Homogenization of the Poisson equation with Dirichlet conditions in random perforated domains

Proceedings of the 13th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2013 24–27 June, 2013. Homogenization of the Poisson equation with Dirichlet conditions in random perforated domains Carmen Calvo-Jurado1, Juan Casado-D´ıaz2and Manuel Luna-Laynez2 1Department of Mathematics, University of Extremadura 2Department of Differential Equations and Numerical Analysis, University of Sevilla emails: [email protected],[email protected],[email protected] Abstract Key words: homogenization, random perforated domains MSC 2000: 35R60, 35B27 1 Extended abstract For a bounded open set O⊂RN, and a sequence of open sets Oε⊂O, we consider the Poisson equation with Dirichlet conditions on ∂Oε. The problem is to study the asymptotic behavior of the solutions, when εtends to zero, by finding the equation satisfied by its limit. This allows us to describe the macroscopic behavior of the material corresponding to the mixture of the solid part Oεand the holes Kε=O\Oε. The obtention of this limit equation is a classical problem which has been considered since the early 80’s. Usually, the set Kε is a union of very small connected components distributed in O. It is well known that the homogenized equation is an elliptic problem in the whole of Owhich contains in general a new term of order zero depending on the distribution and size of the holes, the Cioranescu- Murat “strange term” ([6]). The most classical result corresponds to the homogenization of the Poisson equation with holes of size ε N N−2if N≥3, or ε−1 ε2if N= 2, which are periodically distributed with period ε. In this case the coefficient corresponding to the new term of zero order is a positive constant. The case of arbitrary holes and nonlinear equations has been considered in several papers such as [4], [5], [6], [7], [9], [10], [16]. Our purpose in the present work is to consider the case where the holes are randomly distributed. To simplify, we assume N≥3, and similarly to the classical periodic setting, we consider holes of size ε N N−2such that the distance between two holes is of order ε. Namely: c CMMSE ISBN: 978-84-616-2723-3 Page 364 of 1797 Homogenization of Dirichlet problems in random perforated domains We consider a probability space (Ω,F, P ), a subset ˜ Ω⊂Ω, and a function T(x):Ω→Ω, for every x∈RN, which defines an ergodic measure preserving dynamical system in RN. Then, for a compact set K⊂RN, we define the sequence of random holes as Kε(ω) = [ z∈RN, T (z)ω∈˜ Ωεz +ε N N−2K, P-a.e. ω∈Ω. Denoting by Oε(ω) the open set obtained from Oby removing the random holes, Kε(ω), we want to study the asymptotic behavior of the solutions uεof (−∆xuε(ω, x) = fε(ω, x) in Oε(ω) uε(ω, x) = 0 on ∂Oε(ω)P-a.e. ω∈Ω,(0.1) where fεconverges strongly in L2 P(Ω; H−1(O)) to a function f. The homogenization of random problems has been considered in several papers, see e.g. [2], [3], [8], [11], [12], [14], [15]. In particular, the G. Neguetseng and G. Allaire two-scale convergence method ([1], [13]), which is a very useful tool in periodic homogenization, has been extended in [2] to the setting of random homogenization problems. In general, the heterogeneities considered in those papers are given by functions of the form F(T(x ε)ω), with ωtaking values on the probability space, Ta dynamical system as above and Fa random variable. However, in our problem, the homogenization process contains two different sizes (ε N N−2and ε) to describe it, and then, it can not be analyzed by the results of [2]. To solve this difficulty, we introduce in the present paper a new extension of the two-scale method, which is based on some ideas used in [4] for the homogenization of Dirichlet elliptic problems in (deterministic) periodic domains. We show that the solution uεof (0.1) converges weakly in L2 P(Ω; H1 0(O)) to the unique solution uof the problem (−∆xu(ω, x)) + γκu(ω, x) = f(ω, x) in O u(ω, x) = 0 on ∂O P-a.e. ω∈Ω, where the new term γκu is the equivalent for our random problem of the Cioranescu-Murat strange term in the deterministic case ([6]). Similarly to the classical result, it is given by the capacity κof the closed set Kin RN, multiplied by γ, the mean density of holes in O. Thanks to the ergodic theory we prove that γdoes not depend on ω∈Ω or x∈O. From the physical point of view this means that the limit behavior of the material corresponding to the mixture of the solid part Oε(ω) and the holes Kε(ω) is deterministic. Similar results but assuming different assumptions about the random distribution of the holes and using different techniques have been obtained in [3] and [14]. c CMMSE ISBN: 978-84-616-2723-3 Page 365 of 1797 C. Calvo-Jurado, J. Casado-D´ ıaz, M. Luna-Laynez Acknowledgements This work has been partially supported by the project MTM 2011-24457 of the “Ministerio de Ciencia e Innovaci´on” of Spain and the research group FQM-309 of the “Junta de Andaluc´ıa”. References [1] G. Allaire,Homogenization and two-scale convergence, SIAM J. Math Anal. 23 (1992) 1482–151. [2] A. Bourgeat, A. Mikelic, S. Wright,Stochastic two-scale convergence in the mean and applications J. Reine Angew. Math. 456 (1994) 19-51. [3] L.A. Caffarelli, A. Mellet,Random homogenization of an obstacle problem, Ann. Inst. H. Poincar´e Anal. Non Lin`eaire 26 (2009) 2, 375-395. [4] J. Casado-D´ ıaz,Two-Scale convergence for nonlinear Dirichlet problems in perforated domains, Proceedings of the Royal Society of Edinburgh A, 130 (2000) 249–276. [5] J. Casado-D´ ıaz, A. Garroni, Asymptotic behavior of nonlinear elliptic systems on varying domains, SIAM J. Math. Anal. 31 (2000) 581–624. [6] D. Cionarescu, F. 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