A non-periodic and two-dimensional example of elliptic homogenization

Background. When studying the microscale behavior (beyond the reach of numerical solution methods) of physical systems, one is naturally lead to the concept of homogenization, i.e., the theory of the convergence of sequences of partial differential equations. The homogenization of periodic structures using the two-scale convergence technique is well-established due to the pioneering work by Gabriel Nguetseng [10] and the further development work by Grégoire Allaire [1]. Generalizations of the two-scale convergence technique have been developed independently by, e.g., Maria Lúısa Mascarenhas and Anca-Maria Toader [8] (scale convergence), Gabriel Nguetseng [11, 12] (Σ-convergence), and Anders Holmbom, Jeanette Silfver, Nils Svanstedt and Niklas Wellander [4, 6] (“generalized” two-scale convergence). A simple but possibly powerful method of analyzing non-periodic structures is the λ-scale convergence technique introduced by Anders Holmbom and Jeanette Silfver [5]. λ-scale convergence is scale convergence in the special case of using the Lebesgue (i.e., λ) measure and test functions periodic in the second argument [5]. Homogenization techniques based on this approach are developed in the doctoral thesis [13] of Jeanette Silfver. These results are the point of departure for the main contributions in this paper.