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Analyse mathématique de quelques modèles en calcul de structures électroniques et homogénéisation

Abstract : This thesis is divided into two parts. The first part, that coincides with Chapter 2, deals with mathematical models in quantum chemistry, and specifically focuses on Kohn-Sham models with LDA and GGA exchange-correlation functionals. We prove, for a neutral or positively charged system, that the extended Kohn-Sham LDA model admits a minimizer, and that the Kohn-Sham GGA model for a two-electron system admits a minimizer. The second part is concerned with various issues in homogenization. In Chapters 3 and 4, we introduce and study a model in which the material of interest consists of a random perturbation of a periodic material. We propose different approaches, either rigorous or formal, to compute the homogenized behavior of this material up to the second order in the size of the perturbation, in an entirely deterministic way. Numerical experiments show the efficiency of these approaches as compared to the direct stochastic homogenization process. Chapter 5 is devoted to boundary layers in periodic homogenization, in particular in the parabolic setting. It aims at giving a better understanding of how to take into account boundary and initial conditions, and how to correct the two-scale expansion on which homogenization is classically grounded, to obtain fine error estimates
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Arnaud Anantharaman. Analyse mathématique de quelques modèles en calcul de structures électroniques et homogénéisation. Mathématiques générales [math.GM]. Université Paris-Est, 2010. Français. ⟨NNT : 2010PEST1002⟩. ⟨tel-00558618v2⟩



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