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Abstract

Asymptotic approximations of voltage potentials in the presence of diametrically small inhomogeneities are well studied. In particular it is known that one may construct approximations that are accurate to any order (in the diameter) uniformly in the conductivity of the inhomogeneity. The corresponding problem for thin inhomogeneities is not so well understood, in particular as concerns uniformity of the approximations. If the conductivity degenerates to 0 or goes to infinity as the width of the inhomogeneity goes to zero, the voltage potential may converge to different limiting solutions, and so the construction of uniform approximations is not straightforward. For the case of thin two dimensional inhomogeneities with closed mid-curves such approximations were constructed and rigorously verified in (Chinese Annals of Mathematics, Series B 38 (2017) 293–344). The analysis relied heavily on the regularity of the approximate solutions. In this two part paper we continue this line of research, by showing that the same approximations remain valid, even when the mid-curve is open, and the corresponding approximate solutions have singularities at the endpoints of the curve.

Keywords

Elliptic boundary-value problem thin inhomogeneity uniform approximation

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References

Ammari ,

Beretta and

Francini, Reconstruction of thin conductivity imperfections, Applicable Analysis 83 (2004), 63–76. doi:10.1080/00036810310001620090.

Ammari,

Kang,

Lee and

Lim, Enhancement of near cloaking using generalized polarization tensors vanishing structures. Part I: The conductivity problem, Communications in Mathematical Physics 317 (2013), 253–266. doi:10.1007/s00220-012-1615-8.

Beretta and

Francini, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of thin inhomogeneities, in: Inverse Problems: Theory and Applications, Contemp. Math., Vol. 333, American Mathematical Society, Providence, RI, 2003.

Beretta,

Francini and

M.S.

Vogelius, 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. doi:10.1016/S0021-7824(03)00081-3.

Bergh and

Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, Vol. 223, Springer-Verlag, Berlin/Heidelberg/New York, 1976.

Brühl,

Hanke and

Pidcock, Crack detection using electrostatic measurements, M2AN Math. Model. Numer. Anal. 35 (2001), 595–605. doi:10.1051/m2an:2001128.

Bryan and

M.S.

Vogelius, A computational algorithm to determine crack locations from electrostatic boundary measurements. The case of multiple cracks, Int. J. Engng. Sci. 32 (1994), 579–603. doi:10.1016/0020-7225(94)90020-5.

Buttazzo and

R.V.

Kohn, Reinforcement by a thin layer with oscillating thickness, Appl. Math. Optim. 16 (1987), 247–261. doi:10.1007/BF01442194.

Capdeboscq and

M.S.

Vogelius, 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. doi:10.1051/m2an:2003014.

10.

Capdeboscq and

M.S.

Vogelius, Pointwise polarization tensor bounds, and applications to voltage perturbations caused by thin inhomogeneities, Asymptotic Analysis 50 (2006), 175–204.

11.

D.J.

Cedio-Fengya,

Moskow and

M.S.

Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems 14 (1998), 553–595. doi:10.1088/0266-5611/14/3/011.

12.

Charnley, Uniform asymptotic approximation of solutions to the conductivity problem with thin open filaments, PhD Thesis, Rutgers University, 2019.

13.

Charnley and

M.S.

Vogelius, A uniformly valid model for the limiting behaviour of voltage potentials in the presence of thin inhomogeneities II. A local energy approximation result, Asymptotic Analysis (2019).

14.

Costabel and

Dauge, A singularly perturbed mixed boundary value problem, Communications in Partial Differential Equations 21 (2010), 1919–1949.

15.

Dapogny and

M.S.

Vogelius, Uniform asymptotic expansion of the voltage potential in the presence of thin inhomogeneities with arbitrary conductivity, Chinese Annals of Mathematics, Series B 38 (2017), 293–344. doi:10.1007/s11401-016-1072-3.

16.

Friedman and

Vogelius, Determining cracks by boundary measurements, Indiana University Mathematics Journal 38 (1989), 527–556. doi:10.1512/iumj.1989.38.38025.

17.

Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing, Boston, 1985.

18.

R.V.

Kohn,

Shen,

M.S.

Vogelius and

M.I.

Weinstein, Cloaking via change of variables in electrical impedance tomography, Inverse Problems 24 (2008), 015016, 21pp.

19.

J.-L.

Lions and

Magenes, Non-Homogeneous Boundary Value Problems and Applications I, Grundlehren der Mathematischen Wissenschaften, Vol. 181, Springer-Verlag, Berlin/Heidelberg/New York, 1972.

20.

H.-M.

Nguyen and

M.S.

Vogelius, 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. doi:10.1016/j.anihpc.2009.03.005.

21.

Perrussel and

Poignard, Asymptotic expansion of steady-state potential in a high contrast medium with a thin resistive layer, Applied Mathematics and Computation 221 (2013), 48–65. doi:10.1016/j.amc.2013.06.047.

A uniformly valid model for the limiting behaviour of voltage potentials in the presence of thin inhomogeneities I. The case of an open mid-curve

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