Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T17:55:17.808Z Has data issue: false hasContentIssue false
  • English
  • Français

Dimension reduction and homogenization of random degenerate operators. Part I

Part of: Elliptic equations and systems Stochastic processes Partial differential equations Miscellaneous topics - Partial differential equations Stochastic analysis

Published online by Cambridge University Press:  01 January 2012

Abdelaziz Aït Moussa  and
Loubna Zlaïji
Abdelaziz Aït Moussa
Affiliation:
Department of Mathematics and Computer Sciences, University of Mohammed Premier, 60040 Oujda, Morocco
Loubna Zlaïji
Affiliation:
Department of Mathematics and Computer Sciences, University of Mohammed Premier, 60040 Oujda, Morocco (email: l.zlaiji@yahoo.fr)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Our aim in this paper is to identify the limit behavior of the solutions of random degenerate equations of the form −div Aε(x′,∇Uε)+ρεω(x′)Uε=F with mixed boundary conditions on Ωε whenever ε→0, where Ωε is an N-dimensional thin domain with a small thickness h(ε), ρεω(x′)=ρω(x′/ε), where ρω is the realization of a random function ρ(ω) , and Aε(x′,ξ)=a(Tx′ /εω,ξ) , the map a(ω,ξ) being measurable in ω and satisfying degenerated structure conditions with weight ρ in ξ. As usual in dimension reduction problems, we focus on the rescaled equations and we prove that under the condition h(ε)/ε→0 , the sequence of solutions of them converges to a limit u0, where u0 is the solution of an (N−1) -dimensional limit problem with homogenized and auxiliary equations.

MSC classification

Secondary: 35B27: Homogenization; equations in media with periodic structure 35J60: Nonlinear elliptic equations 35R60: Partial differential equations with randomness, stochastic partial differential equations 60G57: Random measures 60H25: Random operators and equations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 1 - 22
Copyright
Copyright © London Mathematical Society 2012

References

[1]Allaire, G., ‘Homogenization and two-scale convergence’, SIAM J. Math. Anal. 23 (1992) 1482–1518.CrossRefGoogle Scholar
[2]Allaire, G. and Briane, M., ‘Multiscale convergence and reiterated homogenisation’, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) no. 2, 297–342.CrossRefGoogle Scholar
[3]Andrews, K. T. and Wright, S., ‘Stochastic homogenization of elliptic boundary-value problem with L p-data’, Asymptotic Anal. 17 (1998) 165–184.Google Scholar
[4]Bal, G., ‘Central limits and homogenization in random media’, Multiscale Model. Simul. 7 (2008) no. 2, 677–702.CrossRefGoogle Scholar
[5]Belyaev, A. Yu. and Chechkin, G. A., ‘Averaging operators with boundary conditions of fine-scaled structure’, Math. Notes 65 (1999) no. 4, 418–429.CrossRefGoogle Scholar
[6]Blanchard, D. and Gaudiello, A., ‘Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem’, ESAIM Control Optim. Calc. Var. 9 (2003) 449–460.CrossRefGoogle Scholar
[7]Blanchard, D., Gaudiello, A. and Mossino, J., ‘Highly oscillating boundaries and reduction of dimension: the critical case’, Anal. Appl. 5 (2007) no. 2, 137–163.CrossRefGoogle Scholar
[8]Bouchitté, G., Buttazzo, G. and Seppecher, P., ‘Energies with respect to a measure and applications to low-dimensional structures’, Calc. Var. Partial Differential Equations 5 (1997) no. 1, 37–54.CrossRefGoogle Scholar
[9]Bouchitté, G. and Fragala, I., ‘Homogenization of thin structures by two-scale method with respect to measures’, SIAM J. Math. Anal. 32 (2001) 1198–1226.CrossRefGoogle Scholar
[10]Bouchitté, G., Buttazzo, G. and Fragala, I., ‘Convergence of Sobolev spaces on varying manifolds’, J. Geom. Anal. 11 (2001) 399–422.CrossRefGoogle Scholar
[11]Bourgeat, A., Mikelić, A. and Piatnitski, A., ‘Modèle de double porosité aléatoire’, C. R. Acad. Sci. Paris Ser. I 327 (1998) 99–104.CrossRefGoogle Scholar
[12]Bourgeat, A., Mikelić, A. and Wright, S., ‘Stochastic two-scale convergence in the mean and applications’, J. reine angew. Math. 456 (1994) 19–51.Google Scholar
[13]Ciarlet, P. G., Mathematical elasticity, vol. II: Theory of plates (North-Holland, Amsterdam, 1997).Google Scholar
[14]Daley, D. J. and Vere-Jones, D., An introduction to the theory of point processes (Springer, New York, 1988).Google Scholar
[15]Engström, J., Persson, L.-E., Piatnitski, A. and Wall, P., ‘Homogenization of random degenerated nonlinear monotone operators’, Glasg. Math. J. 41 (2006) no. 61, 101–114.CrossRefGoogle Scholar
[16]Gaudiello, A., Gustafsson, B., Lefter, C. and Mossino, J., ‘Asymptotic analysis of a class of minimization problems in a thin multidomain’, Calc. Var. Partial Differential Equations 15 (2002) 181–201.CrossRefGoogle Scholar
[17]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.CrossRefGoogle Scholar
[18]Hoang, V. H., ‘Reduction of dimension for diffusion in a perforated thin plate’, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 1037–1057.CrossRefGoogle Scholar
[19]Lukkassen, D., Nguetseng, G. and Wall, P., ‘Two-scale convergence’, Int. J. Pure Appl. Math. 2 (2002) no. 1, 35–86.Google Scholar
[20]Lukkassen, D. and Wall, P., ‘Two-scale convergence with respect to measures and homogenization of monotone operators’, J. Funct. Spaces Appl. 3 (2005) 125–161.CrossRefGoogle Scholar
[21]Nguetseng, G., ‘A general convergence result for a functional related to the theory of homogenization’, SIAM J. Math. Anal. 20 (1989) no. 3, 608–623.CrossRefGoogle Scholar
[22]Zeidler, E., Nonlinear functional analysis and its applications II/B, nonlinear monotone operators (Springer, New York, 1990).Google Scholar
[23]Zhikov, V. V., ‘Connectedness and homogenization. Examples of fractal conductivity’, Mat. Sb. 187 (1996) no. 8, 3–40.Google Scholar
[24]Zhikov, V. V., ‘On the homogenization technique for variational problems’, Funktsional. Anal. i Prilozhen. 33 (1999) no. 1, 14–29.CrossRefGoogle Scholar
[25]Zhikov, V. V., ‘On an extension of the method of two-scale convergence and its applications’, Mat. Sb. 191 (2000) no. 7, 31–72.Google Scholar
[26]Zhikov, V. V., ‘Homogenization of elasticity problems on singular structures’, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002) no. 2, 81–148.Google Scholar
[27]Zhikov, V. V., ‘Diffusion in an incompressible random flow’, Funct. Anal. Appl. 31 (1997) no. 3, 156–166.CrossRefGoogle Scholar
[28]Zhikov, V. V. and Piatnitski, A. L., ‘Homogenization of random singular structures and random measures’, Izv. Ran. Ser. Mat. 70 (2006) no. 1, 23–74.Google Scholar