T. Abboud and J. Nédélec, Electromagnetic waves in an inhomogeneous medium, Journal of Mathematical Analysis and Applications, vol.164, issue.1, pp.40-58, 1992.
DOI : 10.1016/0022-247X(92)90144-3

S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Communications on Pure and Applied Mathematics, vol.29, issue.4, pp.623-727, 1959.
DOI : 10.1002/cpa.3160120405

S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Communications on Pure and Applied Mathematics, vol.18, issue.1, pp.35-92, 1964.
DOI : 10.1002/cpa.3160170104

M. S. Agranovi?, Spectral properties of elliptic pseudodifferential operators on a closed curve, Funktsional. Anal. i Prilozhen, vol.13, issue.4, pp.54-56, 1979.

H. Ammari and S. Moskow, Asymptotic expansions for eigenvalues in the presence of small inhomogeneities, Mathematical Methods in the Applied Sciences, vol.61, issue.1, pp.67-75, 2003.
DOI : 10.1002/mma.343

H. Ammari, E. Beretta, and E. Francini, Reconstruction of thin conductivity imperfections, II. The case of multiple segments, Applicable Analysis, vol.362, issue.1-3, pp.87-105, 2006.
DOI : 10.1088/0266-5611/20/6/010

H. Ammari and H. Kang, Properties of the Generalized Polarization Tensors, Multiscale Modeling & Simulation, vol.1, issue.2, pp.335-348, 2003.
DOI : 10.1137/S1540345902404551

H. Ammari and H. Kang, Reconstruction of conductivity inhomogeneities of small diameter via boundary measurements, Inverse problems and spectral theory, pp.23-32, 2004.
DOI : 10.1090/conm/348/06311

H. Ammari and F. Santosa, Guided waves in a photonic bandgap structure with a line defect, SIAM J. Appl. Math, vol.64, issue.6, pp.2018-2033, 2004.

H. Antoine, A. Barucq, and . Bendali, Bayliss???Turkel-like Radiation Conditions on Surfaces of Arbitrary Shape, Journal of Mathematical Analysis and Applications, vol.229, issue.1, pp.184-211, 1999.
DOI : 10.1006/jmaa.1998.6153

C. Balanis, Advanced Engineering Electromagnetics, 1989.

E. Beretta and E. Francini, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of thin inhomogeneities, Inverse problems : theory and applications, pp.49-62, 2002.
DOI : 10.1090/conm/333/05953

E. Beretta, E. Francini, and M. S. Vogelius, Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities. A??rigorous error analysis, Journal de Math??matiques Pures et Appliqu??es, vol.82, issue.10, pp.821277-1301, 2003.
DOI : 10.1016/S0021-7824(03)00081-3

E. Beretta, A. Mukherjee, and M. Vogelius, Asymptotic formulas for steady state voltage potentials in the presence of conductivity imperfections of small area, Zeitschrift f??r angewandte Mathematik und Physik, vol.52, issue.4, pp.543-572, 2001.
DOI : 10.1007/PL00001561

J. Essex, . Bond, . Xu, S. C. Li, D. Hagness et al., Microwave imaging via space-time beamforming for early detection of breast cancer, IEEE Transactions on Antennas and Propagation, issue.8, p.51, 2003.

H. Brézis, Analyse fonctionnelle Collection Mathématiques Appliquées pour la Ma??triseMa??trise. [Collection of Applied Mathematics for the Master's Degree], Théorie et applications. [Theory and applications], 1983.

Y. Capdeboscq and M. S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, ESAIM: Mathematical Modelling and Numerical Analysis, vol.37, issue.1, pp.159-173, 2003.
DOI : 10.1051/m2an:2003014

L. A. Carvalho, S. , and J. Hounie, Estimates for the Poisson kernel and Hardy spaces on compact manifolds, Journal of Mathematical Analysis and Applications, vol.299, issue.2, pp.465-493, 2004.
DOI : 10.1016/j.jmaa.2004.03.080

M. Cessenat, Mathematical methods in electromagnetism, of Series on Advances in Mathematics for Applied Sciences Linear theory and applications, 1996.
DOI : 10.1142/2938

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Applied Mathematical Sciences, vol.93, 1992.
DOI : 10.1007/978-1-4614-4942-3

B. Doubrovine, S. Novikov, and A. Fomenko, Géométrie contemporaine. Méthodes et applications . I Géométrie des surfaces, des groupes de transformations et des champs. [Geometry of surfaces, groups of transformations and fields], 1982.

