American Institute of Mathematical Sciences

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

Counting function for interior transmission eigenvalues

Luc Robbiano 1,

1. 

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

Received  October 2013 Revised  February 2015 Published  January 2016

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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

F. Cakoni and H. Haddar, Transmission Eigenvalues in Inverse Scattering Theory,, in Inverse Problems and Applications: Inside Out II, 60 (2013), 529.   Google Scholar

[6]

H. Cartan, Théorie Élémentaire Des Fonctions Analytiques,, Hermann., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Dimassi and V. Petkov, Upper bound for the counting function of interior transmission eigenvalues,, preprint, ().   Google Scholar

[10]

M. Faierman, Transmission eigenvalues for parameter-elliptic boundary problems,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Karamata, Neuer Beweis und Verallgemeinerung einiger Tauberian-Sätze,, Math. Z., 33 (1931), 294.  doi: 10.1007/BF01174355.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

E. Lakshtanov and B. Vainberg, Bounds on positive interior transmission eigenvalues,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/10/105005.  Google Scholar

[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.  Google Scholar

[19]

P. Malliavin, Un théorème taubérien relié aux estimations de valeurs propres,, Séminaire Jean Leray, (): 1962.   Google Scholar

[20]

L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738.  doi: 10.1137/070697525.  Google Scholar

[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.  Google Scholar

[22]

L. Robbiano, Spectral analysis on interior transmission eigenvalues,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/10/104001.  Google Scholar

[23]

J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators,, SIAM J. Math. Anal., 44 (2012), 341.  doi: 10.1137/110836420.  Google Scholar

show all references

References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems,, Van Nostrand Mathematical Studies, (1965).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

F. Cakoni and H. Haddar, Transmission Eigenvalues in Inverse Scattering Theory,, in Inverse Problems and Applications: Inside Out II, 60 (2013), 529.   Google Scholar

[6]

H. Cartan, Théorie Élémentaire Des Fonctions Analytiques,, Hermann., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Dimassi and V. Petkov, Upper bound for the counting function of interior transmission eigenvalues,, preprint, ().   Google Scholar

[10]

M. Faierman, Transmission eigenvalues for parameter-elliptic boundary problems,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Karamata, Neuer Beweis und Verallgemeinerung einiger Tauberian-Sätze,, Math. Z., 33 (1931), 294.  doi: 10.1007/BF01174355.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

E. Lakshtanov and B. Vainberg, Bounds on positive interior transmission eigenvalues,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/10/105005.  Google Scholar

[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.  Google Scholar

[19]

P. Malliavin, Un théorème taubérien relié aux estimations de valeurs propres,, Séminaire Jean Leray, (): 1962.   Google Scholar

[20]

L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal., 40 (2008), 738.  doi: 10.1137/070697525.  Google Scholar

[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.  Google Scholar

[22]

L. Robbiano, Spectral analysis on interior transmission eigenvalues,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/10/104001.  Google Scholar

[23]

J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators,, SIAM J. Math. Anal., 44 (2012), 341.  doi: 10.1137/110836420.  Google Scholar

[1]

Ha Pham, Plamen Stefanov. Weyl asymptotics of the transmission eigenvalues for a constant index of refraction. Inverse Problems & Imaging, 2014, 8 (3) : 795-810. doi: 10.3934/ipi.2014.8.795
[2]

Vesselin Petkov, Georgi Vodev. Localization of the interior transmission eigenvalues for a ball. Inverse Problems & Imaging, 2017, 11 (2) : 355-372. doi: 10.3934/ipi.2017017
[3]

Fioralba Cakoni, Drossos Gintides. New results on transmission eigenvalues. Inverse Problems & Imaging, 2010, 4 (1) : 39-48. doi: 10.3934/ipi.2010.4.39
[4]

Andreas Kirsch. On the existence of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (2) : 155-172. doi: 10.3934/ipi.2009.3.155
[5]

David Colton, Lassi Päivärinta, John Sylvester. The interior transmission problem. Inverse Problems & Imaging, 2007, 1 (1) : 13-28. doi: 10.3934/ipi.2007.1.13
[6]

Dmitry Jakobson and Iosif Polterovich. Lower bounds for the spectral function and for the remainder in local Weyl's law on manifolds. Electronic Research Announcements, 2005, 11: 71-77.

[7]

Armin Lechleiter. The factorization method is independent of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (1) : 123-138. doi: 10.3934/ipi.2009.3.123
[8]

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
[9]

Yalin Zhang, Guoliang Shi. Continuous dependence of the transmission eigenvalues in one dimension. Inverse Problems & Imaging, 2015, 9 (1) : 273-287. doi: 10.3934/ipi.2015.9.273
[10]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487
[11]

Fioralba Cakoni, Shari Moskow, Scott Rome. The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions. Inverse Problems & Imaging, 2015, 9 (3) : 725-748. doi: 10.3934/ipi.2015.9.725
[12]

Fioralba Cakoni, Shari Moskow, Scott Rome. Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast. Inverse Problems & Imaging, 2018, 12 (4) : 971-992. doi: 10.3934/ipi.2018041
[13]

Jun Zhang, Xinyue Fan. An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4799-4813. doi: 10.3934/dcdsb.2019031
[14]

Fioralba Cakoni, Houssem Haddar. A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Problems & Imaging, 2007, 1 (3) : 443-456. doi: 10.3934/ipi.2007.1.443
[15]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159
[16]

Yuebin Hao. Electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with a conductive boundary. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1387-1397. doi: 10.3934/cpaa.2020068
[17]

Uri Bader, Alex Furman. Boundaries, Weyl groups, and Superrigidity. Electronic Research Announcements, 2012, 19: 41-48. doi: 10.3934/era.2012.19.41
[18]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939
[19]

Frédéric Naud, Anke Pohl, Louis Soares. Fractal Weyl bounds and Hecke triangle groups. Electronic Research Announcements, 2019, 26: 24-35. doi: 10.3934/era.2019.26.003
[20]

Kurt Vinhage. On the rigidity of Weyl chamber flows and Schur multipliers as topological groups. Journal of Modern Dynamics, 2015, 9: 25-49. doi: 10.3934/jmd.2015.9.25

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences