November  2015, 9(4): 1025-1049. doi: 10.3934/ipi.2015.9.1025

Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem

1. 

Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, United States

2. 

INRIA, CMAP, Ecole polytechnique, Université Paris Saclay, Route de Saclay, 91128 Palaiseau, France

3. 

Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States

Received  September 2014 Revised  March 2015 Published  October 2015

We consider the interior transmission problem associated with the scattering by an inhomogeneous (possibly anisotropic) highly oscillating periodic media. We show that, under appropriate assumptions, the solution of the interior transmission problem converges to the solution of a homogenized problem as the period goes to zero. Furthermore, we prove that the associated real transmission eigenvalues converge to transmission eigenvalues of the homogenized problem. Finally we show how to use the first transmission eigenvalue of the period media, which is measurable from the scattering data, to obtain information about constant effective material properties of the periodic media. The convergence results presented here are not optimal. Such results with rate of convergence involve the analysis of the boundary correction and will be subject of a forthcoming paper.
Citation: 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
References:
[1]

G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482.  doi: 10.1137/0523084.  Google Scholar

[2]

G. Allaire, Shape Optimization by the Homogenization Method,, Springer, (2002).  doi: 10.1007/978-1-4684-9286-6.  Google Scholar

[3]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, AMS Chelsea Publishing, (1978).   Google Scholar

[4]

E. Blåsten, L. Päivärinta and J. Sylvester, Corners always scatter,, Commun. Math. Phys, 331 (2014), 725.  doi: 10.1007/s00220-014-2030-0.  Google Scholar

[5]

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

[6]

F. Cakoni and D. Colton, Qualitative Approach to Inverse Scattering Theory,, Springer, (2014).  doi: 10.1007/978-1-4614-8827-9.  Google Scholar

[7]

F. Cakoni, D. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data,, C. R. Math. Acad. Sci. Paris, 348 (2010), 379.  doi: 10.1016/j.crma.2010.02.003.  Google Scholar

[8]

F. Cakoni and D. Gintides, The interior transmission eigenvalue problem,, SIAM J. Math. Analysis, 42 (2010), 2912.  doi: 10.1137/100793542.  Google Scholar

[9]

F. Cakoni, D. Colton and H. Haddar, The computation of lower bounds for the norm of the index of refraction in an anisotropic media from far field data,, J. Integral Eqns. Appl., 21 (2009), 203.  doi: 10.1216/JIE-2009-21-2-203.  Google Scholar

[10]

F. Cakoni, D. Colton and H. Haddar, The linear sampling method for anisotropic media,, J.Comp. Appl. Math., 146 (2002), 285.  doi: 10.1016/S0377-0427(02)00361-8.  Google Scholar

[11]

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

[12]

F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/7/074004.  Google Scholar

[13]

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

[14]

F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory,, Inverse Problems and Applications, 60 (2013), 529.   Google Scholar

[15]

F. Cakoni and H. Haddar, Interior transmission problem for anisotropic media,, in Mathematical and Numerical Aspects of Wave Propagation-WAVES 2003 (eds. Cohen et al.), (2003), 613.   Google Scholar

[16]

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

[17]

F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem,, Int. Jour. Comp. Sci. Math., 3 (2010), 142.  doi: 10.1504/IJCSM.2010.033932.  Google Scholar

[18]

A. Cossonnière, Valeurs Propres de Transmission et leur Utilisation dans L'identification D'inclusions à Partir de Mesures Électromagnètiques,, PhD thesis, (2011).   Google Scholar

[19]

G. Giovanni and H. Haddar, Computing estimates on material properties from transmission eigenvalues,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/5/055009.  Google Scholar

[20]

I. Harris, Non-destructive Testing of Anisotropic Materials,, Ph.D. Thesis, ().   Google Scholar

[21]

I. Harris, F. Cakoni and J. Sun, Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids,, Inverse Problems, 30 (2014).  doi: 10.1088/0266-5611/30/3/035016.  Google Scholar

[22]

C. E. Kenig, F. Lin and Z. Shen, Estimates of eigenvalues and eigenfunctions in periodic homogenization,, J. Euro. Math. Soc., 15 (2013), 1901.  doi: 10.4171/JEMS/408.  Google Scholar

