August  2014, 8(3): 713-732. doi: 10.3934/ipi.2014.8.713

Stability of the determination of a coefficient for wave equations in an infinite waveguide

1. 

CPT, UMR CNRS 7332, Aix Marseille Université, Campus de Luminy, Case 907, 13288 Marseille, cedex 9, France

Received  September 2013 Revised  June 2014 Published  August 2014

We consider the stability of the inverse problem consisting of the determination of a coefficient of order zero $q$, appearing in the Dirichlet initial-boundary value problem for a wave equation $\partial_t^2u-\Delta u+q(x)u=0$ in $(0,T)\times\Omega$, with $\Omega=\omega\times\mathbb{R}$ an unbounded cylindrical waveguide and $\omega$ a bounded smooth domain of $\mathbb{R}^2$, from boundary observations. The observation is given by the Dirichlet to Neumann map associated to the wave equation. Using suitable geometric optics solutions, we prove a Hölder stability estimate in the determination of $q$ from the Dirichlet to Neumann map. Moreover, provided that the coefficient $q$ is lying in a set of functions $\mathcal A$, where, for any $q_1,q_2\in\mathcal A$, $|q_1-q_2|$ attains its maximum in a fixed bounded subset of $\overline{\Omega}$, we extend this result to the same inverse problem with measurements on a bounded subset of the lateral boundary $(0,T)\times\partial\Omega$.
Citation: Yavar Kian. Stability of the determination of a coefficient for wave equations in an infinite waveguide. Inverse Problems and Imaging, 2014, 8 (3) : 713-732. doi: 10.3934/ipi.2014.8.713
References:
[1]

M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients, Appl. Anal., 83 (2004), 983-1014. doi: 10.1080/0003681042000221678.

[2]

M. Bellassoued, M. Choulli and M. Yamamoto, Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Diff. Equat., 247 (2009), 465-494. doi: 10.1016/j.jde.2009.03.024.

[3]

M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal., 85 (2006), 1219-1243. doi: 10.1080/00036810600787873.

[4]

M. Bellassoued, D. Jellali and M. Yamamoto, Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, J. Math. Anal. Appl, 343 (2008), 1036-1046. doi: 10.1016/j.jmaa.2008.01.098.

[5]

A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247.

[6]

M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques, Mathématiques et Applications, Vol. 65, Springer-Verlag, Berlin, 2009.

[7]

M. Choulli and E. Soccorsi, Some inverse anisotropic conductivity problem induced by twisting a homogeneous cylindrical domain, preprint, arXiv:1209.5662.

[8]

G. Eskin, A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831. doi: 10.1088/0266-5611/22/3/005.

[9]

G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. PDE, 32 (2007), 1737-1758. doi: 10.1080/03605300701382340.

[10]

G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 1-18. doi: 10.1063/1.2841329.

[11]

C. Hamaker, K. T. Smith, D. C. Solomonand and S. C. Wagner, The divergent beam x-ray transform, Rocky Mountain J. Math., 10 (1980), 253-283. doi: 10.1216/RMJ-1980-10-1-253.

[12]

M. Ikehata, Inverse conductivity problem in the infinite slab, Inverse Problems, 17 (2001), 437-454. doi: 10.1088/0266-5611/17/3/305.

[13]

O. Yu. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations, Comm. PDE, 26 (2001), 1409-1425. doi: 10.1081/PDE-100106139.

[14]

V. Isakov, An inverse hyperbolic problem with many boundary measurements, Comm. PDE, 16 (1991), 1183-1195. doi: 10.1080/03605309108820794.

[15]

V. Isakov and Z. Sun, Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems, 8 (1992), 193-206. doi: 10.1088/0266-5611/8/2/003.

[16]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596. doi: 10.1088/0266-5611/8/4/009.

[17]

K. Krupchyk , M. Lassas and G. Uhlmann, Inverse Problems With Partial Data for a Magnetic Schrödinger Operator in an Infinite Slab and on a Bounded Domain, Communications in Mathematical Physics, 312 (2012), 87-126. doi: 10.1007/s00220-012-1431-1.

[18]

I. Lasiecka, J.-L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.

[19]

X. Li and G. Uhlmann, Inverse problems with partial data in a slab, Inverse Probl. Imaging, 4 (2010), 449-462. doi: 10.3934/ipi.2010.4.449.

[20]

J-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. I, Dunod, Paris, 1968.

[21]

J-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. II, Dunod, Paris, 1968.

[22]

S-I. Nakamura, Uniqueness for an Inverse Problem for the Wave Equation in the Half Space, Tokyo J. of Math., 19 (1996), 187-195. doi: 10.3836/tjm/1270043228.

[23]

F. Natterer, The Mathematics of Computarized Tomography, John Wiley & Sons, Chichester, 1986.

[24]

Rakesh, Reconstruction for an inverse problem for the wave equation with constant velocity, Inverse Problems, 6 (1990), 91-98. doi: 10.1088/0266-5611/6/1/009.

