Uniform Asymptotic Expansion of the Voltage Potential in the Presence of Thin Inhomogeneities with Arbitrary Conductivity  

作　　者：Charles DAPOGNY Michael S.VOGELIUS 

机构地区：[1]Laboratoire Jean Kuntzmann,CNRS,Université Grenoble-Alpes [2]Department of Mathematics,Rutgers University

出　　处：《Chinese Annals of Mathematics,Series B》2017年第1期293-344,共52页数学年刊（B辑英文版）

基　　金：partially supported by NSF grant DMS-12-11330 while CD was a postdoctoral visitor at Rutgers University and by the NSF IR/D program while MSV served at the National Science Foundation

摘　　要：Asymptotic expansions of the voltage potential in terms of the "radius" of a diametrically small(or several diametrically small) material inhomogeneity(ies) are by now quite well-known. Such asymptotic expansions for diametrically small inhomogeneities are uniform with respect to the conductivity of the inhomogeneities.In contrast, thin inhomogeneities, whose limit set is a smooth, codimension 1 manifold,σ, are examples of inhomogeneities for which the convergence to the background potential,or the standard expansion cannot be valid uniformly with respect to the conductivity, a, of the inhomogeneity. Indeed, by taking a close to 0 or to infinity, one obtains either a nearly homogeneous Neumann condition or nearly constant Dirichlet condition at the boundary of the inhomogeneity, and this difference in boundary condition is retained in the limit.The purpose of this paper is to find a "simple" replacement for the background potential, with the following properties:(1) This replacement may be(simply) calculated from the limiting domain Ω\σ, the boundary data on the boundary of Ω, and the right-hand side.(2) This replacement depends on the thickness of the inhomogeneity and the conductivity,a, through its boundary conditions on σ.(3) The difference between this replacement and the true voltage potential converges to 0 uniformly in a, as the inhomogeneity thickness tends to 0.Asymptotic expansions of the voltage potential in terms of the ＂radius＂ of a diametrically small（or several diametrically small） material inhomogeneity（ies） are by now quite well-known. Such asymptotic expansions for diametrically small inhomogeneities are uniform with respect to the conductivity of the inhomogeneities.In contrast, thin inhomogeneities, whose limit set is a smooth, codimension 1 manifold,σ, are examples of inhomogeneities for which the convergence to the background potential,or the standard expansion cannot be valid uniformly with respect to the conductivity, a, of the inhomogeneity. Indeed, by taking a close to 0 or to infinity, one obtains either a nearly homogeneous Neumann condition or nearly constant Dirichlet condition at the boundary of the inhomogeneity, and this difference in boundary condition is retained in the limit.The purpose of this paper is to find a ＂simple＂ replacement for the background potential, with the following properties：（1） This replacement may be（simply） calculated from the limiting domain Ω／σ, the boundary data on the boundary of Ω, and the right-hand side.（2） This replacement depends on the thickness of the inhomogeneity and the conductivity,a, through its boundary conditions on σ.（3） The difference between this replacement and the true voltage potential converges to 0 uniformly in a, as the inhomogeneity thickness tends to 0.

关 键 词：Uniform asymptotic expansions Conductivity problem Thin inhomogeneities 

分 类 号：O441.1[理学—电磁学]