Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary. (English) Zbl 1348.35314
Summary: We consider the multidimensional inverse problem of determining the conductivity coefficient of a hyperbolic equation in an infinite cylindrical domain, from a single boundary observation of the solution. We prove Hölder stability with the aid of a Carleman estimate specifically designed for hyperbolic waveguides.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35L30 | Initial value problems for higher-order hyperbolic equations |
Keywords:
inverse problem; hyperbolic equation; conductivity coefficient; Carleman estimate; infinite cylindrical domainReferences:
| [1] | M. Bellassoued, Global logarithmic stability in inverse hyperbolic problem by arbitrary boundary observation,, Inverse Problems, 20, 1033 (2004) · Zbl 1061.35162 · doi:10.1088/0266-5611/20/4/003 |
| [2] | M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients,, Applicable Analysis, 83, 983 (2004) · Zbl 1069.35035 · doi:10.1080/0003681042000221678 |
| [3] | M. Bellassoued, Inverse boundary value problem for the dynamical heterogeneous Maxwell’s system,, Inverse Problems, 28 (2012) · Zbl 1250.35181 · doi:10.1088/0266-5611/28/9/095009 |
| [4] | M. Bellassoued, Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem,, J. Diff. Equat., 247, 465 (2009) · Zbl 1171.35123 · doi:10.1016/j.jde.2009.03.024 |
| [5] | M. Bellassoued, Lipschitz stability in in an inverse problem for a hyperbolic equation with a finite set of boundary data,, Applicable Analysis, 87, 1105 (2008) · Zbl 1151.35395 · doi:10.1080/00036810802369231 |
| [6] | M. Bellassoued, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation,, J. Math. Pures Appl., 85, 193 (2006) · Zbl 1091.35112 · doi:10.1016/j.matpur.2005.02.004 |
| [7] | M. Bellassoued, Determination of a coefficient in the wave equation with a single measurement,, Applicable Analysis, 87, 901 (2008) · Zbl 1149.35404 · doi:10.1080/00036810802369249 |
| [8] | A. L. Bukhgeim, Global uniqueness of class of multidimentional inverse problems,, Soviet Math. Dokl., 24, 244 (1981) |
| [9] | L. Cardoulis, Inverse problem for the Schrödinger operator in an unbounded strip,, C. R. Math. Acad. Sci. Paris, 346, 635 (2008) · Zbl 1147.35109 · doi:10.1016/j.crma.2008.04.004 |
| [10] | L. C. Evans, <em>Partial Differential Equations</em>,, Amer. Math. Soc. (2010) · Zbl 1194.35001 · doi:10.1090/gsm/019 |
| [11] | O. Imanuvilov, Global Lipschitz stability in an inverse hyperbolic problem by interior observations,, Inverse Problems, 17, 717 (2001) · Zbl 0983.35151 · doi:10.1088/0266-5611/17/4/310 |
| [12] | O. Imanuvilov, Determination of a coefficient in an acoustic equation with single measurement,, Inverse Problems, 19, 157 (2003) · Zbl 1020.35117 · doi:10.1088/0266-5611/19/1/309 |
| [13] | V. Isakov, Stability estimates for hyperbolic inverse problems with local boundary data,, Inverse Problems, 8, 193 (1992) · Zbl 0754.35184 · doi:10.1088/0266-5611/8/2/003 |
| [14] | V. Isakov, <em>Inverse Problems for Partial Differential Equations</em>,, Springer-Verlag (2006) · Zbl 1092.35001 |
| [15] | Y. Kian, Carleman estimate for infinite cylindrical quantum domains and application to inverse problems,, Inverse Problems, 30 (2014) · Zbl 1309.35136 · doi:10.1088/0266-5611/30/5/055016 |
| [16] | Y. Kian, Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains,, Journal of Mathematical Analysis and Applications, 426, 194 (2015) · Zbl 1328.35313 · doi:10.1016/j.jmaa.2015.01.028 |
| [17] | M. V. Klibanov, Newton-Kantorovich method for 3-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time dependent data,, Inverse Problems, 7, 577 (1991) · Zbl 0744.35065 · doi:10.1088/0266-5611/7/4/007 |
| [18] | M. V. Klibanov, Lipschitz stability of an inverse problem for an accoustic equation,, Applicable Analysis, 85, 515 (2006) · Zbl 1274.35413 · doi:10.1080/00036810500474788 |
| [19] | J.-L. Lions, <em>Problèmes aux Limites non Homogènes et Applications</em>,, vol. 1 (1968) · Zbl 0165.10801 |
| [20] | P. Stefanov, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media,, J. Funct. Anal., 154, 330 (1998) · Zbl 0915.35066 · doi:10.1006/jfan.1997.3188 |
| [21] | M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl., 78, 65 (1999) · Zbl 0923.35200 · doi:10.1016/S0021-7824(99)80010-5 |
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