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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.
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B40 Asymptotic behavior of solutions to PDEs
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[1] J.P. García Azorero, I. Peral Alonso: Existence and nonuniqueness for the p-Laplacian: Nonlinear eigenvalues. Commun. Partial Differ. Equations 12 (1987), 1389–1430.
[2] L. Baffico, C. Conca, M. Rajesh: Homogenization of a class of nonlinear eigenvalue problems. Proc. R. Soc. Edinb. 136A (2006), 7–22. · Zbl 1105.35010
[3] A. Bensoussan, J.-L. Lions, G. Papanicolaou: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam, 1978. · Zbl 0404.35001
[4] V. Chiadò-Piat, G. Dal Maso, A. Defranceschi: G-convergence of monotone operators. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7 (1990), 123–160.
[5] V. Chiadò-Piat, A. Defranceschi: Homogenization of monotone operators. Nonlinear Anal., Theory Methods Appl. 14 (1990), 717–732. · Zbl 0705.35041
[6] G. Dal Maso: An Introduction to {\(\Gamma\)}-convergence. Birkhäuser, Boston, 1992. · Zbl 0816.49001
[7] A. Defranceschi: G-convergence of cyclically monotone operators. Asymptotic Anal. 2 (1989), 21–37. · Zbl 0681.47023
[8] T. Champion, L. de Pascale: Asymptotic behaviour of nonlinear eigenvalue problems involving p-Laplacian type operators. Proc. R. Soc. Edinb. 37A (2007), 1179–1195. · Zbl 1134.35013
[9] N. Dunford, J.T. Schwartz: Linear Operators. Part 1: General Theory. John Wiley & Sons, New York, 1957.
[10] Y. Efendiev A. Pankov: Numerical homogenization of nonlinear random parabolic operators. Multiscale Model. Simul. 2 (2004), 237–268. · Zbl 1181.76113
[11] P. Lindqvist: On a nonlinear eigenvalue problem. Fall School in Analysis, Jyväskylä 1994, Finland. Report 68. Univ. Jyväskylä, Jyväskylä, 1995, pp. 33–54. · Zbl 0838.35094
[12] N. Svanstedt: G-convergence of parabolic operators. Nonlinear Anal., Theory Methods Appl. 36 (1999), 807–842. · Zbl 0933.35020
[13] N. Svanstedt: Multiscale stochastic homogenization of monotone operators. Netw. Heterog. Media 2 (2007), 181–192. · Zbl 1140.35341
[14] N. Svanstedt: Multiscale stochastic homogenization of convection-diffusion equations. Appl. Math. 53 (2008), 143–155. · Zbl 1199.35017
[15] V.V. Zhikov, S.M. Kozlov, O.A. Oleinik: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin, 1994.
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