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A new spectral method for $l_1$-regularized minimization
Continuous dependence of the transmission eigenvalues in one dimension
1. | Department of Mathematics, Tianjin University, Tianjin, 300072, China, China |
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), 115004, 17pp.
doi: 10.1088/0266-5611/27/11/115004. |
[2] |
R. P. Boas, Entire Functions, Academic Press, New York, 1954. |
[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), 647-651.
doi: 10.1016/j.crma.2011.05.008. |
[4] |
F. Cakoni, D. Colton and H. Hadder, The interior transmission problem for regions with cavities, SIAM J. Math. Anal., 42 (2010), 145-162.
doi: 10.1137/090754637. |
[5] |
F. Cakoni, A. Cossonnière and H. Haddar, Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Probl. Imaging, 6 (2012), 373-398.
doi: 10.3934/ipi.2012.6.373. |
[6] |
F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.
doi: 10.1137/090769338. |
[7] |
F. Cakoni and H. Hadder, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493.
doi: 10.1080/00036810802713966. |
[8] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $3^{nd}$ edition, Springer-Verlag, Berlin, 2013.
doi: 10.1007/978-1-4614-4942-3. |
[9] |
F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem, Int. J. Comput. Sci. Math., 3 (2010), 142-167.
doi: 10.1137/100793542. |
[10] |
D. Colton, A. Kirsch and L. Päivärinta, Far-field patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal., 20 (1989), 1472-1483.
doi: 10.1137/0520096. |
[11] |
D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Probl. Imaging, 1 (2007), 13-28.
doi: 10.3934/ipi.2007.1.13. |
[12] |
K. S. Hickmann, Interior transmission eigenvalue problem with refractive index having $C^{2}$-transition to the background medium, Appl. Anal., 91 (2012), 1675-1690.
doi: 10.1080/00036811.2011.577741. |
[13] |
M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients, SIAM J. Math. Anal., 42 (2010), 2965-2986.
doi: 10.1137/100793748. |
[14] |
A. Kirsch, On the existence of transmission eigenvalues, Inverse Probl. Imaging, 3 (2009), 155-172.
doi: 10.3934/ipi.2009.3.155. |
[15] |
Q. Kong and A. Zettl, Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations, 131 (1996), 1-19.
doi: 10.1006/jdeq.1996.0154. |
[16] |
Q. Kong and A. Zettl, Linear ordinary differential equations, in Inequalities and Applications (eds. R. P. Agarwal), WSSIAA, 3 (1994), 381-397.
doi: 10.1142/9789812798879_0031. |
[17] |
Y. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media, Inverse Problems, 28 (2012), 075005, 9pp.
doi: 10.1088/0266-5611/28/7/075005. |
[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), 351-382.
doi: 10.1006/jdeq.1994.1017. |
[19] |
J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Boston, 1987. |
[20] |
J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems, 29 (2013), 104009, 11pp.
doi: 10.1088/0266-5611/29/10/104009. |
[21] |
A. Zettl, Sturm-Liouville Theory, vol. 121 of Mathematical Surveys and Monographs, American Mathematical Society, 2005.
doi: 10.1090/surv/121. |
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), 115004, 17pp.
doi: 10.1088/0266-5611/27/11/115004. |
[2] |
R. P. Boas, Entire Functions, Academic Press, New York, 1954. |
[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), 647-651.
doi: 10.1016/j.crma.2011.05.008. |
[4] |
F. Cakoni, D. Colton and H. Hadder, The interior transmission problem for regions with cavities, SIAM J. Math. Anal., 42 (2010), 145-162.
doi: 10.1137/090754637. |
[5] |
F. Cakoni, A. Cossonnière and H. Haddar, Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Probl. Imaging, 6 (2012), 373-398.
doi: 10.3934/ipi.2012.6.373. |
[6] |
F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.
doi: 10.1137/090769338. |
[7] |
F. Cakoni and H. Hadder, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493.
doi: 10.1080/00036810802713966. |
[8] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $3^{nd}$ edition, Springer-Verlag, Berlin, 2013.
doi: 10.1007/978-1-4614-4942-3. |
[9] |
F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem, Int. J. Comput. Sci. Math., 3 (2010), 142-167.
doi: 10.1137/100793542. |
[10] |
D. Colton, A. Kirsch and L. Päivärinta, Far-field patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal., 20 (1989), 1472-1483.
doi: 10.1137/0520096. |
[11] |
D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Probl. Imaging, 1 (2007), 13-28.
doi: 10.3934/ipi.2007.1.13. |
[12] |
K. S. Hickmann, Interior transmission eigenvalue problem with refractive index having $C^{2}$-transition to the background medium, Appl. Anal., 91 (2012), 1675-1690.
doi: 10.1080/00036811.2011.577741. |
[13] |
M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients, SIAM J. Math. Anal., 42 (2010), 2965-2986.
doi: 10.1137/100793748. |
[14] |
A. Kirsch, On the existence of transmission eigenvalues, Inverse Probl. Imaging, 3 (2009), 155-172.
doi: 10.3934/ipi.2009.3.155. |
[15] |
Q. Kong and A. Zettl, Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations, 131 (1996), 1-19.
doi: 10.1006/jdeq.1996.0154. |
[16] |
Q. Kong and A. Zettl, Linear ordinary differential equations, in Inequalities and Applications (eds. R. P. Agarwal), WSSIAA, 3 (1994), 381-397.
doi: 10.1142/9789812798879_0031. |
[17] |
Y. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media, Inverse Problems, 28 (2012), 075005, 9pp.
doi: 10.1088/0266-5611/28/7/075005. |
[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), 351-382.
doi: 10.1006/jdeq.1994.1017. |
[19] |
J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Boston, 1987. |
[20] |
J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems, 29 (2013), 104009, 11pp.
doi: 10.1088/0266-5611/29/10/104009. |
[21] |
A. Zettl, Sturm-Liouville Theory, vol. 121 of Mathematical Surveys and Monographs, American Mathematical Society, 2005.
doi: 10.1090/surv/121. |
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