August  2012, 6(3): 373-398. doi: 10.3934/ipi.2012.6.373

Transmission eigenvalues for inhomogeneous media containing obstacles

1. 

Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716-2553, United States

2. 

CERFACS, 42 avenue Gaspard Coriolis, 31057 Toulouse Cedex 01, France

3. 

INRIA Saclay Ile de France/CMAP Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France

Received  November 2011 Revised  June 2012 Published  September 2012

We consider the interior transmission problem corresponding to the inverse scattering by an inhomogeneous (possibly anisotropic) media in which an impenetrable obstacle with Dirichlet boundary conditions is embedded. Our main focus is to understand the associated eigenvalue problem, more specifically to prove that the transmission eigenvalues form a discrete set and show that they exist. The presence of Dirichlet obstacle brings new difficulties to already complicated situation dealing with a non-selfadjoint eigenvalue problem. In this paper, we employ a variety of variational techniques under various assumptions on the index of refraction as well as the size of the Dirichlet obstacle.
Citation: Fioralba Cakoni, Anne Cossonnière, Houssem Haddar. Transmission eigenvalues for inhomogeneous media containing obstacles. Inverse Problems & Imaging, 2012, 6 (3) : 373-398. doi: 10.3934/ipi.2012.6.373
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show all references

References:
[1]

C. R. Acad. Sci., Ser. I, 340 (2011). Google Scholar

[2]

J. Comput. Appl. Math, 234 (2010), 1912-1919. doi: 10.1016/j.cam.2009.08.041.  Google Scholar

[3]

Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, 2006.  Google Scholar

[4]

SIAM J. Math. Anal., 42 (2010), 2912-2921. doi: 10.1137/100793542.  Google Scholar

[5]

J. Integral Equations and Applications, 21 (2009), 203-227. doi: 10.1216/JIE-2009-21-2-203.  Google Scholar

[6]

SIAM J. Math. Anal., 42 (2010), 145-162. doi: 10.1137/090754637.  Google Scholar

[7]

C. R. Acad. Sci. Paris, 348 (2010), 379-383. doi: 10.1016/j.crma.2010.02.003.  Google Scholar

[8]

Inverse Problems, 23 (2007), 507-522. doi: 10.1088/0266-5611/23/2/004.  Google Scholar

[9]

Inverse Problems and Imaging, 4 (2010), 39-48. doi: 10.3934/ipi.2010.4.39.  Google Scholar

[10]

SIAM J. Math. Anal., 42 (2010), 237-255. doi: 10.1137/090769338.  Google Scholar

[11]

Applicable Analysis, 88 (2008), 475-493. doi: 10.1080/00036810802713966.  Google Scholar

[12]

Int. Jour. Comp. Sci. Math, 3 (2010), 142-167.  Google Scholar

[13]

Springer, New York, 2nd edition, 1998.  Google Scholar

[14]

Inverse Problems and Imaging, 1 (2007), 13-28. doi: 10.3934/ipi.2007.1.13.  Google Scholar

[15]

SIAM J. Math. Anal., 43 (2011), 1698-1715. doi: 10.1137/100813890.  Google Scholar

[16]

SIAM J. Math. Analysis, 42 (2010), 619-651. doi: 10.1137/100793748.  Google Scholar

[17]

Math Research Letters, 18 (2011), 279-293.  Google Scholar

[18]

Inverse Problems and Imaging, 3 (2009), 155-172.  Google Scholar

[19]

in "Mathematics and its Applications," Oxford Lecture Series, 36, Oxford University Press, Oxford, 2008.  Google Scholar

[20]

Math. Meth. Appl. Sci., 21 (1998), 2965-2986. doi: 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P.  Google Scholar

[21]

SIAM J. Math. Anal., 40 (2008), 738-753. doi: 10.1137/070697525.  Google Scholar

[22]

SIAM J. Math. Anal., 44 (2012), 341-354. doi: 10.1137/110836420.  Google Scholar

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