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- W2066125816 abstract "We consider a second-order elliptic equation in a bounded periodic heterogeneous medium and study the asymptotic behavior of its spectrum, as the structure period goes to zero. We use a new method of Bloch wave homogenization which, unlike the classical homogenization method, characterizes a renormalized limit of the spectrum, namely sequences of eigenvalues of the order of the square of the medium period. We prove that such a renormalized limit spectrum is made of two parts: the so-called Bloch spectrum , which is explicitly defined as the spectrum of a family of limit problems, and the so-called boundary layer spectrum , which is made of limit eigenvalues corresponding to sequences of eigenvectors concentrating on the boundary of the domain. This analysis relies also on a notion of Bloch measures which can be seen as ad hoc Wigner measures in the context of semi-classical analysis. Finally, for rectangular domains made of entire periodicity cells, a variant of the Bloch wave homogenization method gives an explicit characterization of the boundary layer spectrum too. On considère uneéquation elliptique du deuxième ordre dans un milieu périodique hétérogène borné, et onétudie le comportement asymptotique de son spectre lorsque la période tend vers zéro. On utilise une nouvelle méthode d' homogénéisation par ondes de Bloch qui, contrairement aux méthodes classiques d'homogénéisation, caractérise la limite renormalisée du spectre, et plus précisément les suites de valeurs propres de l'ordre du carréde la période. On démontre que le spectre limite renormaliséest constitutéde deux parties: un spectre de Bloch , qui est explicitement caractérisécomme le spectre d'une famille de problèmes limites, et un spectre de couche limite , qui est l'ensemble des limites de suites de valeurs propres dont les vecteurs propres correspondants se concentrent sur le bord du domaine. L'analyse présentée repose sur une notion de mesures de Bloch qui peuventeˆtre vues comme des versions ad hoc des mesures de Wigner utilisées en analyse semi-classique. Enfin, pour des domaines rectangulaires constitués uniquement de cellules de périodicitéentières, une variante de la méthode d'homogénéisation par ondes de Bloch permet de donner aussi une chractérisation explicite du spectre de couche limite." @default.
- W2066125816 created "2016-06-24" @default.
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- W2066125816 date "1998-02-01" @default.
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- W2066125816 title "Bloch wave homogenization and spectral asymptotic analysis" @default.
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- W2066125816 doi "https://doi.org/10.1016/s0021-7824(98)80068-8" @default.
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