Stability estimate in determination of a coefficient in transmission wave equation by boundary observation

References

• Lions J-L, Magenes E. Non-homogenous boundary value problems and applications. Vol. I and II. Berlin: Springer-Verlag; 1972.
   Google Scholar
• Dautray R, Lions J-L. Analyse Mathématique et Calcul Numérique pour les Siences et les teechniques, evolution: semi-groupe, variationnel [Mathematics analysis and numerical computation for sciences and techniques, evolution: semigroup, variational]. Vol. 8. Paris: Masson; 1988.
   Google Scholar
• Ikawa M. Hyperbolique partial differential equations and wave phenomena. Providence (RI): American Mathematical Society; 2000.
   Google Scholar
• Baudouin L, Mercado A, Osses A. A global Carleman estimate in a transmission wave equation and application to one-measurement inverse problem. Inverse Probl. 2007;23:257–278.
   Web of Science ®Google Scholar
• Immanuvilov-M OYu, Yamamoto M. Determination of a coefficient in an acoustic equation with single measurement. Inverse Probl. 2003;19:157–171.
   Web of Science ®Google Scholar
• Bellassoued M, Yamamoto M. Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation. J. Math. Pures Appl. 2006;85:193–224.
   Web of Science ®Google Scholar
• Bellassoued M. Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients. Appl. Anal. 2004;83:983–1014.
   Google Scholar
• Robbiano L. Fonction de coût et contrôle des solutions des equations hyperboliques [Cost function and control solutions of hyperbolic equations]. Asymp. Anal. 1995;10:95–115.
   Google Scholar
• Robbiano L. Théorème d’unicité adapté au contrôle des solutions des problèmes hyperboliques [Uniqueness theorem adapted to control solutions of hyperbolic problems]. Comm. Par. Differ. Equ. 1991;16:789–800.
   Web of Science ®Google Scholar
• Jellali D. An inverse problem for the acoustic wave equation with finite sets of Boundary data. J. Inverse III-Posed Probl. 2006;14:665–684.
   Google Scholar
• Bellassoued M, Jellali D, Yamamoto M. Lipschitz stability in an inverse problem for a hyperbolic equation with a finite set of boundary data. Appl. Anal. 2008;87:1105–1119.
   Web of Science ®Google Scholar
• Bellassoued M. Global logarithmic stability in inverse hyperbolic problem by arbitrary boundary observation. Inverse probl. 2004;20:1033–1052.
   Web of Science ®Google Scholar
• Bellassoued M, Yamamoto M. Determination of a coefficient in the wave equation with a single measurement. Appl. Anal. 2008;87:901–920.
   Web of Science ®Google Scholar
• Immanuvilov OYu, Yamamoto M. Global uniqueness and stability in determining coefficients of wave equations. Comm. Part. Differ. Equ. 2001;26:1409–1425.
   Web of Science ®Google Scholar
• Immanuvilov OYu, Yamamoto M. Lipshitz stability in inverse parabolic problems by Carleman estimate. Inverse Probl. 1998;14:1229–1249.
   Web of Science ®Google Scholar
• Bukhgeim AL, Cheng J, Isakov V, Yamamoto M. Uniqueness in determining damping coefficients in hyperbolic equations. Int. Soc. Anal. Appl. Comput. 2001;9:27–46.
   Google Scholar
• Isakov V. Inverse problems for partial differential equations. Berlin: Springer-Verlag; 1998.
   Google Scholar
• Khaidarov A. On stability estimates in multidimensional inverse problems for differential equation. Soviet Math. Dokl. 1989;38:614–617.
   Google Scholar
• Isakov V, Yamamoto M. Carleman estimates with the Neumann boundary condition and its applications to the observability inequality and inverse problems. Comptemp. Math. 2000;268:191–225.
   Google Scholar
• Kubo M. Uniqueness in inverse hyperbolic problems – Carleman estimate for boundary value problems. J. Math. Kyoto Univ. 2000;40–3:451–473.
   Web of Science ®Google Scholar
• Puel-M JP, Yamamoto M. On a global estimate in a linear inverse hyperbolic problem. Inverse Probl. 1996;12:995–1002.
   Web of Science ®Google Scholar
• Yamamoto M. Uniqueness and stability in multidimentional hyperbolic inverse problems. J. Math. Pures App. 1999;78:65–98.
   Web of Science ®Google Scholar
• Bugheim AL, Klibanov MV. Global uniqueness of class of multidimensional inverse problems. Soviet Math. Dokl. 1981;24:244–247.
   Google Scholar
• Klibanov MV. Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J. Inverse Ill-Posed Probl. 2013;21:477–560.
   Web of Science ®Google Scholar
• Marcia F, Zuazua E. On the lack of observability for wave equations a gaussian approach. Asymptotic Anal. 2002;32:1–26.
   Web of Science ®Google Scholar
• Klibanov MV, Yamamoto M. Lipschitz stability estimate of an inverse problem for an acoustic equation. Appl. Anal. 2006;85:515–538.
   Google Scholar
• Lavrent’ev MM, Romonov VG, Shishatskii SP. III-posed problems of mathematical physics and analysis. Providence, RI: American Mathematical Society; 1986.
   Google Scholar
• Klibanov MV. Inverse problems and Carleman estimates. Inverse Probl. 1992;8:575–596.
   Web of Science ®Google Scholar
• Tataru D. Carleman estimates, unique continuation and applications. Forthcoming 1999.
   Google Scholar
• Klibanov MV, Timonov AA. Carleman estimates for coefficient inverse and numerical applications. Inverse and ill-posed problems, series. Utrecht: VSP; 2004.
   Google Scholar
• Beilina L, Klibanov MV. Approximate global convergence and adaptivity for coefficient inverse problems. New York (NY): Springer; 2012.
   Google Scholar
• Kazemi MA, Klibanov MV. Stability estimates for ill-posed Cauchy problems involving hyperbolic equations and inequality. Appl. Anal. 1993;50:93–102.
   Google Scholar
• Klibanov MV, Malinsky J. Newton-Kantorovich method for 3-dimensional potential inverse scattering problem and stability for the hyperbolic Cauchy problem with time dependent data. Inverse Probl. 1991;7:577–596.
   Web of Science ®Google Scholar
• Carleman T. Sur un problème d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendants [A problem of uniqueness for systems of partial differential equations with two independent variables]. Ark. Mat. Astr. Fys. 1939;2B:1–9.
   Google Scholar
• Bukhgeim AL. Introduction to the theory of inverse problems. Norosibirsk: Nauka; 1988.
   Google Scholar
• Tataru D. Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures App. 1996;75:367–408.
   Web of Science ®Google Scholar