omega }_{\epsilon }\subset {ℝ}^{N}$$\left(N\ge 2\right)$. The multidomain ${\Omega }_{\epsilon }$ is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness ${h}_{\epsilon }$ in the ${x}_{N}$ direction, as $\epsilon \to 0$. The second one is a “forest” of cylinders distributed with $\epsilon$-periodicity in the first $N-1$ directions on the upper side of the plate. Each cylinder has a small cross section of size $\epsilon$ and fixed height (for the case $N=3$, see the figure). We identify the limit problem, under the assumption: ${lim}_{\epsilon \to 0}\frac{{\epsilon }^{p}}{{h}_{\epsilon }}=0$. After rescaling the equation, with respect to ${h}_{\epsilon }$, 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.

How to cite

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Blanchard, Dominique, and Gaudiello, Antonio. "Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 449-460. <http://eudml.org/doc/246069>.

@article{Blanchard2003,
abstract = {We investigate the asymptotic behaviour, as $\varepsilon \rightarrow 0$, of a class of monotone nonlinear Neumann problems, with growth $p-1$ ($p\in ]1,+\infty [$), on a bounded multidomain $\Omega _\varepsilon \subset \mathbb \{R\}^N$$(N\ge 2). The multidomain \Omega _\varepsilon is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness h_\varepsilon in the x_N direction, as \varepsilon \rightarrow 0. The second one is a “forest” of cylinders distributed with \varepsilon -periodicity in the first N-1 directions on the upper side of the plate. Each cylinder has a small cross section of size \varepsilon 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 h_\varepsilon , 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.}, affiliation = {Università di Cassino, Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale, via G. Di Biasio 43, 03043 Cassino (FR), Italy;}, author = {Blanchard, Dominique, Gaudiello, Antonio}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, keywords = {homogenization; oscillating boundaries; multidomain; monotone problem; monotone nonlinear Neumann problems; Dirichlet transmission condition}, language = {eng}, pages = {449-460}, publisher = {EDP-Sciences}, title = {Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem}, url = {http://eudml.org/doc/246069}, volume = {9}, year = {2003}, } 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 PB - EDP-Sciences VL - 9 SP - 449 EP - 460 AB - We investigate the asymptotic behaviour, as \varepsilon \rightarrow 0, of a class of monotone nonlinear Neumann problems, with growth p-1 (p\in ]1,+\infty [), on a bounded multidomain \Omega _\varepsilon \subset \mathbb {R}^N$$(N\ge 2)$. The multidomain $\Omega _\varepsilon$ is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness $h_\varepsilon$ in the $x_N$ direction, as $\varepsilon \rightarrow 0$. The second one is a “forest” of cylinders distributed with $\varepsilon$-periodicity in the first $N-1$ directions on the upper side of the plate. Each cylinder has a small cross section of size $\varepsilon$ 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 $h_\varepsilon$, 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.
LA - eng
KW - homogenization; oscillating boundaries; multidomain; monotone problem; monotone nonlinear Neumann problems; Dirichlet transmission condition
UR - http://eudml.org/doc/246069
ER -

References

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1. [1] G. Allaire, Homogenization and Two-Scale Convergence. SIAM J. Math Anal. 23 (1992) 1482-1518. Zbl0770.35005MR1185639
2. [2] G. Allaire and M. Amar, Boundary Layer Tails in Periodic Homogenization. ESAIM: COCV 4 (1999) 209-243. Zbl0922.35014MR1696289
3. [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. Zbl1005.35028MR1884513
4. [4] N. Ansini and A. Braides, Homogenization of Oscillating Boundaries and Applications to Thin Films. J. Anal. Math. 83 (2001) 151-183. Zbl0983.49008MR1828490
5. [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. Zbl0942.35071MR1726724
6. [6] R. Brizzi and J.P. Chalot, Boundary Homogenization and Neumann Boundary Value Problem. Ricerche Mat. 46 (1997) 341-387. Zbl0959.35014MR1760382
7. [7] G. Buttazzo and R.V. Kohn, Reinforcement by a Thin Layer with Oscillating Thickness. Appl. Math. Optim. 16 (1987) 247-261. MR901816
8. [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. Zbl0938.35049MR1676697
9. [9] P.G. Ciarlet and P. Destuynder, A Justification of the Two-Dimensional Linear Plate Model. J. Mécanique 18 (1979) 315-344. Zbl0415.73072MR533827
10. [10] D. Cioranescu and J. Saint Jean Paulin, Homogenization in Open Sets with Holes. J. Math. Anal. Appl. 71 (1979) 590-607. Zbl0427.35073MR548785
11. [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. Zbl0892.35017MR1485630
12. [12] A. Gaudiello, Asymptotic Behaviour of non-Homogeneous Neumann Problems in Domains with Oscillating Boundary. Ricerche Mat. 43 (1994) 239-292. Zbl0938.35511MR1324751
13. [13] A. Gaudiello, Homogenization of an Elliptic Transmission Problem. Adv. Math Sci. Appl. 5 (1995) 639-657. Zbl0852.35015MR1361009
14. [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. Zbl1034.35020MR1895899
15. [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). Zbl1085.37500
16. [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. Zbl1032.74037MR1941655
17. [17] H. Le Dret, Problèmes variationnels dans les multi-domaines : modélisation des jonctions et applications. Masson, Paris (1991). Zbl0744.73027MR1130395
18. [18] J.L. Lions, Quelques méthodes de résolution de problèmes aux limites non linéaires. Dunod, Paris (1969). Zbl0189.40603MR259693
19. [19] T.A. Mel’nyk, Homogenization of the Poisson Equations in a Thick Periodic Junction. ZAA J. Anal. Appl. 18 (1999) 953-975. Zbl0938.35021
20. [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. Zbl0930.35114
21. [21] G. Nguetseng, A General Convergence Result for a Functional Related to the Theory of Homogenization. SIAM J. Math Anal. 20 (1989) 608-623. Zbl0688.35007MR990867
22. [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. Zbl0920.35019

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