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

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• 作者：Shiah-Sen Wang ^{swang@math.nctu.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; Li-Ming Yeh ; ^{liming@math.nctu.edu.tw" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J15 ; 35J25 ; 35J70 ; 35R05 ; 35R60
• 刊名：Nonlinear Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：148
• 期：Complete
• 页码：61-87
• 全文大小：910 K

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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302188&_mathId=si1.gif&_user=111111111&_pii=S0362546X16302188&_rdoc=1&_issn=0362546X&md5=32f4fe1ffdf15b2f2cdbcd7ec53f092f" title="Click to view the MathML source">ω,ϵ,λclass="mathContainer hidden">class="mathCode">$\omega ,ϵ,\lambda$ for the elliptic solutions is derived, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302188&_mathId=si2.gif&_user=111111111&_pii=S0362546X16302188&_rdoc=1&_issn=0362546X&md5=6f05e21622d709b0d4f5fdc68ca754f6" title="Click to view the MathML source">ωclass="mathContainer hidden">class="mathCode">$\omega$ is a realization of the random ensemble, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302188&_mathId=si3.gif&_user=111111111&_pii=S0362546X16302188&_rdoc=1&_issn=0362546X&md5=6a8bdd810bf02a853e1c40523aca702f" title="Click to view the MathML source">ϵ∈(0,1]class="mathContainer hidden">class="mathCode">$ϵ\in (0,1]$ is the length scale of the interfaces, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302188&_mathId=si4.gif&_user=111111111&_pii=S0362546X16302188&_rdoc=1&_issn=0362546X&md5=8778151219c8c34a43d84408af1cd4af" title="Click to view the MathML source">λ²∈(0,1]class="mathContainer hidden">class="mathCode">${\lambda }^2\in (0,1]$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302188&_mathId=si1.gif&_user=111111111&_pii=S0362546X16302188&_rdoc=1&_issn=0362546X&md5=32f4fe1ffdf15b2f2cdbcd7ec53f092f" title="Click to view the MathML source">ω,ϵ,λclass="mathContainer hidden">class="mathCode">$\omega ,ϵ,\lambda$ 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.