TY - JOUR

T1 - A Hölder estimate for non-uniform elliptic equations in a random medium

AU - Wang, Shiah-Sen

AU - Yeh, Li-Ming

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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 λ2∈(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.

AB - 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 λ2∈(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.

KW - Conductivity

KW - Diffeomorphism

KW - Random media

KW - Realization

KW - Stationary–ergodic

UR - http://www.scopus.com/inward/record.url?scp=85006914055&partnerID=8YFLogxK

U2 - 10.1016/j.na.2016.09.009

DO - 10.1016/j.na.2016.09.009

M3 - Article

AN - SCOPUS:85006914055

VL - 148

SP - 61

EP - 87

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

ER -