TY - JOUR

T1 - Uniform bound and convergence for elliptic homogenization problems

AU - Yeh, Li-Ming

N1 - Publisher Copyright:
© 2015, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - Uniform bound and convergence for the solutions of elliptic homogenization problems are concerned. The problem domain has a periodic microstructure; it consists of a connected subregion with high permeability and a disconnected matrix block subset with low permeability. Let ϵ∈ (0 , 1) denote the size ratio of the period to the whole domain, and let ω2∈ (0 , 1) denote the permeability ratio of the disconnected matrix block subset to the connected subregion. For elliptic equations with diffusion depending on the permeability, the elliptic solutions are smooth in the connected subregion but change rapidly in the disconnected matrix block subset. More precisely, the solutions in the connected subregion can be bounded uniformly in ω, ϵ in Hölder norm, but not in the matrix block subset. It is known that the elliptic solutions converge to a solution of some homogenized elliptic equation as ω, ϵ converge to 0. In this work, the Lp convergence rate for p∈ (2 , ∞] is derived. Depending on strongly coupled or weakly coupled case, the convergence rate is related to the factors ω,ϵ,ωϵ for the former and related to the factors ω, ϵ for the latter.

AB - Uniform bound and convergence for the solutions of elliptic homogenization problems are concerned. The problem domain has a periodic microstructure; it consists of a connected subregion with high permeability and a disconnected matrix block subset with low permeability. Let ϵ∈ (0 , 1) denote the size ratio of the period to the whole domain, and let ω2∈ (0 , 1) denote the permeability ratio of the disconnected matrix block subset to the connected subregion. For elliptic equations with diffusion depending on the permeability, the elliptic solutions are smooth in the connected subregion but change rapidly in the disconnected matrix block subset. More precisely, the solutions in the connected subregion can be bounded uniformly in ω, ϵ in Hölder norm, but not in the matrix block subset. It is known that the elliptic solutions converge to a solution of some homogenized elliptic equation as ω, ϵ converge to 0. In this work, the Lp convergence rate for p∈ (2 , ∞] is derived. Depending on strongly coupled or weakly coupled case, the convergence rate is related to the factors ω,ϵ,ωϵ for the former and related to the factors ω, ϵ for the latter.

KW - Elliptic homogenization problem

KW - Permeability

KW - Two-phase media

UR - http://www.scopus.com/inward/record.url?scp=84940979745&partnerID=8YFLogxK

U2 - 10.1007/s10231-015-0530-y

DO - 10.1007/s10231-015-0530-y

M3 - Article

AN - SCOPUS:84940979745

SN - 0373-3114

VL - 195

SP - 1803

EP - 1832

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

IS - 6

ER -