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Homogenization of a class of nonlinear eigenvalue problems. (English) Zbl 1105.35010
The authors consider the asymptotic behavior of the eigenvalues of a family of nonlinear monotone operators $${\mathcal A}_\epsilon = -\text{div}(a_\epsilon(x,\nabla u))$$, which are subdifferentials of even positively homogeneous convex functionals. Under the assumption that $${\mathcal A}_\epsilon$$ $$G$$-converges to $${\mathcal A}_{\text{hom}} = -\text{div}(a_{\text{hom}}(x,\nabla u))$$, the authors prove that any limit point $$\lambda$$ of a sequence $$\lambda_\epsilon$$ of eigenvalues of the operators $${\mathcal A}_\epsilon$$ is an eigenvalue of $${\mathcal A}_{\text{hom}}$$: Moreover the sequence of the first eigenvalues of $${\mathcal A}_\epsilon$$ converges to the first eigenvalue of $${\mathcal A}_{\text{hom}}$$.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
##### Keywords:
Homogenization; Eigenvalues of nonlinear operators
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