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Article

Keywords:
stochastic; homogenization; eigenvalue
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 $$ -\div \Bigl (a\Bigl (T_1\Bigl (\frac x{\varepsilon _1}\Bigr )\omega _1,T_2 \Bigl (\frac x{\varepsilon _2}\Bigr )\omega _2, \nabla u^\omega _{\varepsilon }\Bigr )\Bigr ) =\lambda _\varepsilon ^\omega \mathcal C(u^\omega _{\varepsilon }). $$ 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 $$ - {\rm div}( b(\nabla u) ) = \lambda {\overline {\mathcal C}}(u). $$ For the case of $p$-Laplacian type maps we characterize $b$ explicitly.
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