Mathematics of Invisibility

Greenleaf, Allan; Kurylev, Yaroslav; Lassas, Matti; Uhlmann, Gunther

doi:10.5802/jedp.44

Greenleaf, Allan ¹ ; Kurylev, Yaroslav ² ; Lassas, Matti ³ ; Uhlmann, Gunther ⁴

¹ Department of Mathematics, University of Rochester, Rochester, NY 14627, USA. Partially supported by NSF grant DMS-0551894.
² Department of Mathematics, University College London, Gower Street, London, WC1E 5BT, UK
³ Helsinki University of Technology, Institute of Mathematics, P.O.Box 1100, FIN-02015, Finland. Partially supported by Academy of Finland CoE Project 213476.
⁴ Department of Mathematics, University of Washington, Seattle, WA 98195, USA. Partially supported by the NSF and a Walker Family Endowed Professorship.

Journées équations aux dérivées partielles (2007), article no. 5, 11 p.

Résumé

We will describe recent some of the recent theoretical progress on making objects invisible to electromagnetic waves based on singular transformations.

DOI : 10.5802/jedp.44

Affiliations des auteurs :

Greenleaf, Allan ¹ ; Kurylev, Yaroslav ² ; Lassas, Matti ³ ; Uhlmann, Gunther ⁴

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     pages = {1--11},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2007},
     doi = {10.5802/jedp.44},
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     url = {http://www.numdam.org/articles/10.5802/jedp.44/}
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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://www.numdam.org/articles/10.5802/jedp.44/

Bibliographie
Cité par

[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 $L^{p}, p > 2 n,$ 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.

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