B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern geometry?methods and applications . Part I, volume 93 of Graduate Texts in Mathematics The geometry of surfaces, transformation groups, and fields, 1992.
DOI : 10.1007/978-1-4612-4474-5

P. Dular, C. Geuzaine, F. Henrotte, and W. Legros, A general environment for the treatment of discrete problems and its application to the finite element method, IEEE Transactions on Magnetics, vol.34, issue.5, pp.3395-3398, 1998.
DOI : 10.1109/20.717799

E. C. Fear and M. A. Stuchly, Modeling assemblies of biological cells exposed to electric fields, IEEE Transactions on Biomedical Engineering, vol.45, issue.10, pp.1259-1271, 1998.
DOI : 10.1109/10.720204

E. C. Fear and M. A. Stuchly, A novel equivalent circuit model for gap-connected cells, Physics in Medicine and Biology, vol.43, issue.6, pp.1439-1448, 1998.
DOI : 10.1088/0031-9155/43/6/005

H. Flanders, Differential forms wih applications to the physical sciences, 1963.

K. R. Foster and H. P. Schwan, Dielectric properties of tissues and biological materials : a critical review, CRC in Biomedical Engineering, vol.17, issue.1, pp.25-104, 1989.

S. Gabriel, R. Lau, and E. Corthout, The dielectric properties of biological tissues: I. Literature survey, Physics in Medicine and Biology, vol.41, issue.11, pp.2231-2249, 1996.
DOI : 10.1088/0031-9155/41/11/001

S. Gabriel, R. Lau, and C. Gabriel, The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz, Physics in Medicine and Biology, vol.41, issue.11, pp.2251-2269, 1996.
DOI : 10.1088/0031-9155/41/11/002

S. Gabriel, R. Lau, and C. Gabriel, The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues, Physics in Medicine and Biology, vol.41, issue.11, pp.2271-2293, 1996.
DOI : 10.1088/0031-9155/41/11/003

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences, 1983.

B. Peter, J. V. Gilkey, J. Leahy, and . Park, Spectral geometry, Riemannian submersions , and the Gromov-Lawson conjecture, Studies in Advanced Mathematics. Chapman & Hall/CRC, 1999.

V. Girault and P. Raviart, Finite element methods for Navier-Stokes equations Theory and algorithms, of Springer Series in Computational Mathematics, 1986.

L. Halpern and J. Rauch, Absorbing boundary conditions for diffusion equations, Numerische Mathematik, vol.71, issue.2, pp.185-224, 1995.
DOI : 10.1007/s002110050141

L. Halpern and J. Rauch, Error analysis for absorbing boundary conditions, Numerische Mathematik, vol.176, issue.4, pp.459-467, 1987.
DOI : 10.1007/BF01397547

D. Hansen and M. Vogelius, High frequency perturbation formulas for the effect of small inhomogeneities, Journal of Physics: Conference Series, vol.135, 2006.
DOI : 10.1088/1742-6596/135/1/012106

D. J. Hansen, C. Poignard, and M. S. Vogelius, Asymptotically precise norm estimates of scattering from a small circular inhomogeneity, Applicable Analysis, vol.316, issue.4, 2006.
DOI : 10.1051/m2an:2000101

URL : https://hal.archives-ouvertes.fr/inria-00334779

E. Hille, Ordinary differential equations in the complex domain, 1997.

L. Hörmander, The analysis of linear partial differential operators. III, volume 274 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences, 1985.

Y. Katznelson, An introduction to harmonic analysis, 1976.

D. Olivier and . Lafitte, Diffraction in the high frequency regime by a thin layer of dielectric material . I. The equivalent impedance boundary condition, SIAM J. Appl. Math, vol.59, issue.3, pp.1028-1052, 1999.

D. Olivier and . Lafitte, Diffraction in the high frequency regime by a thin layer of dielectric material. II. The trace of the wave in the shadow of the obstacle, SIAM J. Appl. Math, vol.59, issue.3, pp.1053-1079, 1999.

. Xu, S. C. Li, and . Hagness, A conformal microwave imaging algorithm for breast cancer detection, IEEE Microwave and wireless components letters, vol.11, issue.3, 2001.

Y. Y. , L. , and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Ration. Mech. Anal, vol.153, issue.2, pp.91-151, 2000.

J. Lions and E. Magenes, Probì emes aux limites non homogènes et applications, Travaux et Recherches Mathématiques, vol.1, issue.17, 1968.

T. Severino and . Melo, Characterizations of pseudodifferential operators on the circle, Proc. Amer. Math. Soc, vol.125, issue.5, pp.1407-1412, 1997.

S. Muñoz, J. L. Sebastián, M. Sancho, and J. M. Miranda, Transmembrane voltage induced on altered erythrocyte shapes exposed to RF fields, Bioelectromagnetics, vol.56, issue.8, pp.631-633, 2004.
DOI : 10.1002/bem.20065

L. Nicolas, N. Burais, F. Buret, O. Fabrègue, L. Krähenbühl et al., Interactions between electromagnetic field and biological tissues : Questions, some answers and future trends, 2003.

