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April  2013, 33(4): 1615-1631. doi: 10.3934/dcds.2013.33.1615

Fractal bodies invisible in 2 and 3 directions

Alexander Plakhov 1, and  Vera Roshchina 2,

1. 

Department of Mathematics, University of Aveiro, Aveiro 3810-193
2. 

Collaborative Research Network, University of Ballarat, VIC 3353, Australia

Received  October 2011 Revised  January 2012 Published  October 2012

We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [1,12], where two-dimensional bodies invisible in one direction and three-dimensional bodies invisible in one and two orthogonal directions were constructed.
Keywords: billiards., problems of minimal resistance, Invisibility in geometrical optics.
Mathematics Subject Classification: Primary: 78A05, 37D50; Secondary: 76G2.
Citation: Alexander Plakhov, Vera Roshchina. Fractal bodies invisible in 2 and 3 directions. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1615-1631. doi: 10.3934/dcds.2013.33.1615
References:
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, "Invisibility,'', Wikipedia article. Available from: , ().   Google Scholar

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show all references

References:
[1]

Nonlinearity, 22 (2009), 1247-1258. doi: 10.1088/0951-7715/22/6/001.  Google Scholar

[2]

J. Modern Dynam., 5 (2011), 33-48.  Google Scholar

[3]

Birkhäuser (2005).  Google Scholar

[4]

Math. Intell., 15 (1993), 7-12. doi: 10.1007/BF03024318.  Google Scholar

[5]

Calc. Var. Partial Differ. Equ., 12 (2001), 173-211. doi: 10.1007/PL00009911.  Google Scholar

[6]

SIAM J. Optim., 16 (2006), 368-379. doi: 10.1137/040608039.  Google Scholar

[7]

Math. Nachr., 226 (2001), 153-176. doi: 10.1002/1522-2616(200106)226:1<153::AID-MANA153>3.3.CO;2-U.  Google Scholar

[8]

(1687). Google Scholar

[9]

Russ. Math. Surv., 64 (2009), 873-938. doi: 10.1070/RM2009v064n05ABEH004642.  Google Scholar

[10]

Discr. Contin. Dynam. Syst.-A, 30 (2011), 1211-1235. doi: 10.3934/dcds.2011.30.1211.  Google Scholar

[11]

ESAIM Control Optim. Calc. Var. 16 (2010), 206-220.  Google Scholar

[12]

Nonlinearity, 24 (2011), 847-854. doi: 10.1088/0951-7715/24/3/007.  Google Scholar

[13]

, "Invisibility,'', Wikipedia article. Available from: , ().   Google Scholar

[14]

, "Unsichtbarkeit,'', Wikipedia article. Available from: , ().   Google Scholar

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