Abstract
This chapter is devoted to the homogenization of boundary value problems in a periodically perforated domain by an approach which is alternative to those of asymptotic analysis and of classical homogenization theory. The domain has a periodic structure, and the size of each cell is determined by a positive parameter δ. The relative size of each periodic perforation is instead determined by a positive parameter ε. We analyze the behavior of a family of solutions as δ and ε degenerate to zero.
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References
Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs. In Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. II (Paris, 1979/1980), volume 60 of Res. Notes in Math., 98–138, 389–390. Pitman, Boston, Mass. (1982)
Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs. II. In Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. III (Paris, 1980/1981), volume 70 of Res. Notes in Math., 154–178, 425–426. Pitman, Boston, Mass. (1982)
Dalla Riva, M.: Stokes flow in a singularly perturbed exterior domain. Complex Var. Elliptic Equ. 58, 231–257 (2013)
Dalla Riva, M., Lanza de Cristoforis, M.: Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem: a functional analytic approach. Complex Var. Elliptic Equ. 55, 771–794 (2010)
Dalla Riva, M., Lanza de Cristoforis, M.: Weakly singular and microscopically hypersingular load perturbation for a nonlinear traction boundary value problem: a functional analytic approach. Complex Anal. Oper. Theory 5, 811–833 (2011)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, Springer Verlag, Berlin (1983)
Lanza de Cristoforis, M.: Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces. Comput. Methods Funct. Theory 2, 1–27 (2002)
Lanza de Cristoforis, M.: Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach. Analysis (Munich) 28, 63–93 (2008)
Lanza de Cristoforis, M.: Asymptotic behaviour of the solutions of a non-linear transmission problem for the Laplace operator in a domain with a small hole. A functional analytic approach. Complex Var. Elliptic Equ. 55, 269–303 (2010)
Lanza de Cristoforis, M.: Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole. A functional analytic approach. Rev. Mat. Complut. 25, 369–412 (2012)
Lanza de Cristoforis, M., Musolino, P.: A singularly perturbed nonlinear Robin problem in a periodically perforated domain: a functional analytic approach. Complex Var. Elliptic Equ. 58, 511–536 (2013)
Lanza de Cristoforis, M., Musolino, P.: A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach. Commun. Pure Appl. Anal. 13, 2509–2542 (2014)
Lanza de Cristoforis, M., Musolino, P.: Two-parameter anisotropic homogenization for a Dirichlet problem for the Poisson equation in an unbounded periodically perforated domain. A functional analytic approach. Typewritten manuscript (2014)
Marčenko, V.A., Khruslov, E.Y.: Boundary value problems in domains with a fine-grained boundary. Izdat. “Naukova Dumka”, Kiev (1974) (in Russian)
Maz’ya, V., Movchan, A.: Asymptotic treatment of perforated domains without homogenization. Math. Nachr. 283, 104–125 (2010)
Musolino, P.: A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach. Math. Methods Appl. Sci. 35, 334–349 (2012)
Acknowledgements
The authors acknowledge the support of “Progetto di Ateneo: Singular perturbation problems for differential operators, CPDA120171/12,” University of Padova, and of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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de Cristoforis, M.L., Musolino, P. (2015). A Functional Analytic Approach to Homogenization Problems. In: Constanda, C., Kirsch, A. (eds) Integral Methods in Science and Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-16727-5_30
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DOI: https://doi.org/10.1007/978-3-319-16727-5_30
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-16726-8
Online ISBN: 978-3-319-16727-5
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