A functional analytic approach to homogenization problems

We plan to illustrate a functional analytic approach to analyze homogenization problems, which has already been developed for singular perturbation problems in bounded domains with small holes (cf. e.g., Lanza de Cristoforis (Comput Methods Funct Theory 2:1– 27, 2002), Lanza de Cristoforis (Analysis (Munich) 28:63–93, 2008), Lanza de Cristoforis (Complex Var Elliptic Equ 55:269–303, 2010) Lanza de Cristoforis (Rev Mat Comput 25:369–412, 2012)). In the frame of linearized elastostatics and of the Stokes equations, we mention Dalla Riva and Lanza de Cristoforis (Complex Var Elliptic Equ 55:771–794, 2010; Complex Anal Oper Theory 5:811–833, 2011), and Dalla Riva (Complex Var Elliptic Equ 58:231–257, 2013). Later on, such an approach has been exploited for the analysis of problems in unbounded perforated domains with a fixed periodic structure, for example in Lanza de Cristoforis and Musolino (Complex Var Elliptic Equ 58:511–536, 2013; Commun Pure Appl Anal 13:2509–2542, 2014; Math Methods Appl Sci 35:334–349, 2012). Instead, here we consider the case where also the size of the periodicity cell tends to zero. The results in this chapter are based on the work of Lanza de Cristoforis and Musolino (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).