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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 \({\mathcal A}_{\text{hom}}\).

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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