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Determining the potential in a wave equation without a geometric condition. Extension to the heat equation

Authors: Kaïs Ammari, Mourad Choulli and Faouzi Triki
Journal: Proc. Amer. Math. Soc. 144 (2016), 4381-4392
MSC (2010): Primary 35R30
Published electronically: April 20, 2016
MathSciNet review: 3531187
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Abstract: We prove a logarithmic stability estimate for the inverse problem of determining the potential in a wave equation from boundary measurements obtained by varying the first component of the initial condition. The novelty of the present work is that no geometric condition is imposed to the sub-boundary where the measurements are made. Our results improve those obtained by the first and second authors in an earlier work. We also show how the analysis for the wave equation can be adapted to an inverse coefficient problem for the heat equation.

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  • [1] Giovanni Alessandrini and John Sylvester, Stability for a multidimensional inverse spectral theorem, Comm. Partial Differential Equations 15 (1990), no. 5, 711-736. MR 1070844 (91i:35198),
  • [2] Carlos Alves, Ana Leonor Silvestre, Takéo Takahashi, and Marius Tucsnak, Solving inverse source problems using observability. Applications to the Euler-Bernoulli plate equation, SIAM J. Control Optim. 48 (2009), no. 3, 1632-1659. MR 2516181 (2010m:93045),
  • [3] Kaïs Ammari and Mourad Choulli, Logarithmic stability in determining two coefficients in a dissipative wave equation. Extensions to clamped Euler-Bernoulli beam and heat equations, J. Differential Equations 259 (2015), no. 7, 3344-3365. MR 3360675,
  • [4] K. Ammari and M. Choulli, Logarithmic stability in determining a boundary coefficient in an IBVP for the wave equation, arXiv:1505.07248.
  • [5] Gang Bao and Kihyun Yun, On the stability of an inverse problem for the wave equation, Inverse Problems 25 (2009), no. 4, 045003, 7. MR 2482154 (2010a:35270),
  • [6] M. I. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problems 13 (1997), no. 5, R1-R45. MR 1474359 (98k:58073),
  • [7] Mourad Bellassoued, Mourad Choulli, and Masahiro Yamamoto, Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Differential Equations 247 (2009), no. 2, 465-494. MR 2523687 (2010k:35523),
  • [8] Mourad Bellassoued, Global logarithmic stability in inverse hyperbolic problem by arbitrary boundary observation, Inverse Problems 20 (2004), no. 4, 1033-1052. MR 2087978 (2005f:35311),
  • [9] M. Bellassoued, D. Jellali, and M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal. 85 (2006), no. 10, 1219-1243. MR 2263922 (2007h:35342),
  • [10] 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), no. 2, 1036-1046. MR 2417121 (2009f:35342),
  • [11] Fernando Cardoso and Ramón Mendoza, On the hyperbolic Dirichlet to Neumann functional, Comm. Partial Differential Equations 21 (1996), no. 7-8, 1235-1252. MR 1399197 (97g:35016),
  • [12] Mourad Choulli, Une introduction aux problèmes inverses elliptiques et paraboliques, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 65, Springer-Verlag, Berlin, 2009 (French). MR 2554831 (2010m:35003)
  • [13] Victor Isakov and Zi Qi Sun, Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems 8 (1992), no. 2, 193-206. MR 1158175 (93g:35140)
  • [14] Alexander Katchalov, Yaroslav Kurylev, and Matti Lassas, Inverse boundary spectral problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 123, Chapman & Hall/CRC, Boca Raton, FL, 2001. MR 1889089 (2003e:58045)
  • [15] Otared Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 13, Springer-Verlag, Paris, 1993 (French, with French summary). MR 1276944 (95e:58036)
  • [16] Yaroslav V. Kurylev and Matti Lassas, Hyperbolic inverse problem with data on a part of the boundary, Differential equations and mathematical physics (Birmingham, AL, 1999), AMS/IP Stud. Adv. Math., vol. 16, Amer. Math. Soc., Providence, RI, 2000, pp. 259-272. MR 1764756 (2001f:58061)
  • [17] Vilmos Komornik and Masahiro Yamamoto, Estimation of point sources and applications to inverse problems, Inverse Problems 21 (2005), no. 6, 2051-2070. MR 2183667 (2006i:35372),
  • [18] Jean-Pierre Puel and Masahiro Yamamoto, Applications de la contrôlabilité exacte à quelques problèmes inverses hyperboliques, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 10, 1171-1176 (French, with English and French summaries). MR 1336250 (96b:93014)
  • [19] Rakesh, Reconstruction for an inverse problem for the wave equation with constant velocity, Inverse Problems 6 (1990), no. 1, 91-98. MR 1036380 (91d:35232)
  • [20] Rakesh and William W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations 13 (1988), no. 1, 87-96. MR 914815 (89f:35208),
  • [21] L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques, Asymptotic Anal. 10 (1995), no. 2, 95-115 (French, with French summary). MR 1324385 (96c:35130)
  • [22] Zi Qi Sun, On continuous dependence for an inverse initial-boundary value problem for the wave equation, J. Math. Anal. Appl. 150 (1990), no. 1, 188-204. MR 1059582 (91i:35024),
  • [23] Marius Tucsnak and George Weiss, From exact observability to identification of singular sources, Math. Control Signals Systems 27 (2015), no. 1, 1-21. MR 3306623,

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Additional Information

Kaïs Ammari
Affiliation: UR Analysis and Control of Pde, UR 13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisia

Mourad Choulli
Affiliation: Institut Élie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine, Boulevard des Aiguillettes, BP 70239, 54506 Vandoeuvre les Nancy cedex - Ile du Saulcy, 57045 Metz cedex 01, France

Faouzi Triki
Affiliation: Laboratoire Jean Kuntzmann, UMR CNRS 5224, Université de Joseph Fourier, 38041 Grenoble Cedex 9, France

Keywords: Inverse problem, wave equation, potential, boundary measurements
Received by editor(s): September 28, 2015
Received by editor(s) in revised form: December 19, 2015
Published electronically: April 20, 2016
Additional Notes: The third author was partially supported by Labex PERSYVAL-Lab (ANR-11-LABX-0025-01)
Communicated by: Joachim Krieger
Article copyright: © Copyright 2016 American Mathematical Society

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