Abstract
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 Ωσ, 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.
Similar content being viewed by others
References
Acerbi, E. and Buttazzo, G., Reinforcement problems in the calculus of variations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4, 1986, 273–284.
Allaire, G., Dapogny, C., Delgado, G. and Michailidis, G., Multi-phase optimization via a level set method, ESAIM: Control, Optimization and Calculus of Variations, 20(2), 2014, 576–611.
Ammari, H., Beretta, E. and Francini, E., Reconstruction of thin conductivity imperfections, Appl. Anal., 83, 2004, 63–76.
Ammari, H. and Kang, H., Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, Vol. 1846, Springer-Verlag, New York, 2004.
Attouch, H., Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman, London, 1984.
Beretta, E. and Francini, E., Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of thin inhomogeneities, Inverse Problems: Theory and Applications, Contemp. Math., 333, AMS, Providence, RI,2003.
Beretta, E., Francini, E. and Vogelius, M., Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities,A rigorous error analysis, J. Math. Pures Appl., 82, 2003, 1277–1301.
Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2000.
Buttazzo, G. and Kohn, R. V., Reinforcement by a thin layer with oscillating thickness, Appl. Math. Optim., 16, 1987, 247–261.
Capdeboscq, Y. and Vogelius, M. S., A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, ESAIM: Math. Mod. Numer. Anal., 37, 2003, 159–173.
Cedio-Fengya, D. J., Moskow, S. and Vogelius, M. S., Identification of conductivity imperfections of small diameter by boundary measurements, Continuous dependence and computational reconstruction, Inverse Problems, 14, 1998, 553–595.
Chavel, I., Riemannian Geometry, A Modern Introduction, 2nd Edition, Cambridge University Press, Cambridge, 2006.
Delfour, M. C. and Zolesio, J.-P., Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd Edition, SIAM, Philadelphia, 2011.
Ekeland, I. and Temam, R., Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976.
Evans, L. C. and Gariepy, R. F., Measure Theory and Fine Properties of Functions, CRC Press, Florida, 1992.
Folland, G. B., Introduction to Partial Differential Equations, 2nd Edition, Princeton University Press, Princeton, 1995.
Friedman, A. and Vogelius, M., Determining cracks by boundary measurements, Indiana Univ. Math. J., 38, 1989, 527–556.
Henrot, A. and Pierre, M., Variation et Optimisation de Formes, une Analyse Géométrique, Springer-Verlag, New York, 2005.
Kohn, R. V. and Milton, G. W., On bounding the effective conductivity of anisotropic composites, Homogenization and Effective Moduli of Materials and Media, J. L. Ericksen, D. Kinderlehrer, R. Kohn and J.-L. Lions (eds.), IMA Volumes in Mathematics and Its Applications, 1, Springer-Verlag, New York, 1986, 97–125.
Morgenstern, D. and Szabo, I., Vorlesungen über theoretische Mechanik, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 112, Springer-Verlag, Berlin/Göttingen/Heidelberg, 1961.
Nguyen, H.-M. and Vogelius, M. S., A representation formula for the voltage perturbations caused by diametrically small conductivity inhomogeneities, Proof of uniform validity, Annales de l’Institut Henri Poincaré Non Linear Analysis, 26, 2009, 2283–2315.
Perrussel, R. and Poignard, C., Asymptotic expansion of steady-state potential in a high contrast medium with a thin resistive layer, Appl. Math. Comp., 221, 2013, 48–65.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Haim Brezis on the occasion of his 70th birthday
This work was 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.
Rights and permissions
About this article
Cite this article
Dapogny, C., Vogelius, M.S. Uniform asymptotic expansion of the voltage potential in the presence of thin inhomogeneities with arbitrary conductivity. Chin. Ann. Math. Ser. B 38, 293–344 (2017). https://doi.org/10.1007/s11401-016-1072-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-016-1072-3