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

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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

Abstract
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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 $(0,T)\times\partial\Omega$.

Keywords: scalar potential, wave equation, Inverse problem, infinite cylindrical domain..

Mathematics Subject Classification: 35R3.

Citation: Yavar Kian. Stability of the determination of a coefficient for wave equations in an infinite waveguide. Inverse Problems & Imaging, 2014, 8 (3) : 713-732. doi: 10.3934/ipi.2014.8.713

References:

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[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. doi: 10.1080/00036810600787873. Google Scholar
[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. doi: 10.1016/j.jmaa.2008.01.098. Google Scholar*
[5]	A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimensional inverse problems,, Sov. Math. Dokl., 24 (1981), 244. Google Scholar
[6]	M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques,, Mathématiques et Applications, (2009). Google Scholar
[7]	M. Choulli and E. Soccorsi, Some inverse anisotropic conductivity problem induced by twisting a homogeneous cylindrical domain,, preprint, (). Google Scholar
[8]	G. Eskin, A new approach to hyperbolic inverse problems,, Inverse Problems, 22 (2006), 815. doi: 10.1088/0266-5611/22/3/005. Google Scholar
[9]	G. Eskin, Inverse hyperbolic problems with time-dependent coefficients,, Comm. PDE, 32 (2007), 1737. doi: 10.1080/03605300701382340. Google Scholar
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[13]	O. Yu. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations,, Comm. PDE, 26 (2001), 1409. doi: 10.1081/PDE-100106139. Google Scholar
[14]	V. Isakov, An inverse hyperbolic problem with many boundary measurements,, Comm. PDE, 16 (1991), 1183. doi: 10.1080/03605309108820794. Google Scholar
[15]	V. Isakov and Z. Sun, Stability estimates for hyperbolic inverse problems with local boundary data,, Inverse Problems, 8 (1992), 193. doi: 10.1088/0266-5611/8/2/003. Google Scholar
[16]	M. V. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009. Google Scholar
[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. doi: 10.1007/s00220-012-1431-1. Google Scholar*
[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. Google Scholar
[19]	X. Li and G. Uhlmann, Inverse problems with partial data in a slab,, Inverse Probl. Imaging, 4 (2010), 449. doi: 10.3934/ipi.2010.4.449. Google Scholar
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[22]	S-I. Nakamura, Uniqueness for an Inverse Problem for the Wave Equation in the Half Space,, Tokyo J. of Math., 19 (1996), 187. doi: 10.3836/tjm/1270043228. Google Scholar
[23]	F. Natterer, The Mathematics of Computarized Tomography,, John Wiley & Sons, (1986). Google Scholar
[24]	Rakesh, Reconstruction for an inverse problem for the wave equation with constant velocity,, Inverse Problems, 6 (1990), 91. doi: 10.1088/0266-5611/6/1/009. Google Scholar
[25]	Rakesh, An inverse problem for the wave equation in the half plane,, Inverse Problems, 9 (1993), 433. doi: 10.1088/0266-5611/9/3/005. Google Scholar
[26]	Rakesh and W. Symes, Uniqueness for an inverse problem for the wave equation,, Comm. PDE, 13 (1988), 87. doi: 10.1080/03605308808820539. Google Scholar
[27]	A. Ramm and J. Sjöstrand, An inverse problem of the wave equation,, Math. Z., 206* (1991), 119. doi: 10.1007/BF02571330. Google Scholar*
[28]	M. Salo and J. N. Wang, Complex spherical waves and inverse problems in unbounded domains,, Inverse Problems, 22 (2006), 2299. doi: 10.1088/0266-5611/22/6/023. Google Scholar
[29]	P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media,, J. Funct. Anal., 154 (1998), 330. doi: 10.1006/jfan.1997.3188. Google Scholar

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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. doi: 10.1080/0003681042000221678. Google Scholar
[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. doi: 10.1016/j.jde.2009.03.024. Google Scholar*
[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. doi: 10.1080/00036810600787873. Google Scholar
[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. doi: 10.1016/j.jmaa.2008.01.098. Google Scholar*
[5]	A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimensional inverse problems,, Sov. Math. Dokl., 24 (1981), 244. Google Scholar
[6]	M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques,, Mathématiques et Applications, (2009). Google Scholar
[7]	M. Choulli and E. Soccorsi, Some inverse anisotropic conductivity problem induced by twisting a homogeneous cylindrical domain,, preprint, (). Google Scholar
[8]	G. Eskin, A new approach to hyperbolic inverse problems,, Inverse Problems, 22 (2006), 815. doi: 10.1088/0266-5611/22/3/005. Google Scholar
[9]	G. Eskin, Inverse hyperbolic problems with time-dependent coefficients,, Comm. PDE, 32 (2007), 1737. doi: 10.1080/03605300701382340. Google Scholar
[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. doi: 10.1063/1.2841329. Google Scholar
[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. doi: 10.1216/RMJ-1980-10-1-253. Google Scholar
[12]	M. Ikehata, Inverse conductivity problem in the infinite slab,, Inverse Problems, 17 (2001), 437. doi: 10.1088/0266-5611/17/3/305. Google Scholar
[13]	O. Yu. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations,, Comm. PDE, 26 (2001), 1409. doi: 10.1081/PDE-100106139. Google Scholar
[14]	V. Isakov, An inverse hyperbolic problem with many boundary measurements,, Comm. PDE, 16 (1991), 1183. doi: 10.1080/03605309108820794. Google Scholar
[15]	V. Isakov and Z. Sun, Stability estimates for hyperbolic inverse problems with local boundary data,, Inverse Problems, 8 (1992), 193. doi: 10.1088/0266-5611/8/2/003. Google Scholar
[16]	M. V. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009. Google Scholar
[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. doi: 10.1007/s00220-012-1431-1. Google Scholar*
[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. Google Scholar
[19]	X. Li and G. Uhlmann, Inverse problems with partial data in a slab,, Inverse Probl. Imaging, 4 (2010), 449. doi: 10.3934/ipi.2010.4.449. Google Scholar
[20]	J-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications,, Vol. I, (1968). Google Scholar
[21]	J-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications,, Vol. II, (1968). Google Scholar
[22]	S-I. Nakamura, Uniqueness for an Inverse Problem for the Wave Equation in the Half Space,, Tokyo J. of Math., 19 (1996), 187. doi: 10.3836/tjm/1270043228. Google Scholar
[23]	F. Natterer, The Mathematics of Computarized Tomography,, John Wiley & Sons, (1986). Google Scholar
[24]	Rakesh, Reconstruction for an inverse problem for the wave equation with constant velocity,, Inverse Problems, 6 (1990), 91. doi: 10.1088/0266-5611/6/1/009. Google Scholar
[25]	Rakesh, An inverse problem for the wave equation in the half plane,, Inverse Problems, 9 (1993), 433. doi: 10.1088/0266-5611/9/3/005. Google Scholar
[26]	Rakesh and W. Symes, Uniqueness for an inverse problem for the wave equation,, Comm. PDE, 13 (1988), 87. doi: 10.1080/03605308808820539. Google Scholar
[27]	A. Ramm and J. Sjöstrand, An inverse problem of the wave equation,, Math. Z., 206* (1991), 119. doi: 10.1007/BF02571330. Google Scholar*
[28]	M. Salo and J. N. Wang, Complex spherical waves and inverse problems in unbounded domains,, Inverse Problems, 22 (2006), 2299. doi: 10.1088/0266-5611/22/6/023. Google Scholar
[29]	P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media,, J. Funct. Anal., 154 (1998), 330. doi: 10.1006/jfan.1997.3188. Google Scholar

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