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Stochastic homogenization of a class of monotone eigenvalue problems. (English) Zbl 1224.35026
Summary: Stochastic homogenization (with multiple fine scales) is studied for a class of nonlinear monotone eigenvalue problems. More specifically, we are interested in the asymptotic behaviour of a sequence of realizations of the form \[ -\operatorname {div} \Bigl (a\Bigl (T_1\Bigl (\frac {x}{\varepsilon _1}\Bigr )\omega _1,T_2 \Bigl (\frac {x}{\varepsilon _2}\Bigr )\omega _2, \nabla u^\omega _{\varepsilon }\Bigr )\Bigr ) =\lambda _\varepsilon ^\omega \mathcal C(u^\omega _{\varepsilon }). \] It is shown, under certain structure assumptions on the random map \(a(\omega _1,\omega _2,\xi )\), that the sequence \(\{\lambda _\varepsilon ^{\omega ,k},u^{\omega ,k}_\varepsilon \}\) of \(k\)th eigenpairs converges to the \(k\)th eigenpair \(\{\lambda ^k,u^k\}\) of the homogenized eigenvalue problem \( - \operatorname {div}( {b}(\nabla u) ) = \lambda {\overline {\mathcal C}}(u). \) For the case of \(p\)-Laplacian type maps we characterize \(b\) explicitly.
MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B40 Asymptotic behavior of solutions to PDEs
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