[23]

C. E. Kenig, F. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems,, Arch. Rat. Mech. Anal., 203 (2012), 1009.  doi: 10.1007/s00205-011-0469-0.  Google Scholar

[24]

C. E. Kenig, F. Lin and Z. Shen, Homogenization of elliptic systems with Neumann boundary conditions,, J. Amer. Math. Soc., 26 (2013), 901.  doi: 10.1090/S0894-0347-2013-00769-9.  Google Scholar

[25]

S. Kesavan, Homogenization of elliptic eigenvalue problems: Part 1,, Appl. Math. Optim., 5 (1979), 153.  doi: 10.1007/BF01442551.  Google Scholar

[26]

S. Kesavan, Homogenization of elliptic eigenvalue problems: Part 2,, Appl. Math. Optim., 5 (1979), 197.  doi: 10.1007/BF01442554.  Google Scholar

[27]

A. Kirsch and A. Lechleiter, The inside-outside duality for scattering problems by inhomogeneous media,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/10/104011.  Google Scholar

[28]

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

[29]

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

[30]

S. Moskow and M. Vogelius, First-order corrections to the homogenized eigenvalues of periodic composite material. A convergence proof,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1263.  doi: 10.1017/S0308210500027050.  Google Scholar

[31]

S. Moskow and M. Vogelius, First-order corrections to the homogenized eigenvalues of periodic composite material. The case of Neumann boundary conditions,, preprint, (1997).   Google Scholar

[32]

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

[33]

F. Santosa and M. Vogelius, First-order corrections to the homogenized eigenvalues of periodic composite medium,, SIAM J. Appl. Math., 53 (1993), 1636.  doi: 10.1137/0153076.  Google Scholar

[34]

J. Sun, Iterative methods for transmission eigenvalues,, SIAM J. Numer. Anal., 49 (2011), 1860.  doi: 10.1137/100785478.  Google Scholar

[35]

J. Sun and L. Xu, Computation of Maxwell's transmission eigenvalues and its applications in inverse medium problems,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/10/104013.  Google Scholar

[36]

A. Wautier and B. Guzina, On the second-order homogenization of wave motion in periodic media and the sound of chessboard,, J. Mech. Phys. Solids, 78 (2015), 382.  doi: 10.1016/j.jmps.2015.03.001.  Google Scholar

[37]

J. Wloka, Partial Differential Equations,, Cambridge University Press, (1987).  doi: 10.1017/CBO9781139171755.  Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482.  doi: 10.1137/0523084.  Google Scholar

[2]

G. Allaire, Shape Optimization by the Homogenization Method,, Springer, (2002).  doi: 10.1007/978-1-4684-9286-6.  Google Scholar

[3]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, AMS Chelsea Publishing, (1978).   Google Scholar

[4]

E. Blåsten, L. Päivärinta and J. Sylvester, Corners always scatter,, Commun. Math. Phys, 331 (2014), 725.  doi: 10.1007/s00220-014-2030-0.  Google Scholar

[5]

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

[6]

F. Cakoni and D. Colton, Qualitative Approach to Inverse Scattering Theory,, Springer, (2014).  doi: 10.1007/978-1-4614-8827-9.  Google Scholar

[7]

F. Cakoni, D. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data,, C. R. Math. Acad. Sci. Paris, 348 (2010), 379.  doi: 10.1016/j.crma.2010.02.003.  Google Scholar

[8]

F. Cakoni and D. Gintides, The interior transmission eigenvalue problem,, SIAM J. Math. Analysis, 42 (2010), 2912.  doi: 10.1137/100793542.  Google Scholar

[9]

F. Cakoni, D. Colton and H. Haddar, The computation of lower bounds for the norm of the index of refraction in an anisotropic media from far field data,, J. Integral Eqns. Appl., 21 (2009), 203.  doi: 10.1216/JIE-2009-21-2-203.  Google Scholar

[10]

F. Cakoni, D. Colton and H. Haddar, The linear sampling method for anisotropic media,, J.Comp. Appl. Math., 146 (2002), 285.  doi: 10.1016/S0377-0427(02)00361-8.  Google Scholar