A. Nikiforov and V. Ouvarov, ´ Eléments de la théorie des fonctions spéciales. ´ Editions Mir, 1976.

C. Poignard, Impedance boundary condition in a biological cell submitted to a high frequency field

C. Poignard, Asymptotic estimates of the electric field in a biological cell at low frequencies, The 7th International Conference on Mathematical and Numerical Aspects of Waves (WAVES'05, pp.20-24, 2005.

C. Poignard, P. Dular, L. Krähenbühl, L. Nicolas, and M. Schatzman, Méthodes asymptotiques pour le calcul de champs, 5` eme Conférence Européenne sur les Méthodes Numériques en Electromagnétisme, pp.29-30, 2006.

C. Poignard, M. Schatzman, L. Nicolas, L. Krähenbühl, F. Musy et al., Asymptotic estimates of the electric field in a circular biological cell at low frequencies, The 15th Conference on the Computation of Electromagnetic Fields (Compumag'2005), pp.26-30, 2005.

C. Poignard, Asymptotics expansions of the electric field in the biological cell at low frequencies : the circular case

C. Poignard, Rigorous asymptotics for steady state voltage potentials in a bidimensionnal highly contrasted medium. Submitted to M2AN, 2006.

C. Poignard, Rigorous asymptotics for the electric field in tm mode at mid-frequency in a bidimensional medium with a thin layer, Submitted to SIAM J. APPL. MATH, 2006.

G. Pucihar, . Kotnik, D. Vali?, and . Miklav?i?, Numerical Determination of Transmembrane Voltage Induced on Irregularly Shaped Cells, Annals of Biomedical Engineering, vol.32, issue.4, pp.642-652, 2006.
DOI : 10.1007/s10439-005-9076-2

W. Rudin, Real and complex analysis, 1987.

J. Saranen and G. Vainikko, Periodic integral and pseudodifferential equations with numerical approximation, 2002.
DOI : 10.1007/978-3-662-04796-5

J. L. Sebastián, S. Muñoz, M. Sancho, and J. M. Miranda, Analysis of the influence of the cell geometry, orientation and cell proximity effects on the electric field distribution from direct RF exposure, Physics in Medicine and Biology, vol.46, issue.1, pp.213-225, 2001.
DOI : 10.1088/0031-9155/46/1/315

D. Sel, D. Cukjati, D. Batiuskaite, T. Slivnik, L. M. Mir et al., Sequential Finite Element Model of Tissue Electropermeabilization, IEEE Transactions on Biomedical Engineering, vol.52, issue.5, pp.816-827, 2005.
DOI : 10.1109/TBME.2005.845212

URL : https://hal.archives-ouvertes.fr/hal-00319701

M. Elias and . Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, issue.30, 1970.

M. Elias, G. Stein, and N. J. Weiss, Introduction to Fourier analysis on Euclidean spaces, 1971.

M. A. Stuchly and S. S. Stuchly, Electrical properties of biological substances. Biological Effects and Medical Applications of Electromagnetic Energy, 1990.

S. I. Sukharev, V. Klenchin, S. M. Nad-chernomordik, L. V. Serov, and C. Y. , Electroporation and electrophoretic DNA transfer into cells. The effect of DNA interaction with electropores, Biophysical Journal, vol.63, issue.5, pp.1320-1327, 1992.
DOI : 10.1016/S0006-3495(92)81709-5

URL : http://doi.org/10.1016/s0006-3495(92)81709-5

E. Michael, L. Taylor, and . Hormander, Reviews : The Analysis of Linear Partial Differential Operators, Vols I & II, 1985.

F. Trèves, Introduction to pseudodifferential and Fourier integral operators, 1980.
DOI : 10.1007/978-1-4684-8780-0

T. Y. Tsong, Electroporation of cell membranes, Biophys J, issue.60, pp.297-306, 1991.

V. Turunen, Commutator Characterization of Periodic Pseudodifferential Operators, Zeitschrift f??r Analysis und ihre Anwendungen, vol.19, issue.1, pp.95-108, 2000.
DOI : 10.4171/ZAA/940

V. Turunen and G. Vainikko, On symbol analysis of periodic pseudodifferential operators, Journal for Analysis and its Appl, vol.17, issue.1, pp.9-22, 1998.

F. Karl, D. V. Warnick, and . Arnold, Electromagnetic green functions using differential forms, Journal of. Electromagnetic Waves and Applications, vol.10, issue.3, pp.427-438, 1996.

G. N. Watson, A Treatise on the Theory of Bessel Functions, The Mathematical Gazette, vol.18, issue.231, 1944.
DOI : 10.2307/3605513