American Institute of Mathematical Sciences

Advanced
February  2015, 9(1): 273-287. doi: 10.3934/ipi.2015.9.273

Continuous dependence of the transmission eigenvalues in one dimension

Yalin Zhang 1, and  Guoliang Shi 1,

1. 

Department of Mathematics, Tianjin University, Tianjin, 300072, China, China

Received  May 2014 Revised  September 2014 Published  January 2015

The transmission eigenvalue problem on the interval $\left[ a,b\right] $ is considered. We show that the isolated transmission eigenvalues are continuous functions of the coefficients of the problem and that the transmission eigenfunctions can be normalized so that they depend continuously on all coefficients in the uniform norm. Throughout this work, our results are established without assumptions on the sign of the contrasts.
Keywords: inhomogeneous medium, transmission eigenfunctions, continuous dependence., transmission eigenvalues, Interior transmission problem.
Mathematics Subject Classification: Primary: 34B40, 34L25; Secondary: 78A2.
Citation: 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
References:
[1]

T. Aktosun, D. Gintides and V. G. Papanicolaou, The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/11/115004.  Google Scholar

[2]

R. P. Boas, Entire Functions,, Academic Press, (1954).   Google Scholar

[3]

A. S. Bonnet-BenDhia, L. Chesnel and H. Haddar, On the use of T-coercivity to study the interior transmission eigenvalue problem,, C. R. Math. Acad. Sci., 349 (2011), 647.  doi: 10.1016/j.crma.2011.05.008.  Google Scholar

[4]

F. Cakoni, D. Colton and H. Hadder, The interior transmission problem for regions with cavities,, SIAM J. Math. Anal., 42 (2010), 145.  doi: 10.1137/090754637.  Google Scholar

[5]

F. Cakoni, A. Cossonnière and H. Haddar, Transmission eigenvalues for inhomogeneous media containing obstacles,, Inverse Probl. Imaging, 6 (2012), 373.  doi: 10.3934/ipi.2012.6.373.  Google Scholar

[6]

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

[7]

F. Cakoni and H. Hadder, On the existence of transmission eigenvalues in an inhomogeneous medium,, Appl. Anal., 88 (2009), 475.  doi: 10.1080/00036810802713966.  Google Scholar

[8]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, $3^{nd}$ edition, (2013).  doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[9]

F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem,, Int. J. Comput. Sci. Math., 3 (2010), 142.  doi: 10.1137/100793542.  Google Scholar

[10]

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

[11]

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Probl. Imaging, 1 (2007), 13.  doi: 10.3934/ipi.2007.1.13.  Google Scholar

[12]

K. S. Hickmann, Interior transmission eigenvalue problem with refractive index having $C^{2}$-transition to the background medium,, Appl. Anal., 91 (2012), 1675.  doi: 10.1080/00036811.2011.577741.  Google Scholar

[13]

M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients,, SIAM J. Math. Anal., 42 (2010), 2965.  doi: 10.1137/100793748.  Google Scholar

[14]

A. Kirsch, On the existence of transmission eigenvalues,, Inverse Probl. Imaging, 3 (2009), 155.  doi: 10.3934/ipi.2009.3.155.  Google Scholar

[15]

Q. Kong and A. Zettl, Eigenvalues of regular Sturm-Liouville problems,, J. Differential Equations, 131 (1996), 1.  doi: 10.1006/jdeq.1996.0154.  Google Scholar

[16]

Q. Kong and A. Zettl, Linear ordinary differential equations,, in Inequalities and Applications (eds. R. P. Agarwal), 3 (1994), 381.  doi: 10.1142/9789812798879_0031.  Google Scholar

[17]

Y. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/7/075005.  Google Scholar

[18]

J. McLaughlin and P. L. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues,, J. Differential Equations, 107 (1994), 351.  doi: 10.1006/jdeq.1994.1017.  Google Scholar

[19]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory,, Academic Press, (1987).   Google Scholar

[20]

J. Sylvester, Transmission eigenvalues in one dimension,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/10/104009.  Google Scholar

[21]

