Stochastic homogenization of a class of monotone eigenvalue problems

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 $$ - div\left( {a\left( {T_1 \left( {\frac{x} {{\varepsilon _1 }}} \right)\omega _1 ,T_2 \left( {\frac{x} {{\varepsilon _2 }}} \right)\omega _2 ,\nabla u_\varepsilon ^\omega } \right)} \right) = \lambda _\varepsilon ^\omega \mathcal{C}\left( {u_\varepsilon ^\omega } \right) $$ . It is shown, under certain structure assumptions on the random map a(ω 1, ω 2, ξ), that the sequence {λ , u } of kth eigenpairs converges to the kth eigenpair {λ k , u k } of the homogenized eigenvalue problem . For the case of p-Laplacian type maps we characterize b explicitly.