[11]

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

[12]

F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/7/074004.  Google Scholar

[13]

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

[14]

F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory,, Inverse Problems and Applications, 60 (2013), 529.   Google Scholar

[15]

F. Cakoni and H. Haddar, Interior transmission problem for anisotropic media,, in Mathematical and Numerical Aspects of Wave Propagation-WAVES 2003 (eds. Cohen et al.), (2003), 613.   Google Scholar

[16]

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

[17]

F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem,, Int. Jour. Comp. Sci. Math., 3 (2010), 142.  doi: 10.1504/IJCSM.2010.033932.  Google Scholar

[18]

A. Cossonnière, Valeurs Propres de Transmission et leur Utilisation dans L'identification D'inclusions à Partir de Mesures Électromagnètiques,, PhD thesis, (2011).   Google Scholar

[19]

G. Giovanni and H. Haddar, Computing estimates on material properties from transmission eigenvalues,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/5/055009.  Google Scholar

[20]

I. Harris, Non-destructive Testing of Anisotropic Materials,, Ph.D. Thesis, ().   Google Scholar

[21]

I. Harris, F. Cakoni and J. Sun, Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids,, Inverse Problems, 30 (2014).  doi: 10.1088/0266-5611/30/3/035016.  Google Scholar

[22]

C. E. Kenig, F. Lin and Z. Shen, Estimates of eigenvalues and eigenfunctions in periodic homogenization,, J. Euro. Math. Soc., 15 (2013), 1901.  doi: 10.4171/JEMS/408.  Google Scholar

[23]

C. E. Kenig, F. Lin and Z. Shen, Convergence rates in $L^2$ for elliptic homogenization problems,, Arch. Rat. Mech. Anal., 203 (2012), 1009.  doi: 10.1007/s00205-011-0469-0.  Google Scholar

[24]

C. E. Kenig, F. Lin and Z. Shen, Homogenization of elliptic systems with Neumann boundary conditions,, J. Amer. Math. Soc., 26 (2013), 901.  doi: 10.1090/S0894-0347-2013-00769-9.  Google Scholar

[25]

S. Kesavan, Homogenization of elliptic eigenvalue problems: Part 1,, Appl. Math. Optim., 5 (1979), 153.  doi: 10.1007/BF01442551.  Google Scholar

[26]

S. Kesavan, Homogenization of elliptic eigenvalue problems: Part 2,, Appl. Math. Optim., 5 (1979), 197.  doi: 10.1007/BF01442554.  Google Scholar

[27]

A. Kirsch and A. Lechleiter, The inside-outside duality for scattering problems by inhomogeneous media,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/10/104011.  Google Scholar

[28]

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

[29]

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

[30]

S. Moskow and M. Vogelius, First-order corrections to the homogenized eigenvalues of periodic composite material. A convergence proof,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1263.  doi: 10.1017/S0308210500027050.  Google Scholar

[31]

S. Moskow and M. Vogelius, First-order corrections to the homogenized eigenvalues of periodic composite material. The case of Neumann boundary conditions,, preprint, (1997).   Google Scholar

[32]

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

[33]

F. Santosa and M. Vogelius, First-order corrections to the homogenized eigenvalues of periodic composite medium,, SIAM J. Appl. Math., 53 (1993), 1636.  doi: 10.1137/0153076.  Google Scholar

[34]

J. Sun, Iterative methods for transmission eigenvalues,, SIAM J. Numer. Anal., 49 (2011), 1860.  doi: 10.1137/100785478.  Google Scholar

[35]

J. Sun and L. Xu, Computation of Maxwell's transmission eigenvalues and its applications in inverse medium problems,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/10/104013.  Google Scholar

[36]

A. Wautier and B. Guzina, On the second-order homogenization of wave motion in periodic media and the sound of chessboard,, J. Mech. Phys. Solids, 78 (2015), 382.  doi: 10.1016/j.jmps.2015.03.001.  Google Scholar

[37]

J. Wloka, Partial Differential Equations,, Cambridge University Press, (1987).  doi: 10.1017/CBO9781139171755.  Google Scholar

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