Counting function for interior transmission eigenvalues

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March 2016, 6(1): 167-183. doi: 10.3934/mcrf.2016.6.167

Counting function for interior transmission eigenvalues

1.	Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France

Received October 2013 Revised February 2015 Published January 2016

Abstract
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In this paper we give results on the counting function associated with the interior transmission eigenvalues. For a complex refraction index we estimate of the counting function by $Ct^{n}$. In the case where the refraction index is positive we give an equivalent of the counting function.

Keywords: Weyl law., Interior transmission eigenvalues.

Mathematics Subject Classification: Primary: 35P10, 35P20, 35J5.

Citation: Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control & Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167

References:

[1]	S. Agmon, Lectures on Elliptic Boundary Value Problems,, Van Nostrand Mathematical Studies, (1965).
[2]	F. Cakoni, M. Çayören and D. Colton, Transmission eigenvalues and the nondestructive testing of dielectrics,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/6/065016.
[3]	F. Cakoni, D. Colton and H. Haddar, The interior transmission problem for regions with cavities,, SIAM J. Math. Anal., 42 (2010), 145. doi: 10.1137/090754637.
[4]	F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math. Anal., 42 (2010), 237. doi: 10.1137/090769338.
[5]	F. Cakoni and H. Haddar, Transmission Eigenvalues in Inverse Scattering Theory,, in Inverse Problems and Applications: Inside Out II, 60 (2013), 529.
[6]	H. Cartan, Théorie Élémentaire Des Fonctions Analytiques,, Hermann., ().
[7]	D. Colton, A. Kirsch and L. Päivärinta, Far-field patterns for acoustic waves in an inhomogeneous medium,, SIAM J. Math. Anal., 20 (1989), 1472. doi: 10.1137/0520096.
[8]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Applied Mathematical Sciences, 93 (1992). doi: 10.1007/978-3-662-02835-3.
[9]	M. Dimassi and V. Petkov, Upper bound for the counting function of interior transmission eigenvalues,, preprint, ().
[10]	M. Faierman, Transmission eigenvalues for parameter-elliptic boundary problems,, preprint., ().
[11]	M. Hitrik, K. Krupchyk, P. Ola Petri and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients,, SIAM J. Math. Anal., 42 (2010), 2965. doi: 10.1137/100793748.
[12]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for elliptic operators,, SIAM J. Math. Anal., 43 (2011), 2630. doi: 10.1137/110827867.
[13]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, The interior transmission problem and bounds on transmission eigenvalues,, Math. Res. Lett., 18 (2011), 279. doi: 10.4310/MRL.2011.v18.n2.a7.
[14]	J. Karamata, Neuer Beweis und Verallgemeinerung einiger Tauberian-Sätze,, Math. Z., 33 (1931), 294. doi: 10.1007/BF01174355.
[15]	E. Lakshtanov and B. Vainberg, Ellipticity in the interior transmission problem in anisotropic media,, SIAM J. Math. Anal., 44 (2012), 1165. doi: 10.1137/11084738X.
[16]	E. Lakshtanov and B. Vainberg, Remarks on interior transmission eigenvalues, Weyl formula and branching billiards,, J. Phys. A: Math. Theor, 45 (2012). doi: 10.1088/1751-8113/45/12/125202.
[17]	E. Lakshtanov and B. Vainberg, Bounds on positive interior transmission eigenvalues,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/10/105005.
[18]	E. Lakshtanov and B. Vainberg, Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104003.
[19]	P. Malliavin, Un théorème taubérien relié aux estimations de valeurs propres,, Séminaire Jean Leray, (): 1962.
[20]	L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738. doi: 10.1137/070697525.
[21]	H. Pham and P. Stefanov, Weyl asymptotics of the transmission eigenvalues for a constant index of refraction,, Inverse Problems and Imaging, 8 (2014), 795. doi: 10.3934/ipi.2014.8.795.
[22]	L. Robbiano, Spectral analysis on interior transmission eigenvalues,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104001.
[23]	J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators,, SIAM J. Math. Anal., 44 (2012), 341. doi: 10.1137/110836420.

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References:

[1]	S. Agmon, Lectures on Elliptic Boundary Value Problems,, Van Nostrand Mathematical Studies, (1965).
[2]	F. Cakoni, M. Çayören and D. Colton, Transmission eigenvalues and the nondestructive testing of dielectrics,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/6/065016.
[3]	F. Cakoni, D. Colton and H. Haddar, The interior transmission problem for regions with cavities,, SIAM J. Math. Anal., 42 (2010), 145. doi: 10.1137/090754637.
[4]	F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math. Anal., 42 (2010), 237. doi: 10.1137/090769338.
[5]	F. Cakoni and H. Haddar, Transmission Eigenvalues in Inverse Scattering Theory,, in Inverse Problems and Applications: Inside Out II, 60 (2013), 529.
[6]	H. Cartan, Théorie Élémentaire Des Fonctions Analytiques,, Hermann., ().
[7]	D. Colton, A. Kirsch and L. Päivärinta, Far-field patterns for acoustic waves in an inhomogeneous medium,, SIAM J. Math. Anal., 20 (1989), 1472. doi: 10.1137/0520096.
[8]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Applied Mathematical Sciences, 93 (1992). doi: 10.1007/978-3-662-02835-3.
[9]	M. Dimassi and V. Petkov, Upper bound for the counting function of interior transmission eigenvalues,, preprint, ().
[10]	M. Faierman, Transmission eigenvalues for parameter-elliptic boundary problems,, preprint., ().
[11]	M. Hitrik, K. Krupchyk, P. Ola Petri and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients,, SIAM J. Math. Anal., 42 (2010), 2965. doi: 10.1137/100793748.
[12]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for elliptic operators,, SIAM J. Math. Anal., 43 (2011), 2630. doi: 10.1137/110827867.
[13]	M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, The interior transmission problem and bounds on transmission eigenvalues,, Math. Res. Lett., 18 (2011), 279. doi: 10.4310/MRL.2011.v18.n2.a7.
[14]	J. Karamata, Neuer Beweis und Verallgemeinerung einiger Tauberian-Sätze,, Math. Z., 33 (1931), 294. doi: 10.1007/BF01174355.
[15]	E. Lakshtanov and B. Vainberg, Ellipticity in the interior transmission problem in anisotropic media,, SIAM J. Math. Anal., 44 (2012), 1165. doi: 10.1137/11084738X.
[16]	E. Lakshtanov and B. Vainberg, Remarks on interior transmission eigenvalues, Weyl formula and branching billiards,, J. Phys. A: Math. Theor, 45 (2012). doi: 10.1088/1751-8113/45/12/125202.
[17]	E. Lakshtanov and B. Vainberg, Bounds on positive interior transmission eigenvalues,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/10/105005.
[18]	E. Lakshtanov and B. Vainberg, Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104003.
[19]	P. Malliavin, Un théorème taubérien relié aux estimations de valeurs propres,, Séminaire Jean Leray, (): 1962.
[20]	L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738. doi: 10.1137/070697525.
[21]	H. Pham and P. Stefanov, Weyl asymptotics of the transmission eigenvalues for a constant index of refraction,, Inverse Problems and Imaging, 8 (2014), 795. doi: 10.3934/ipi.2014.8.795.
[22]	L. Robbiano, Spectral analysis on interior transmission eigenvalues,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104001.
[23]	J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators,, SIAM J. Math. Anal., 44 (2012), 341. doi: 10.1137/110836420.

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