Mathematics of Invisibility

Allan Greenleaf[1]; Yaroslav Kurylev[2]; Matti Lassas[3]; Gunther Uhlmann[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)

  • Volume: 275, Issue: 3, page 1-11
  • ISSN: 0752-0360

Abstract

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

How to cite

top

Greenleaf, Allan, et al. "Mathematics of Invisibility." Journées Équations aux dérivées partielles 275.3 (2007): 1-11. <http://eudml.org/doc/10632>.

@article{Greenleaf2007,
abstract = {We will describe recent some of the recent theoretical progress on making objects invisible to electromagnetic waves based on singular transformations.},
affiliation = {Department of Mathematics, University of Rochester, Rochester, NY 14627, USA. Partially supported by NSF grant DMS-0551894.; Department of Mathematics, University College London, Gower Street, London, WC1E 5BT, UK; Helsinki University of Technology, Institute of Mathematics, P.O.Box 1100, FIN-02015, Finland. Partially supported by Academy of Finland CoE Project 213476.; Department of Mathematics, University of Washington, Seattle, WA 98195, USA. Partially supported by the NSF and a Walker Family Endowed Professorship.},
author = {Greenleaf, Allan, Kurylev, Yaroslav, Lassas, Matti, Uhlmann, Gunther},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
month = {6},
number = {3},
pages = {1-11},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Mathematics of Invisibility},
url = {http://eudml.org/doc/10632},
volume = {275},
year = {2007},
}

TY - JOUR
AU - Greenleaf, Allan
AU - Kurylev, Yaroslav
AU - Lassas, Matti
AU - Uhlmann, Gunther
TI - Mathematics of Invisibility
JO - Journées Équations aux dérivées partielles
DA - 2007/6//
PB - Groupement de recherche 2434 du CNRS
VL - 275
IS - 3
SP - 1
EP - 11
AB - We will describe recent some of the recent theoretical progress on making objects invisible to electromagnetic waves based on singular transformations.
LA - eng
UR - http://eudml.org/doc/10632
ER -

References

top
  1. K. Astala and L. Päivärinta: Calderón’s inverse conductivity problem in the plane. Annals of Math., 163 (2006), 265-299. Zbl1111.35004MR2195135
  2. 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. Zbl1129.35483MR2131051
  3. K. Astala, M. Lassas, and L. Päivärinta, Limits of visibility and invisibility for Calderón’s inverse problem in the plane, in preparation. Zbl06542780
  4. R. Brown and R. Torres, Uniqueness in the inverse conductivity problem for conductivities with 3 / 2 derivatives in L p , p &gt; 2 n , J. Fourier Analysis Appl., 9 (2003), 1049-1056. Zbl0867.35111MR2026763
  5. R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem with less regular conductivities in two dimensions, Comm. PDE, 22 (1997), 1009-10027. Zbl0884.35167MR1452176
  6. 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. MR590275
  7. 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. 
  8. A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann: Full-wave invisibility of active devices at all frequencies. Comm. Math. Phys.275, (2007), 749-789. Zbl1151.78006MR2336363
  9. A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann: Electromagnetic wormholes and virtual magnetic monopoles from metamaterials. Phys. Rev. Lett.99, 183901 (2007). 
  10. A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann: Effectiveness and improvement of cylindrical cloaking with the SHS lining. Optics Express15, (2007), 12717-12734. 
  11. A. Greenleaf, M. Lassas, and G. Uhlmann, The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction, Comm. Pure Appl. Math56, (2003), no. 3, 328–352. Zbl1061.35165MR1941812
  12. 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. 
  13. A. Greenleaf, M. Lassas, and G. Uhlmann, On nonuniqueness for Calderón’s inverse problem, Math. Res. Lett.10 (2003), no. 5-6, 685-693. Zbl1054.35127MR2024725
  14. T. Kilpeläinen, J. Kinnunen, and O. Martio, Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12 (2000), no. 3, 233–247. Zbl0962.46021MR1752853
  15. R. Kohn, H. Shen, M. Vogelius, and M. Weinstein, Cloaking via change of variables in Electrical Impedance Tomography, preprint 2007. Zbl1153.35406MR2384775
  16. 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). Zbl0573.35084MR773707
  17. M. Lassas and G. Uhlmann, Determining Riemannian manifold from boundary measurements, Ann. Sci. École Norm. Sup., 34 (2001), no. 5, 771–787. Zbl0992.35120MR1862026
  18. 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. Zbl1077.58012MR2014876
  19. J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097–1112. Zbl0702.35036MR1029119
  20. U. Leonhardt, Optical Conformal Mapping, Science, 312, 1777-1780. Zbl1226.78001MR2237569
  21. 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. 
  22. G. Milton and N.-A. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. Royal Soc. A, 462 (2006), 3027–3059. Zbl1149.00310MR2263683
  23. A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. Zbl0857.35135MR1370758
  24. P. Ola, L. Päivärinta, E. Somersalo, An inverse boundary value problem in electrodynamics. Duke Math. J., 70 (1993), no. 3, 617–653. Zbl0804.35152MR1224101
  25. L. Päivärinta, A. Panchenko and G. Uhlmann, Complex geometrical optics for Lipschitz conductivities, Revista Matematica Iberoamericana, 19(2003), 57-72. Zbl1055.35144MR1993415
  26. J.B. Pendry, D. Schurig, and D.R. Smith, Controlling electromagnetic fields, Science, 312, 1780-1782. Zbl1226.78003MR2237570
  27. J.B. Pendry, D. Schurig, and D.R. Smith, Calculation of material properties and ray tracing in transformation media, Opt. Exp., 14, (2006), 9794. 
  28. 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. 
  29. 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 
  30. Z. Sun and G. Uhlmann, Anisotropic inverse problems in two dimensions", Inverse Problems, 19 (2003), 1001-1010. Zbl1054.35139MR2024685
  31. J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math.43 (1990), 201–232. Zbl0709.35102MR1038142
  32. J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem. Ann. of Math., 125 (1987), 153–169. Zbl0625.35078MR873380
  33. R. Weder, A Rigorous Time-Domain Analysis of Full–Wave Electromagnetic Cloaking (Invisibility), preprint 2007. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.