{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:38:34Z","timestamp":1787323114598,"version":"build-2736575974"},"reference-count":15,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Math. Anal."],"published-print":{"date-parts":[[2003,1]]},"abstract":"\n This article is devoted to the asymptotic study, as $\\varepsilon\\to 0$, of the Dirichlet problem \\[ \\left\\{\\begin{array}{@{}rll} -\\,\\mbox{div}\\left(A_\\varepsilon({\\textstyle{x\\over\\varepsilon}})\\nabla u_\\varepsilon\\right) & \\kern -.5em=f & \\mbox{in }\\Omega, \\\\*[.4em] u_\\varepsilon & \\kern -.5em=0 & \\mbox{on }\\partial\\Omega, \\end{array} \\right. \\] where $\\Omega$ is an x\n 3<\/jats:sub>\n -axis bounded open cylinder of ${\\mathbb R}^3$, and $A_\\varepsilon$ is a positive measurable function which does not depend on the variable x\n 3<\/jats:sub>\n , periodic with respect to the two-dimensional torus Y\n 2<\/jats:sub>\n . The conductivity $A_\\varepsilon$ is not uniformly bounded in an open set of small measure $Q_\\varepsilon\\subset Y\n 2<\/jats:sub>\n and is equal to 1 elsewhere.\n <\/jats:p>\n We propose a new approach to solving this high-conductivity homogenization problem. It is based on the study of the asymptotic behavior of the periodic spectral problem weighted by the conductivity function $A_\\varepsilon$: \\[ -\\,\\mbox{div}\\left(A_\\varepsilon\\nabla V_{k,\\varepsilon}\\right) =\\Lambda_k(\\varepsilon)\\,A_\\varepsilon\\,V_{k,\\varepsilon}\\quad\\mbox{in }Y_2,\\quad k\\in{\\mathbb N}, \\] where the eigenfunctions $V_{k,\\varepsilon}$ are $Y_2$-periodic.<\/jats:p>\n On the one hand, under suitable conditions on $Q_\\varepsilon$ we prove that nonlocal effects appear through a coupling in the limit problem if and only if the sequence $({\\Lambda_1(\\varepsilon)\\over\\varepsilon^2})_{\\varepsilon > 0}$ is bounded, where $\\Lambda_1(\\varepsilon)$ is the first nonzero eigenvalue of the previous spectral problem.<\/jats:p>\n On the other hand, when $Q_\\varepsilon$ is composed of N smooth connected open subsets of small diameter, we prove that the limit problem is a coupled system of second order linear PDEs whose size is $n\\leq N+1$. The number n is equal to the smallest integer such that the sequence $({\\Lambda_n(\\varepsilon)\\over\\varepsilon^2})_{\\varepsilon > 0}$ tends to $+\\infty$ as $\\varepsilon$ tends to 0. We illustrate this result by studying the case of N=2 highly conducting cylinders in the period cell of the same radius $r_\\varepsilon \\ll 1$ and separated by distance $d_\\varepsilon>0$.<\/jats:p>","DOI":"10.1137\/s0036141001398666","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"33-60","source":"Crossref","is-referenced-by-count":17,"title":["Homogenization of High-Conductivity Periodic Problems: Application to a General Distribution of One-Directional Fibers"],"prefix":"10.1137","volume":"35","author":[{"given":"Marc","family":"Briane","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,8,1]]},"reference":[{"key":"R1","first-page":"407","volume":"26","author":"Bellieud Michel","year":"1998","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci. 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Khruslov,\n Homogenized models of composite media\n , in Composite Media and Homogenization Theory, G. Dal Maso and G. F. Dell\u2019Antonio, eds., Progr. Nonlinear Differential Equations Appl. 5, Birkh\u00e4user Boston, Boston, 1991, pp. 159\u2013182.","DOI":"10.1007\/978-1-4684-6787-1_10"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1006\/jfan.1994.1093"},{"key":"R14","unstructured":"Fran\u00e7ois Murat, Luc Tartar, H\u2010convergence, Progr. Nonlinear Differential Equations Appl., Vol. 31, Birkh\u00e4user Boston, Boston, MA, 1997, 21\u2013431493039"},{"key":"R15","unstructured":"Fran\u00e7ois Murat, Luc Tartar, H\u2010convergence, Progr. 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