A. Zettl, Sturm-Liouville Theory,, vol. 121 of Mathematical Surveys and Monographs, (2005).  doi: 10.1090/surv/121.  Google Scholar

show all references

References:
[1]

T. Aktosun, D. Gintides and V. G. Papanicolaou, The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/11/115004.  Google Scholar

[2]

R. P. Boas, Entire Functions,, Academic Press, (1954).   Google Scholar

[3]

A. S. Bonnet-BenDhia, L. Chesnel and H. Haddar, On the use of T-coercivity to study the interior transmission eigenvalue problem,, C. R. Math. Acad. Sci., 349 (2011), 647.  doi: 10.1016/j.crma.2011.05.008.  Google Scholar

[4]

F. Cakoni, D. Colton and H. Hadder, The interior transmission problem for regions with cavities,, SIAM J. Math. Anal., 42 (2010), 145.  doi: 10.1137/090754637.  Google Scholar

[5]

F. Cakoni, A. Cossonnière and H. Haddar, Transmission eigenvalues for inhomogeneous media containing obstacles,, Inverse Probl. Imaging, 6 (2012), 373.  doi: 10.3934/ipi.2012.6.373.  Google Scholar

[6]

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

[7]

F. Cakoni and H. Hadder, On the existence of transmission eigenvalues in an inhomogeneous medium,, Appl. Anal., 88 (2009), 475.  doi: 10.1080/00036810802713966.  Google Scholar

[8]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, $3^{nd}$ edition, (2013).  doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[9]

F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem,, Int. J. Comput. Sci. Math., 3 (2010), 142.  doi: 10.1137/100793542.  Google Scholar

[10]

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

[11]

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Probl. Imaging, 1 (2007), 13.  doi: 10.3934/ipi.2007.1.13.  Google Scholar

[12]

K. S. Hickmann, Interior transmission eigenvalue problem with refractive index having $C^{2}$-transition to the background medium,, Appl. Anal., 91 (2012), 1675.  doi: 10.1080/00036811.2011.577741.  Google Scholar

[13]

M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients,, SIAM J. Math. Anal., 42 (2010), 2965.  doi: 10.1137/100793748.  Google Scholar

[14]

A. Kirsch, On the existence of transmission eigenvalues,, Inverse Probl. Imaging, 3 (2009), 155.  doi: 10.3934/ipi.2009.3.155.  Google Scholar

[15]

Q. Kong and A. Zettl, Eigenvalues of regular Sturm-Liouville problems,, J. Differential Equations, 131 (1996), 1.  doi: 10.1006/jdeq.1996.0154.  Google Scholar

[16]

Q. Kong and A. Zettl, Linear ordinary differential equations,, in Inequalities and Applications (eds. R. P. Agarwal), 3 (1994), 381.  doi: 10.1142/9789812798879_0031.  Google Scholar

[17]

Y. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/7/075005.  Google Scholar

[18]

J. McLaughlin and P. L. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues,, J. Differential Equations, 107 (1994), 351.  doi: 10.1006/jdeq.1994.1017.  Google Scholar

[19]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory,, Academic Press, (1987).   Google Scholar

[20]

J. Sylvester, Transmission eigenvalues in one dimension,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/10/104009.  Google Scholar

[21]

A. Zettl, Sturm-Liouville Theory,, vol. 121 of Mathematical Surveys and Monographs, (2005).  doi: 10.1090/surv/121.  Google Scholar

[1]

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
[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]

Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control & Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167
[4]

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
[5]

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
[6]

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
[7]

Fioralba Cakoni, Drossos Gintides. New results on transmission eigenvalues. Inverse Problems & Imaging, 2010, 4 (1) : 39-48. doi: 10.3934/ipi.2010.4.39
[8]

Andreas Kirsch. On the existence of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (2) : 155-172. doi: 10.3934/ipi.2009.3.155
[9]

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
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

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
[16]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551
[17]

David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems & Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004
[18]

Hui Wan, Jing-An Cui. A model for the transmission of malaria. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 479-496. doi: 10.3934/dcdsb.2009.11.479
[19]

Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025
[20]

Mahin Salmani, P. van den Driessche. A model for disease transmission in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 185-202. doi: 10.3934/dcdsb.2006.6.185

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences