American Institute of Mathematical Sciences

Advanced
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

Yavar Kian 1,

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 $(0,T)\times\partial\Omega$.
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:
[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

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.  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

[1]

Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057
[2]

Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems & Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019
[3]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427
[4]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387
[5]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[6]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems & Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117
[7]

Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307
[8]

Lauri Oksanen. Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements. Inverse Problems & Imaging, 2011, 5 (3) : 731-744. doi: 10.3934/ipi.2011.5.731
[9]

Jussi Korpela, Matti Lassas, Lauri Oksanen. Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation. Inverse Problems & Imaging, 2019, 13 (3) : 575-596. doi: 10.3934/ipi.2019027
[10]

A. Ducrot. Travelling wave solutions for a scalar age-structured equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 251-273. doi: 10.3934/dcdsb.2007.7.251
[11]

Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007
[12]

Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control & Related Fields, 2015, 5 (1) : 177-189. doi: 10.3934/mcrf.2015.5.177
[13]

Anna Doubova, Enrique Fernández-Cara. Some geometric inverse problems for the linear wave equation. Inverse Problems & Imaging, 2015, 9 (2) : 371-393. doi: 10.3934/ipi.2015.9.371
[14]

Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176
[15]

Patrick Martinez, Judith Vancostenoble. The cost of boundary controllability for a parabolic equation with inverse square potential. Evolution Equations & Control Theory, 2019, 8 (2) : 397-422. doi: 10.3934/eect.2019020
[16]

Bernard Brighi, S. Guesmia. Asymptotic behavior of solution of hyperbolic problems on a cylindrical domain. Conference Publications, 2007, 2007 (Special) : 160-169. doi: 10.3934/proc.2007.2007.160
[17]

Xiaomei Sun, Yimin Zhang. Elliptic equations with cylindrical potential and multiple critical exponents. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1943-1957. doi: 10.3934/cpaa.2013.12.1943
[18]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131
[19]

Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37
[20]

Julien Jimenez. Scalar conservation law with discontinuous flux in a bounded domain. Conference Publications, 2007, 2007 (Special) : 520-530. doi: 10.3934/proc.2007.2007.520

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences