A lower bound for the principal eigenvalue of the Stokes operator in a random domain
Abstract
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.
Keywords
Cite
@article{arxiv.0804.1415,
title = {A lower bound for the principal eigenvalue of the Stokes operator in a random domain},
author = {V. V. Yurinsky},
journal= {arXiv preprint arXiv:0804.1415},
year = {2008}
}
Comments
Published in at http://dx.doi.org/10.1214/07-AIHP136 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org)