Open Access
2013 Transmission Eigenvalues for Non-regular Cases
Valery Serov
Commun. Math. Anal. 14(2): 129-142 (2013).
Abstract

We prove the existence of transmission eigenvalues in the case when the perturbation of the index of refraction may have singularity or degeneration on the boundary of its support. This singularity or degeneration is measured in terms of the distance to the boundary.

References

1.

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Academic Press, 2003.  MR2424078 R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Academic Press, 2003.  MR2424078

2.

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer, Berlin, 2006.  MR2256477 1099.78008 F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer, Berlin, 2006.  MR2256477 1099.78008

3.

F. Cakoni, D. Colton and P. Monk, On the use of transmission eigenvalues to estimate the index of refraction from far field data. Inverse Problems 23 (2007), pp 507-522.  MR2309661 10.1088/0266-5611/23/2/004 F. Cakoni, D. Colton and P. Monk, On the use of transmission eigenvalues to estimate the index of refraction from far field data. Inverse Problems 23 (2007), pp 507-522.  MR2309661 10.1088/0266-5611/23/2/004

4.

F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues. SIAM J. Math. Analysis 42, no. 1 (2010), pp 237-255.  MR2596553 1210.35282 10.1137/090769338 F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues. SIAM J. Math. Analysis 42, no. 1 (2010), pp 237-255.  MR2596553 1210.35282 10.1137/090769338

5.

F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem. Int. Journal Comp. Sci. Math. 3, no. 1-2 (2010) pp 142-167.  MR2682279 1204.78008 F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem. Int. Journal Comp. Sci. Math. 3, no. 1-2 (2010) pp 142-167.  MR2682279 1204.78008

6.

F. Cakoni, D. Colton and H. Haddar, The interior transmission problem for regions with cavities. SIAM J. Math. Analysis 42, no.1 (2010), pp 145-162.  MR2596549 1209.35135 10.1137/090754637 F. Cakoni, D. Colton and H. Haddar, The interior transmission problem for regions with cavities. SIAM J. Math. Analysis 42, no.1 (2010), pp 145-162.  MR2596549 1209.35135 10.1137/090754637

7.

D. Colton, A. Kirsch and L. Päivärinta, Far field patterns for acoustic waves in an inhomogeneous medium. SIAM J. Math. Analysis 20 (1989), pp 1472-1483.  MR1019312 0681.76084 10.1137/0520096 D. Colton, A. Kirsch and L. Päivärinta, Far field patterns for acoustic waves in an inhomogeneous medium. SIAM J. Math. Analysis 20 (1989), pp 1472-1483.  MR1019312 0681.76084 10.1137/0520096

8.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edition, Applied Mathematical Sciencies, Vol. 93, Springer, New York, 1998.  MR1635980 D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edition, Applied Mathematical Sciencies, Vol. 93, Springer, New York, 1998.  MR1635980

9.

D. Colton and P. Monk, The inverse scattering problem for acoustic waves in an inhomogeneous medium. Quart. Jour. Mech. Applied Math. 41 (1988), pp 97-125.  MR934695 10.1093/qjmam/41.1.97 D. Colton and P. Monk, The inverse scattering problem for acoustic waves in an inhomogeneous medium. Quart. Jour. Mech. Applied Math. 41 (1988), pp 97-125.  MR934695 10.1093/qjmam/41.1.97

10.

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Problems and Imaging 1, no. 1 (2007), pp 13-28.  MR2262743 1130.35132 10.3934/ipi.2007.1.13 D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Problems and Imaging 1, no. 1 (2007), pp 13-28.  MR2262743 1130.35132 10.3934/ipi.2007.1.13

11.

L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, New Jersey, 2004.  MR2449250 1148.42001 L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, New Jersey, 2004.  MR2449250 1148.42001

12.

M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients. Math. Res. Lett. 18, no. 2 (2011), pp 279-293.  MR2784672 1241.47057 M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients. Math. Res. Lett. 18, no. 2 (2011), pp 279-293.  MR2784672 1241.47057

13.

M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for elliptic operators. SIAM J. Math. Anal. 43 (2011), pp 2630-2639.  MR2873234 1233.35148 10.1137/110827867 M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for elliptic operators. SIAM J. Math. Anal. 43 (2011), pp 2630-2639.  MR2873234 1233.35148 10.1137/110827867

14.

M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, The interior transmission problem and bounds on transmission eigenvalues. htpp://arxiv.org/abs/1009.5640.  MR2784672 1241.47057 M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, The interior transmission problem and bounds on transmission eigenvalues. htpp://arxiv.org/abs/1009.5640.  MR2784672 1241.47057

15.