[25]

Rakesh, An inverse problem for the wave equation in the half plane, Inverse Problems, 9 (1993), 433-441. doi: 10.1088/0266-5611/9/3/005.

[26]

Rakesh and W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. PDE, 13 (1988), 87-96. doi: 10.1080/03605308808820539.

[27]

A. Ramm and J. Sjöstrand, An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130. doi: 10.1007/BF02571330.

[28]

M. Salo and J. N. Wang, Complex spherical waves and inverse problems in unbounded domains, Inverse Problems, 22 (2006), 2299-2309. doi: 10.1088/0266-5611/22/6/023.

[29]

P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358. doi: 10.1006/jfan.1997.3188.

show all references

References:
[1]

M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients, Appl. Anal., 83 (2004), 983-1014. doi: 10.1080/0003681042000221678.

[2]

M. Bellassoued, M. Choulli and M. Yamamoto, Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Diff. Equat., 247 (2009), 465-494. doi: 10.1016/j.jde.2009.03.024.

[3]

M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal., 85 (2006), 1219-1243. doi: 10.1080/00036810600787873.

[4]

M. Bellassoued, D. Jellali and M. Yamamoto, Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, J. Math. Anal. Appl, 343 (2008), 1036-1046. doi: 10.1016/j.jmaa.2008.01.098.

[5]

A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247.

[6]

M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques, Mathématiques et Applications, Vol. 65, Springer-Verlag, Berlin, 2009.

[7]

M. Choulli and E. Soccorsi, Some inverse anisotropic conductivity problem induced by twisting a homogeneous cylindrical domain, preprint, arXiv:1209.5662.

[8]

G. Eskin, A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831. doi: 10.1088/0266-5611/22/3/005.

[9]

G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. PDE, 32 (2007), 1737-1758. doi: 10.1080/03605300701382340.

[10]

G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 1-18. doi: 10.1063/1.2841329.

[11]

C. Hamaker, K. T. Smith, D. C. Solomonand and S. C. Wagner, The divergent beam x-ray transform, Rocky Mountain J. Math., 10 (1980), 253-283. doi: 10.1216/RMJ-1980-10-1-253.

[12]

M. Ikehata, Inverse conductivity problem in the infinite slab, Inverse Problems, 17 (2001), 437-454. doi: 10.1088/0266-5611/17/3/305.

[13]

O. Yu. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations, Comm. PDE, 26 (2001), 1409-1425. doi: 10.1081/PDE-100106139.

[14]

V. Isakov, An inverse hyperbolic problem with many boundary measurements, Comm. PDE, 16 (1991), 1183-1195. doi: 10.1080/03605309108820794.

[15]

V. Isakov and Z. Sun, Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems, 8 (1992), 193-206. doi: 10.1088/0266-5611/8/2/003.

[16]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596. doi: 10.1088/0266-5611/8/4/009.

[17]

K. Krupchyk , M. Lassas and G. Uhlmann, Inverse Problems With Partial Data for a Magnetic Schrödinger Operator in an Infinite Slab and on a Bounded Domain, Communications in Mathematical Physics, 312 (2012), 87-126. doi: 10.1007/s00220-012-1431-1.

[18]

I. Lasiecka, J.-L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.

[19]

X. Li and G. Uhlmann, Inverse problems with partial data in a slab, Inverse Probl. Imaging, 4 (2010), 449-462. doi: 10.3934/ipi.2010.4.449.

[20]

J-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. I, Dunod, Paris, 1968.

[21]

J-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. II, Dunod, Paris, 1968.

[22]

S-I. Nakamura, Uniqueness for an Inverse Problem for the Wave Equation in the Half Space, Tokyo J. of Math., 19 (1996), 187-195. doi: 10.3836/tjm/1270043228.

[23]

F. Natterer, The Mathematics of Computarized Tomography, John Wiley & Sons, Chichester, 1986.

[24]

Rakesh, Reconstruction for an inverse problem for the wave equation with constant velocity, Inverse Problems, 6 (1990), 91-98. doi: 10.1088/0266-5611/6/1/009.

[25]

Rakesh, An inverse problem for the wave equation in the half plane, Inverse Problems, 9 (1993), 433-441. doi: 10.1088/0266-5611/9/3/005.

[26]

Rakesh and W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. PDE, 13 (1988), 87-96. doi: 10.1080/03605308808820539.

[27]

A. Ramm and J. Sjöstrand, An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130. doi: 10.1007/BF02571330.

[28]

M. Salo and J. N. Wang, Complex spherical waves and inverse problems in unbounded domains, Inverse Problems, 22 (2006), 2299-2309. doi: 10.1088/0266-5611/22/6/023.

[29]

P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358. doi: 10.1006/jfan.1997.3188.

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