On the smallest eigenvalue of the Stokes operator in a domain with fine-grained random boundary

This article deals with a problem arising in localization of the principal eigenvalue (PE) of the Stokes operator under the Dirichlet condition on the fine-grained random boundary of a domain contained in a cube of size t >> 1. The random microstructure is assumed identically distributed in distinct unit cubic cells and, in essence, independent. In this setting, the asymptotic behavior of the PE as t -> infinity is deterministic: it proves possible to find nonrandom upper and lower bounds on the PE which apply with probability that converges to 1. It was proved earlier that in two dimensions the nonrandom unilateral bounds on the PE can be chosen asymptotically equivalent, which implies the convergence in probability to a nonrandom limit of the appropriately normalized PE. The present article extends this result to higher dimensions.