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• W147798326 abstract "In this mini-course we will go into the so-called Bloch-Fourier approach to understand homogenization of periodic structures. The aims are to state the main results with motivations, highlight various phenomena in the Fourier space and indicate possible gains over other methods. Main ideas will be given trying to avoid technicalities. It is fair to say that ever since the publication of the book [4], there is a renewed and vigorous activity in homogenization problems which form an important area of Applied Mathematics. In order to tackle the key questions in homogenization, several methods have been devised and there is an enormous literature on the subject. Our goal in these series of lectures is to address certain basic questions which we consider to be fundamental and to present a way to answer them which can be classified inside category (B) below. Homogenization methods can be broadly categorized as follows: (A) Physical space methods, (B) Fourier space methods, and (C) Phase space methods. There are many methods falling in the class (A) and let us mention some successful ones : Multiscale expansion [4], Multiscale convergence [1], [21], Method of oscillating test functions [19], [26], Compensated Compactness [20], Gamma Convergence [11] and so on. Though they have varying domains of applicability, they have some common characteristics. An insight into the kind of oscillations produced by the medium is gained and this information is used in some way or the other (ansatz, test functions, ...). In order to study partial differential equations with variable coefficients, techniques of pseudo-differential operators and Fourier integral operators have been developed [15]. These phase space theories provide tools suitable to study the behaviour of in-homogeneous media which are qualitatively similar to homogeneous ones and their small perturbations. To deal with heterogeneities possessing qualitatively different properties, we need more sophisticated tools such as H-measures [25], micro-local defect measures [13], Wigner measures [14], [16] and their generalizations and refinements (see [18]). In the middle ground between the classes (A) and (C) lie the methods based purely on Fourier techniques and this mini-course address them. In problems involving oscillations, it is natural to seek methods based on Fourier space and this general outlook led us to look for tools suitable to describe oscillations produced by heterogeneities in the Fourier space. My intention is to present some of them and illustrate how they can be effectively used to study classical homogenization problems. Though there are many aspects of homogenization, we will confine ourselves to some of them which I consider to be fundamental. Using the tools of Fourier space, one simple" @default.
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• W147798326 date "2004-01-01" @default.
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• W147798326 title "Bloch-Fourier Approach to Classical Homogenization Problems" @default.
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