Ministerio de Ciencia e InnovaciónJunta de Andalucí

Calvo Jurado, Carmen

Casado Díaz, Juan

Hamilton, I.P. (Coordinador)

Luna Laynez, Manuel

Vigo Aguiar, Jesús (Coordinador)

English

idUS. Depósito de Investigación Universidad de Sevilla

Proceedings of the 13th International Conferenceon Computational and Mathematical Methodsin Science and Engineering, CMMSE 201324–27 June, 2013.Homogenization of the Poisson equation with Dirichletconditions in random perforated domainsCarmen Calvo-Jurado1, Juan Casado-Dı́az2 and Manuel Luna-Laynez21 Department of Mathematics, University of Extremadura2 Department of Differential Equations and Numerical Analysis, University of Sevillaemails: ccalvo@unex.es, jcasadod@us.es, mllaynez@us.esAbstractKey words: homogenization, random perforated domainsMSC 2000: 35R60, 35B271 Extended abstractFor a bounded open set O ⊂ RN , and a sequence of open sets Oε ⊂ O, we consider thePoisson equation with Dirichlet conditions on ∂Oε. The problem is to study the asymptoticbehavior of the solutions, when ε tends to zero, by finding the equation satisfied by its limit.This allows us to describe the macroscopic behavior of the material corresponding to themixture of the solid part Oε and the holes Kε = O\Oε. The obtention of this limit equationis a classical problem which has been considered since the early 80’s. Usually, the set Kεis a union of very small connected components distributed in O. It is well known that thehomogenized equation is an elliptic problem in the whole of O which contains in general anew 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 homogenizationof the Poisson equation with holes of size εNN−2 if N ≥ 3, or ε−1ε2 if N = 2, which areperiodically distributed with period ε. In this case the coefficient corresponding to the newterm of zero order is a positive constant. The case of arbitrary holes and nonlinear equationshas 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 randomlydistributed. To simplify, we assume N ≥ 3, and similarly to the classical periodic setting, weconsider holes of size εNN−2 such that the distance between two holes is of order ε. Namely:c©CMMSE ISBN: 978-84-616-2723-3Page 364 of 1797Homogenization of Dirichlet problems in random perforated domainsWe consider a probability space (Ω,F , P ), a subset Ω̃ ⊂ 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 asKε(ω) =⋃z∈RN , T (z)ω∈Ω̃(εz + εNN−2K), P -a.e. ω ∈ Ω.Denoting by Oε(ω) the open set obtained from O by removing the random holes, Kε(ω),we want to study the asymptotic behavior of the solutions uε of{−∆xuε(ω, x) = fε(ω, x) in Oε(ω)uε(ω, x) = 0 on ∂Oε(ω)P -a.e. ω ∈ Ω, (0.1)where fε converges strongly in L2P (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-scaleconvergence method ([1], [13]), which is a very useful tool in periodic homogenization, hasbeen extended in [2] to the setting of random homogenization problems. In general, theheterogeneities considered in those papers are given by functions of the form F (T (xε )ω), withω taking values on the probability space, T a dynamical system as above and F a randomvariable. However, in our problem, the homogenization process contains two different sizes(εNN−2 and ε) to describe it, and then, it can not be analyzed by the results of [2]. To solvethis 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 problemsin (deterministic) periodic domains. We show that the solution uε of (0.1) converges weaklyin L2P (Ω;H10 (O)) to the unique solution u of the problem{−∆xu(ω, x)) + γκu(ω, x) = f(ω, x) in Ou(ω, x) = 0 on ∂OP -a.e. ω ∈ Ω,where the new term γκu is the equivalent for our random problem of the Cioranescu-Muratstrange term in the deterministic case ([6]). Similarly to the classical result, it is given bythe capacity κ of the closed set K in RN , multiplied by γ, the mean density of holes in O.Thanks to the ergodic theory we prove that γ does not depend on ω ∈ Ω or x ∈ O. Fromthe physical point of view this means that the limit behavior of the material correspondingto the mixture of the solid part Oε(ω) and the holes Kε(ω) is deterministic. Similar resultsbut assuming different assumptions about the random distribution of the holes and usingdifferent techniques have been obtained in [3] and [14].c©CMMSE ISBN: 978-84-616-2723-3Page 365 of 1797C. Calvo-Jurado, J. Casado-D́ıaz, M. Luna-LaynezAcknowledgementsThis work has been partially supported by the project MTM 2011-24457 of the “Ministe-rio de Ciencia e Innovación” of Spain and the research group FQM-309 of the “Junta deAndalućıa”.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 themean 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. Poincaré Anal. Non Linèaire 26 (2009) 2, 375-395.[4] J. Casado-D́ıaz, Two-Scale convergence for nonlinear Dirichlet problems in perfo-rated domains, Proceedings of the Royal Society of Edinburgh A, 130 (2000) 249–276.[5] J. Casado-D́ıaz, A. Garroni, Asymptotic behavior of nonlinear elliptic systems onvarying domains, SIAM J. Math. Anal. 31 (2000) 581–624.[6] D. Cionarescu, F. Murat, Un terme étrange venu d’ailleurs, in Nonlinear PartialDifferential Equations and Their Applications, Collége de France Seminar, Vols. II andIII, Research Notes in Math. 60 and 70 Pitman, London, 1982, 98–138 and 154–178.[7] G. dal Maso, A. Defranceschi, Limits of nonlinear Dirichlet problems in varyingdomains, Manuscr. Math. 61 (1988) 251–278.[8] G. Dal Maso, L. Modica, Nonlinear stochastic homogenization and ergodic theory,J. Reine Angew, Math. 368 (1986) 28–42.[9] G. Dal Maso, U. Mosco, Wiener-criterion and Γ-convergence, Appl. Math. Optim.15 (1987) 15–63.[10] G. Dal Maso, F. Murat, Asymptotic behaviour and correctors for the Dirichletproblems in perforated domains with homogeneous monotone operators, Ann. Sc. Norm.Sup. Pisa 4 (1997) 239–290.[11] V.V. Jikov, S.M. Kozlov, O.A. Oleinik Homogenization of differential operatorsand integral functionals, Springer-Verlag, Berlin, 1994.c©CMMSE ISBN: 978-84-616-2723-3Page 366 of 1797Homogenization of Dirichlet problems in random perforated domains[12] S.M. Kozlov, Homogenization of random operators, Math. U.S.S.R. Sb.37 (1980)167–180.[13] G. Nguetseng, A general convergence result for a functional related to the theory ofhomogenization, SIAM J. Math. Anal. 20 (1989) 608–623.[14] G.C. Papanicolaou, S.R.S. Varadhan, Diffusion in regions with many smallholes, in Stochastic differential systems, Lecture Notes in Control and InformationSci. Springer, Berlin-New York, 25 (1980).[15] G.C. Papanicolaou, S.R.S. Varadhan, Boundary value problems with rapidly os-cillating random coefficients, Colloq. Math. Soc. J. Bolyai, North-Holland, Amsterdam-New York, 27 (1981) 835-873.[16] I.V. Skrypnik, Averaging of nonlinear Dirichlet problems in punctured domains ofgeneral structure, Mat. Sb. 187, 8 (1996) 125–157.c©CMMSE ISBN: 978-84-616-2723-3Page 367 of 1797

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

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

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