We will describe recent some of the recent theoretical progress on making objects invisible to electromagnetic waves based on singular transformations.
@article{JEDP_2007____A5_0, author = {Greenleaf, Allan and Kurylev, Yaroslav and Lassas, Matti and Uhlmann, Gunther}, title = {Mathematics of {Invisibility}}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {5}, pages = {1--11}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2007}, doi = {10.5802/jedp.44}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jedp.44/} }
TY - JOUR AU - Greenleaf, Allan AU - Kurylev, Yaroslav AU - Lassas, Matti AU - Uhlmann, Gunther TI - Mathematics of Invisibility JO - Journées équations aux dérivées partielles PY - 2007 SP - 1 EP - 11 PB - Groupement de recherche 2434 du CNRS UR - http://archive.numdam.org/articles/10.5802/jedp.44/ DO - 10.5802/jedp.44 LA - en ID - JEDP_2007____A5_0 ER -
%0 Journal Article %A Greenleaf, Allan %A Kurylev, Yaroslav %A Lassas, Matti %A Uhlmann, Gunther %T Mathematics of Invisibility %J Journées équations aux dérivées partielles %D 2007 %P 1-11 %I Groupement de recherche 2434 du CNRS %U http://archive.numdam.org/articles/10.5802/jedp.44/ %R 10.5802/jedp.44 %G en %F JEDP_2007____A5_0
Greenleaf, Allan; Kurylev, Yaroslav; Lassas, Matti; Uhlmann, Gunther. Mathematics of Invisibility. Journées équations aux dérivées partielles (2007), article no. 5, 11 p. doi : 10.5802/jedp.44. http://archive.numdam.org/articles/10.5802/jedp.44/
[AP] K. Astala and L. Päivärinta: Calderón’s inverse conductivity problem in the plane. Annals of Math., 163 (2006), 265-299. | MR | Zbl
[ALP] K. Astala, M. Lassas, and L. Päiväirinta, Calderón’s inverse problem for anisotropic conductivity in the plane, Comm. Partial Diff. Eqns. 30 (2005), 207–224. | MR | Zbl
[ALP2] K. Astala, M. Lassas, and L. Päivärinta, Limits of visibility and invisibility for Calderón’s inverse problem in the plane, in preparation.
[BT] R. Brown and R. Torres, Uniqueness in the inverse conductivity problem for conductivities with $3/2$ derivatives in ${L}^{p},p>2n,$ J. Fourier Analysis Appl., 9 (2003), 1049-1056. | MR | Zbl
[BU] R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem with less regular conductivities in two dimensions, Comm. PDE, 22 (1997), 1009-10027. | MR | Zbl
[C] A.P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), 65–73, Soc. Brasil. Mat., Rio de Janeiro, 1980. | MR
[CPSSP] S. Cummer, B.-I. Popa, D. Schurig, D. Smith, and J. Pendry, Full-wave simulations of electromagnetic cloaking structures, Phys. Rev. E, 74 (2006), 036621.
[GKLU1] A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann: Full-wave invisibility of active devices at all frequencies. Comm. Math. Phys. 275, (2007), 749-789. | MR | Zbl
[GKLU2] A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann: Electromagnetic wormholes and virtual magnetic monopoles from metamaterials. Phys. Rev. Lett. 99, 183901 (2007).
[GKLU3] A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann: Effectiveness and improvement of cylindrical cloaking with the SHS lining. Optics Express 15, (2007), 12717-12734.
[GLU1] A. Greenleaf, M. Lassas, and G. Uhlmann, The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction, Comm. Pure Appl. Math 56, (2003), no. 3, 328–352. | MR | Zbl
[GLU2] A. Greenleaf, M. Lassas, and G. Uhlmann, Anisotropic conductivities that cannot detected in EIT, Physiolog. Meas. (special issue on Impedance Tomography), 24, (2003), 413-420.
[GLU3] A. Greenleaf, M. Lassas, and G. Uhlmann, On nonuniqueness for Calderón’s inverse problem, Math. Res. Lett. 10 (2003), no. 5-6, 685-693. | MR | Zbl
[KKM] T. Kilpeläinen, J. Kinnunen, and O. Martio, Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12 (2000), no. 3, 233–247. | MR | Zbl
[KSVW] R. Kohn, H. Shen, M. Vogelius, and M. Weinstein, Cloaking via change of variables in Electrical Impedance Tomography, preprint 2007. | MR
[KV] R. Kohn and M. Vogelius, Identification of an unknown conductivity by means of measurements at the boundary, in Inverse Problems, SIAM-AMS Proc., 14 (1984). | MR | Zbl
[LaU] M. Lassas and G. Uhlmann, Determining Riemannian manifold from boundary measurements, Ann. Sci. École Norm. Sup., 34 (2001), no. 5, 771–787. | Numdam | MR | Zbl
[LTU] M. Lassas, M. Taylor, and G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Geom. Anal., 11 (2003), 207-222. | MR | Zbl
[LeU] J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097–1112. | MR | Zbl
[Le] U. Leonhardt, Optical Conformal Mapping, Science, 312, 1777-1780. | MR
[MBW] G. Milton, M. Briane, and J. Willis, On cloaking for elasticity and physical equations with a transformation invariant form, New J. Phys., 8 (2006), 248.
[MN] G. Milton and N.-A. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. Royal Soc. A, 462 (2006), 3027–3059. | MR | Zbl
[N] A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. | MR | Zbl
[OPS] P. Ola, L. Päivärinta, E. Somersalo, An inverse boundary value problem in electrodynamics. Duke Math. J., 70 (1993), no. 3, 617–653. | MR | Zbl
[PPU] L. Päivärinta, A. Panchenko and G. Uhlmann, Complex geometrical optics for Lipschitz conductivities, Revista Matematica Iberoamericana, 19(2003), 57-72. | MR | Zbl
[PSS1] J.B. Pendry, D. Schurig, and D.R. Smith, Controlling electromagnetic fields, Science, 312, 1780-1782. | MR
[PSS2] J.B. Pendry, D. Schurig, and D.R. Smith, Calculation of material properties and ray tracing in transformation media, Opt. Exp., 14, (2006), 9794.
[Sc] D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Metamaterial electromagnetic cloak at microwave frequencies, Science, 314 (2006), 977-980.
[Sh] V. Shalaev, W. Cai, U. Chettiar, H.-K. Yuan, A. Sarychev, V. Drachev, and A. Kildishev, Negative index of refraction in optical metamaterials Optics Letters, 30 (2005), 3356-3358
[SuU] Z. Sun and G. Uhlmann, Anisotropic inverse problems in two dimensions", Inverse Problems, 19 (2003), 1001-1010. | MR | Zbl
[S] J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math. 43 (1990), 201–232. | MR | Zbl
[SyU] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem. Ann. of Math., 125 (1987), 153–169. | MR | Zbl
[W] R. Weder, A Rigorous Time-Domain Analysis of Full–Wave Electromagnetic Cloaking (Invisibility), preprint 2007.
Cited by Sources: