Inverse problems for general second order hyperbolic equations with time-dependent coefficients

Bulletin of Mathematical Sciences - Tập 7 - Trang 247-307 - 2017
G. Eskin1
1Department of Mathematics, UCLA, Los Angeles, USA

Tóm tắt

We study the inverse problems for the second order hyperbolic equations of general form with time-dependent coefficients assuming that the boundary data are given on a part of the boundary. The main result of this paper is the determination of the time-dependent Lorentzian metric by the boundary measurements. This is achieved by the adaptation of a variant of the boundary control method developed by Eskin (Inverse Probl 22(3):815–833, 2006; Inverse Probl 23:2343–2356, 2007).

Tài liệu tham khảo

Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Opt. 30, 1024–1065 (1992) Belishev, M.: An approach to multidimensional inverse problems for the wave equation. Sov. Math. Dokl. 36(3), 481–484 (1988) Belishev, M.: Boundary control in reconstruction of manifolds and metrics (the BC method). Inverse Probl. 13(5), R1–R45 (1997) Belishev, M.: How to see waves under the Earthsurface (the BC-method for geophysicists). Ill-Posed Inverse Prob. (S.Kabanikhin and V.Romanov (Eds), VSP, Zeist, pp. 67–84 (2002) Belishev, M.: Recent progress in boundary control method. Inverse Probl. 23(5), R1–R67 (2007) Belishev, M., Kurylev, Y.: The reconsruction of the Riemannian manifolds via its spectral data. Commun. Partial Differ. Equ. 17, 767–804 (1992) Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. II. Wiley Interscience, New York (1962) Eskin, G.: A new approach to the hyperbolic inverse problems. Inverse Probl. 22(3), 815–833 (2006) Eskin, G.: A new approach to the hyperbolic inverse problems II: global step. Inverse Probl. 23, 2343–2356 (2007) Eskin, G.: Inverse hyperbolic problems with time-dependent coefficients. Commun. Partial Differ. Equ. 32, 1737–1758 (2007) Eskin, G.: Optical Aharonov–Bohm effect: an inverse hyperbolic problems approach. Commun. Math. Phys. 284, 317–343 (2008) Eskin, G.: Lectures on Linear Partial Differential Equations, Graduate Studies in Mathematics, vol. 123, AMS (2011) Eskin, G.: Mixed initial–boundary value problems for second order hyperbolic equations. Commun. Partial Differ. Equ. 12, 503–587 (1987) Eskin, G.: Inverse problems for Schrödinger equations with Yang–Mills potentials in domains with obstacles and the Aharonov–Bohm effect. Inst. Phys. Conf. Ser. 12, 23–32 (2005) Hormander, L.: The Analysis of Linear Partial Differential Operators III. Springer, Berlin (1985) Hirsch, M.: Differential Topology. Springer, New York (1976) Isakov, V.: An inverse hyperbolic problem with many boundary measurements. Commun. Partial Differ. Equ. 16, 1183–1195 (1991) Isakov, V.: Inverse problems for partial differential equations. Appl. Math. Stud., vol. 127, Springer, 284 pp (1998) Katchalov, A., Kurylev, Y., Lassas, M.: Inverse boundary spectral problems. Chapman&Hall, Boca Baton (2001) Katchalov, A., Kurylev, Y., Lassas, M.: Energy measurements and equivalence of boundary data for inverse problems on noncompact manifolds. IMA Vol. 137, 183–214 (2004) Kurylev, Y., Lassas, M.: Hyperbolic inverse problems with data on a part of the boundary. AMS/1P Stud. Adv. Math. 16, 259–272 (2000) Kurylev, Y., Lassas, M.: Hyperbolic inverse boundary value problems and time-continuation of the non-stationary Dirichlet-to-Neumann map. Proc. R. Soc. Edinb. 132, 931–949 (2002) Lee, J., Uhlmann, G.: Determining anisotropic real analytic conductivity by boundary measurements. Commun. Pure Appl. Math. 42, 1097–112 (1989) Ramm, A., Sjostrand, J.: An inverse problem of the wave equation. Math. Z. 206, 119–130 (1991) Robbiario, L., Zuily, C.: Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients. Invent. Math. 131(3), 493–539 (1998) Salazar, R.: Determination of time-dependent coefficients for a hyperbolic inverse problem. Inverse Probl. 29(9), 095015 (2013) Salazar, R.: Stability estimate for the relativistic Schrödinger equation with time-dependent vector potentials. Inverse Probl. 30(10), 105005 (2014) Stefanov, P.: Uniqueness of multidimensional inverse scattering problem with time-dependent potentials. Math. Z. 201, 541–549 (1989) Tataru, D.: Unique continuation for operators with partially analytic coefficients. J. Math. Pures Appl. 78(5), 505–521 (1999)