L. Hörmander, The Analysis of Linear Partial Differential Equations, Vols. 1-2, Springer-Verlag, New-York, 1983. L. Hörmander, The Analysis of Linear Partial Differential Equations, Vols. 1-2, Springer-Verlag, New-York, 1983.

16.

L. Hörmander, The Analysis of Linear Partial Differential Equations, Vols. 3-4, Springer-Verlag, New-York, 1985. L. Hörmander, The Analysis of Linear Partial Differential Equations, Vols. 3-4, Springer-Verlag, New-York, 1985.

17.

A. Kirsch, Factorization of the far field operator for the inhomogeneous medium case and an application in inverse scattering theory. Inverse Problems 15 (1999), pp 413-429.  MR1684466 10.1088/0266-5611/15/2/005 A. Kirsch, Factorization of the far field operator for the inhomogeneous medium case and an application in inverse scattering theory. Inverse Problems 15 (1999), pp 413-429.  MR1684466 10.1088/0266-5611/15/2/005

18.

A. Kirsch, An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems and Imaging 1, no. 1 (2007), pp 159-179.  MR2262751 1129.35080 10.3934/ipi.2007.1.159 A. Kirsch, An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems and Imaging 1, no. 1 (2007), pp 159-179.  MR2262751 1129.35080 10.3934/ipi.2007.1.159

19.

A. Kirsch, On the existence of transmission eigenvalues. Inverse Problems and Imaging 3, no. 2 (2009), pp 155-172.  MR2558284 1186.35122 10.3934/ipi.2009.3.155 A. Kirsch, On the existence of transmission eigenvalues. Inverse Problems and Imaging 3, no. 2 (2009), pp 155-172.  MR2558284 1186.35122 10.3934/ipi.2009.3.155

20.

J. R. McLaughlin and P. L. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues. J. Diff. Equations 107, no. 2 (1994), pp 351-382.  MR1264527 0803.35163 10.1006/jdeq.1994.1017 J. R. McLaughlin and P. L. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues. J. Diff. Equations 107, no. 2 (1994), pp 351-382.  MR1264527 0803.35163 10.1006/jdeq.1994.1017

21.

J. R. McLaughlin, P. L. Polyakov and P. E. Sacks, Reconstruction of a spherically symmetric speed of sound. SIAM J. Appl. Math. 54, no. 5 (1994), pp 1203-1223.  MR1293096 0809.34024 10.1137/S0036139992238218 J. R. McLaughlin, P. L. Polyakov and P. E. Sacks, Reconstruction of a spherically symmetric speed of sound. SIAM J. Appl. Math. 54, no. 5 (1994), pp 1203-1223.  MR1293096 0809.34024 10.1137/S0036139992238218

22.

J. Necas, Sur une methode pour resourde les equations aux derivees partielle du type elliptique, voisine de la variationelle. Ann.Scuola Norm. Sup. Pisa 16 (1962), pp 305-326.  MR163054 J. Necas, Sur une methode pour resourde les equations aux derivees partielle du type elliptique, voisine de la variationelle. Ann.Scuola Norm. Sup. Pisa 16 (1962), pp 305-326.  MR163054

23.

L. Päivärinta and V. Serov, New mapping properties for the resolvent of the Laplacian and recovery of singularities of a multi-dimensional scattering potential. Inverse Problems 17 (2001), pp 1321-1326.  MR1862193 10.1088/0266-5611/17/5/306 L. Päivärinta and V. Serov, New mapping properties for the resolvent of the Laplacian and recovery of singularities of a multi-dimensional scattering potential. Inverse Problems 17 (2001), pp 1321-1326.  MR1862193 10.1088/0266-5611/17/5/306

24.

L. Päivärinta and J. Sylvester, Transmission eigenvalues. SIAM J. Math. Analysis 40, no. 2 (2008), pp 738-753.  MR2438784 10.1137/070697525 L. Päivärinta and J. Sylvester, Transmission eigenvalues. SIAM J. Math. Analysis 40, no. 2 (2008), pp 738-753.  MR2438784 10.1137/070697525

25.

B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton University Press, Princeton, N.J., 1971.  MR455975 0232.47053 B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton University Press, Princeton, N.J., 1971.  MR455975 0232.47053

26.

H. Triebel, Interpolation Theory. Function Spaces. Differential Operators., Mir, Moscow, 1980.  MR2284819 H. Triebel, Interpolation Theory. Function Spaces. Differential Operators., Mir, Moscow, 1980.  MR2284819
Copyright © 2013 Mathematical Research Publishers
Valery Serov "Transmission Eigenvalues for Non-regular Cases," Communications in Mathematical Analysis 14(2), 129-142, (2013). https://doi.org/
Published: 2013
Vol.14 • No. 2 • 2013
Back to Top