Geodesic
Tartar's method of compensated compactness in averaging the spectrum of a mixed problem for an elliptic equation in a perforated domain with third boundary condition
Sbornik. Mathematics, Tome 186 (1995) no. 5, pp. 753-770 
S. E. Pastukhova. Tartar's method of compensated compactness in averaging the spectrum of a mixed problem for an elliptic equation in a perforated domain with third boundary condition. Sbornik. Mathematics, Tome 186 (1995) no. 5, pp. 753-770. http://geodesic.mathdoc.fr/item/SM_1995_186_5_a7/
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     author = {S. E. Pastukhova},
     title = {Tartar's method of compensated compactness in averaging the~spectrum of a~mixed problem for an~elliptic equation in a~perforated domain with third boundary condition},
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     volume = {186},
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     url = {http://geodesic.mathdoc.fr/item/SM_1995_186_5_a7/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We study the problem described in the title of this paper in the domain $\Omega_\varepsilon$ obtained from a domain $\Omega\in\mathbb R^d$ by periodic perforation with period $\varepsilon Q$, where $Q$ is the unit cube in $\mathbb R^d$. For this problem we use the method of compensated compactness to obtain the first two terms of the asymptotics of the $k$-th eigenvalue in powers of $\varepsilon$ as $\varepsilon\to0$: $\lambda_{\varepsilon,k}=\varepsilon^{-1}\Lambda+\lambda_k+\dotsb$, where $\Lambda$ is a constant independent of $k$ and $\lambda_k$ is the $k$-th eigenvalue of the averaged problem (which turns out to be the Dirichlet problem in the domain $\Omega$) for $k\in\mathbb N$.

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