A Hölder estimate for non-uniform elliptic equations in a random medium. (January 2017)

• Record Type: Journal Article
• Authors: Wang, Shiah-Sen; Yeh, Li-Ming
• Is Part Of: Nonlinear analysis. Volume 148(2017)
• Page Start: 61
• Page End: 87
• Publication Date: 2017-01
• Subjects: 35J15 -- 35J25 -- 35J70 -- 35R05 -- 35R60; Random media -- Conductivity -- Stationary–ergodic -- Realization -- Diffeomorphism; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.09.009
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Uniform regularity for second order elliptic equations in a highly heterogeneous random medium is concerned. The medium is separated by a random ensemble of simply closed interfaces into a connected sub-region with high conductivity and a disconnected subset with low conductivity. The elliptic equations, whose diffusion coefficients depend on the conductivity, have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Without a stationary–ergodic assumption, a uniform Hölder estimate in ω, ϵ, λ for the elliptic solutions is derived, where ω is a realization of the random ensemble, ϵ ∈ ( 0, 1 ] is the length scale of the interfaces, and λ 2 ∈ ( 0, 1 ] is the conductivity ratio of the disconnected subset to the connected sub-region. Results show that if external sources are small enough in the disconnected subset, the uniform Hölder estimate in ω, ϵ, λ holds in the whole domain. If not, it holds only in the connected sub-region. Meanwhile, the elliptic solutions change rapidly in the disconnected subset.