Geodesic
Stochastic homogenization of a class of monotone eigenvalue problems
Applications of Mathematics, Tome 55 (2010) no. 5, pp. 385-404 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.1007/s10492-010-0014-8
Classification : 35B27, 35B40, 35J25, 35J62, 35J92, 35P30
Keywords: stochastic; homogenization; eigenvalue 
@article{10_1007_s10492_010_0014_8,
     author = {Svanstedt, Nils},
     title = {Stochastic homogenization of a class of monotone eigenvalue problems},
     journal = {Applications of Mathematics},
     pages = {385--404},
     year = {2010},
     volume = {55},
     number = {5},
     doi = {10.1007/s10492-010-0014-8},
     mrnumber = {2737719},
     zbl = {1224.35026},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-010-0014-8/}
}
TY  - JOUR
AU  - Svanstedt, Nils
TI  - Stochastic homogenization of a class of monotone eigenvalue problems
JO  - Applications of Mathematics
PY  - 2010
SP  - 385
EP  - 404
VL  - 55
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10492-010-0014-8/
DO  - 10.1007/s10492-010-0014-8
LA  - en
ID  - 10_1007_s10492_010_0014_8
ER  - 
%0 Journal Article
%A Svanstedt, Nils
%T Stochastic homogenization of a class of monotone eigenvalue problems
%J Applications of Mathematics
%D 2010
%P 385-404
%V 55
%N 5
%U http://geodesic.mathdoc.fr/articles/10.1007/s10492-010-0014-8/
%R 10.1007/s10492-010-0014-8
%G en
%F 10_1007_s10492_010_0014_8
Svanstedt, Nils. Stochastic homogenization of a class of monotone eigenvalue problems. Applications of Mathematics, Tome 55 (2010) no. 5, pp. 385-404. doi: 10.1007/s10492-010-0014-8

Cité par Sources :