Stochastic homogenization of a class of monotone eigenvalue problems

Svanstedt, Nils. "Stochastic homogenization of a class of monotone eigenvalue problems." Applications of Mathematics 55.5 (2010): 385-404. <http://eudml.org/doc/116469>.

@article{Svanstedt2010,
abstract = {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 $\lbrace \lambda _\varepsilon ^\{\omega ,k\},u^\{\omega ,k\}_\varepsilon \rbrace $ of $k$th eigenpairs converges to the $k$th eigenpair $\lbrace \lambda ^k,u^k\rbrace $ 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.},
author = {Svanstedt, Nils},
journal = {Applications of Mathematics},
keywords = {stochastic; homogenization; eigenvalue; stochastic; homogenization; eigenvalue},
language = {eng},
number = {5},
pages = {385-404},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Stochastic homogenization of a class of monotone eigenvalue problems},
url = {http://eudml.org/doc/116469},
volume = {55},
year = {2010},
}

TY - JOUR
AU - Svanstedt, Nils
TI - Stochastic homogenization of a class of monotone eigenvalue problems
JO - Applications of Mathematics
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 5
SP - 385
EP - 404
AB - 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 $\lbrace \lambda _\varepsilon ^{\omega ,k},u^{\omega ,k}_\varepsilon \rbrace $ of $k$th eigenpairs converges to the $k$th eigenpair $\lbrace \lambda ^k,u^k\rbrace $ 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.
LA - eng
KW - stochastic; homogenization; eigenvalue; stochastic; homogenization; eigenvalue
UR - http://eudml.org/doc/116469
ER -