Abstract

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			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. Let a denote the conductivity inside the inhomogeneity. The voltage potential converges (in the far field) to a limit ``background'' potential, which is independent of the conductivity a; this convergence (and for that matter the approximation rate of any finite number of terms in the asymtotic expansion) is uniform with respect to a. The existence of the first two terms of the asymptotic expansion carries over to a situation much more general than that of a finite collection of diametrically small inhomogeneities, namely that of an arbitrary set whose Lebesgue measure converges to zero. The convergence statement for the voltage potential is now modulo the extraction of a subsequence, and so it is really a compactness result. Furthermore the convergence is not generally uniform with respect to the inhomogeneity conductivity, a. Thin inhomogeneities, whose limit set is a smooth, codimension 1 manifold, are indeed examples of inhomogeneities for which the convergence to the background potential, or the standard expansion cannot be valid uniformly in a. 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. This boundary, however, does not shrink to a single point when the thickness goes to 0, as is the case when the inhomogeneity is of small radius, but rather it ``converges" to a codimension 1 manifold sigma, which has positive capacity. Neither the problem with homogeneous Neumann boundary condition nor the one with constant Dirichlet condition on sigma has the background potential as its solution; consequently, the convergence of the voltage potential towards the background potential cannot take place uniformly in a. The purpose of this paper is to find a ``simple" replacement for the background potential, with the properties that: (1) this replacement may be (simply) calculated from the limiting domain Omega minus sigma, the boundary data on the boundary of Omega, and the right hand side, (2) this replacement depends on the thickness of the inhomogeneity and the conductivity, a, through its boundary conditions on sigma, (3) the difference between this replacement and the true voltage potential converges to 0 uniformly in a, as the inhomogeneity thickness tends to 0. View/Download Publication in pdf format Back to Preprint list Michael Vogelius 2-13-98