Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2016, Article ID 2438253, 9 pages
http://dx.doi.org/10.1155/2016/2438253
Research Article

Inverse Uniqueness in Interior Transmission Problem and Its Eigenvalue Tunneling in Simple Domain

Department of Mathematics, National Chung Cheng University, 168 University Road, Min-Hsiung, Chia-Yi County 621, Taiwan

Received 7 October 2015; Accepted 6 December 2015

Academic Editor: Ivan Avramidi

Copyright © 2016 Lung-Hui Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Kirsch, “The denseness of the far field patterns for the transmission problem,” IMA Journal of Applied Mathematics, vol. 37, no. 3, pp. 213–225, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. D. Colton and P. Monk, “The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium,” The Quarterly Journal of Mechanics & Applied Mathematics, vol. 41, no. 1, pp. 97–125, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. 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, vol. 27, Article ID 115004, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. T. Aktosun, D. Gintides, and V. G. Papanicolaou, “Reconstruction of the wave speed from transmission eigenvalues for the spherically symmetric variable-speed wave equation,” Inverse Problems, vol. 29, no. 6, Article ID 065007, 2013. View at Google Scholar
  5. F. Cakoni, D. Colton, and H. Haddar, “The interior transmission eigenvalue problem for absorbing media,” Inverse Problems, vol. 28, no. 4, Article ID 045005, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. F. Cakoni, D. Colton, and D. Gintides, “The interior transmission eigenvalue problem,” SIAM Journal on Mathematical Analysis, vol. 42, no. 6, pp. 2912–2921, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. D. Colton, L. Päivärinta, and J. Sylvester, “The interior transmission problem,” Inverse Problems and Imaging, vol. 1, no. 1, pp. 13–28, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  8. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied Mathematical Sciences, Springer, New York, NY, USA, 3rd edition, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  9. M. Hitrik, K. Krupchyk, P. Ola, and L. P. Päivärinta, “The interior transmission problem and bounds on transmission eigenvalues,” Mathematical Research Letters, vol. 18, no. 2, pp. 279–293, 2011. View at Google Scholar
  10. M. Hitrik, K. Krupchyk, P. Ola, and L. P. Päivärinta, “Transmission eigenvalues for elliptic operators,” The SIAM Journal on Mathematical Analysis, vol. 43, no. 6, pp. 2630–2639, 2011. View at Google Scholar
  11. A. Kirsch, “On the existence of transmission eigenvalues,” Inverse Problems and Imaging, vol. 3, no. 2, pp. 155–172, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  12. H. Y. Liu, “Schiffer's conjecture, interior transmission eigenvalues and invisibility cloaking: singular problem vs. nonsingular problem,” in Geometric Analysis and Integral Geometry, vol. 598 of Contemporary Mathematics, pp. 147–154, American Mathematical Society, Providence, RI, USA, 2013. View at Google Scholar
  13. B. P. Rynne and B. D. Sleeman, “The interior transmission problem and inverse scattering from inhomogeneous media,” SIAM Journal on Mathematical Analysis, vol. 22, no. 6, pp. 1755–1762, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  14. L.-H. Chen, “An uniqueness result with some density theorems with interior transmission eigenvalues,” Applicable Analysis, vol. 94, no. 8, pp. 1527–1544, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. E. Lakshtanov and B. Vainberg, “Bounds on positive interior transmission eigenvalues,” Inverse Problems, vol. 28, no. 10, Article ID 105005, 2012. View at Publisher · View at Google Scholar
  16. E. Lakshtanov and B. Vainberg, “Weyl type bound on positive interior transmission eigenvalues,” Communications in Partial Differential Equations, vol. 39, no. 9, pp. 1729–1740, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. E. Lakshtanov and B. Vainberg, “Remarks on interior transmission eigenvalues, Weyl formula and branching billiards,” Journal of Physics A: Mathematical and Theoretical, vol. 45, no. 12, pp. 125–202, 2012. View at Publisher · View at Google Scholar
  18. J. R. McLaughlin and P. L. Polyakov, “On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues,” Journal of Differential Equations, vol. 107, no. 2, pp. 351–382, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. M. L. Cartwright, “On the directions of Borel of functions which are regular and of finite order in an angle,” Proceedings of the London Mathematical Society, Series 2, vol. 38, pp. 503–541, 1933. View at Google Scholar
  20. M. L. Cartwright, Integral Functions, Cambridge University Press, Cambridge, UK, 1956. View at MathSciNet
  21. P. Koosis, The Logarithmic Integral I, Cambridge University Press, New York, NY, USA, 1997.
  22. B. Ja. Levin, Distribution of Zeros of Entire Functions, Translations of Mathematical Mongraphs, American Mathematical Society, Providence, RI, USA, 1972. View at MathSciNet
  23. B. Ja. Levin, Lectures on Entire Functions, Translation of Mathematical Monographs, vol. 150, AMS, Providence, RI, USA, 1996.
  24. J. Sun, “Estimation of transmission eigenvalues and the index of refraction from Cauchy data,” Inverse Problems, vol. 27, no. 1, Article ID 015009, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. F. Zeng, T. Turner, and J. Sun, “Some results on electromagnetic transmission eigenvalues,” Mathematical Methods in the Applied Sciences, vol. 38, no. 1, pp. 155–163, 2015. View at Publisher · View at Google Scholar
  26. L.-H. Chen, “A uniqueness theorem on the eigenvalues of spherically symmetric interior transmission problem in absorbing medium,” Complex Variables and Elliptic Equations, vol. 60, no. 2, pp. 145–167, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. L.-H. Chen, “On the inverse spectral theory in a non-homogeneous interior transmission problem,” Complex Variables and Elliptic Equations, vol. 60, no. 5, pp. 707–731, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  28. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions, Applied Mathematics Series 55, National Bureau of Standards, Washington, DC, USA, 1956.
  29. R. Carlson, “Inverse spectral theory for some singular Sturm-Liouville problems,” Journal of Differential Equations, vol. 106, no. 1, pp. 121–140, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. R. Carlson, “A Borg-Levinson theorem for Bessel operators,” Pacific Journal of Mathematics, vol. 177, no. 1, pp. 1–26, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. R. Carlson, “Inverse Sturm-Liouville problems with a singularity at zero,” Inverse Problems, vol. 10, no. 4, pp. 851–864, 1994. View at Google Scholar · View at MathSciNet
  32. D. Colton, Y.-J. Leung, and S. Meng, “Distribution of complex transmission eigenvalues for spherically stratified media,” Inverse Problems, vol. 31, no. 3, Article ID 035006, 2015. View at Publisher · View at Google Scholar
  33. J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Orlando, Fla, USA, 1987. View at MathSciNet
  34. R. P. Boas, Entire Functions, Academic Press, New York, NY, USA, 1954. View at MathSciNet
  35. V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhaeuser, Basel, Switzerland, 1986.