Mathematics – Probability
Scientific paper
2008-04-09
Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques 2008, Vol. 44, No. 1, 1-18
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/07-AIHP136 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiqu
Scientific paper
10.1214/07-AIHP136
This article is dedicated to localization of the principal eigenvalue (PE) of the Stokes operator acting on solenoidal vector fields that vanish outside a large random domain modeling the pore space in a cubic block of porous material with disordered micro-structure. Its main result is an asymptotically deterministic lower bound for the PE of the sum of a low compressibility approximation to the Stokes operator and a small scaled random potential term, which is applied to produce a similar bound for the Stokes PE. The arguments are based on the method proposed by F. Merkl and M. V. W\"{u}trich for localization of the PE of the Schr\"{o}dinger operator in a similar setting. Some additional work is needed to circumvent the complications arising from the restriction to divergence-free vector fields of the class of test functions in the variational characterization of the Stokes PE.
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