文摘
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 2546X16302188&_mathId=si1.gif&_user=111111111&_pii=S0362546X16302188&_rdoc=1&_issn=0362546X&md5=32f4fe1ffdf15b2f2cdbcd7ec53f092f" title="Click to view the MathML source">ω,ϵ,λ for the elliptic solutions is derived, where 2546X16302188&_mathId=si2.gif&_user=111111111&_pii=S0362546X16302188&_rdoc=1&_issn=0362546X&md5=6f05e21622d709b0d4f5fdc68ca754f6" title="Click to view the MathML source">ω is a realization of the random ensemble, 2546X16302188&_mathId=si3.gif&_user=111111111&_pii=S0362546X16302188&_rdoc=1&_issn=0362546X&md5=6a8bdd810bf02a853e1c40523aca702f" title="Click to view the MathML source">ϵ∈(0,1] is the length scale of the interfaces, and 2546X16302188&_mathId=si4.gif&_user=111111111&_pii=S0362546X16302188&_rdoc=1&_issn=0362546X&md5=8778151219c8c34a43d84408af1cd4af" title="Click to view the MathML source">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 2546X16302188&_mathId=si1.gif&_user=111111111&_pii=S0362546X16302188&_rdoc=1&_issn=0362546X&md5=32f4fe1ffdf15b2f2cdbcd7ec53f092f" title="Click to view the MathML source">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.
