On Reconstruction of the Diffusion and of the Principal Coefficient of a Hyperbolic Equation

  • Victor Isakov
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 90)


Consider the second order hyperbolic equation
$${u_{{tt}}} - div\left( {a\nabla u} \right) + b{u_{t}} = 0{\text{ }}in{\text{ }}Q = \times \left( {0,T} \right)$$
with zero initial conditions
$$u = {u_{t}} = 0{\text{ }}on{\text{ }}\Omega \left\{ 0 \right\}$$
and the lateral Neumann boundary condition
$$a{\partial _{v}}u = h{\text{ }}on{\text{ }}\partial \Omega \times \left( {0,T} \right)$$
Under rather general assumptions: a, 6 do not depend on x, are in L∞(Ω), a is C 1 near ∂Ω, and > ∈ > 0 this initial boundary value problem has a unique solution u, u t ∈ H 1 2 (Q) provided h, h t ∈ L2(öΩ × (0, T)), h = 0 on öΩ × {0}, and ∞ is a bounded domain in ℝ n with the boundary öΩ ∈ C 1 or fi is a half-space in ℝ n (and then h is assumed to be compactly supported). A solution u is understood in the generalized sense as a function satisfying the following integral equality \(\int\limits_{Q} {\left( {{u_{t}}{v_{t}} - a\nabla u \cdot \nabla v - b{u_{t}}v} \right)dx{\text{ }}dt = {\text{ }}\int\limits_{{\partial \Omega \times \left( {0,T} \right)}} {h{\text{ }}v{\text{ }}dS\left( {x,t} \right)} }\) for any function v from the same class as u but satisfying the zero Cauchy data on ∞ × {T}. In geophysics of fundamental importance is the following inverse problem: find a,b given either for one h (single boundary measurement) or for all (regular) h supported in F × (0,T). Here Γ is a part of öΩ which can coincide with öΩ, but in most of geophysical applications it is relatively small part of it. In fact, F is the observation area.


Manifold Geophysics Acoustics 


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  1. [B]
    Belishev, M., Wave bases in multidimensional inverse problems, Mat. Sb., 180 (1989), 584–602.Google Scholar
  2. [BK]
    Belishev, M., Kurylev, Ya., Boundary Control, wave field continuation, and inverse problems for the wave equation, Comput. Math. Appl., 22 (1991), 27–52.MathSciNetMATHCrossRefGoogle Scholar
  3. [BuK]
    Bukhgeim, A.L., Klibanov, M.V., Uniqueness in the large of a class of multidimensional inverse problems, Soviet Math. Dokl., 24 (1981), 244–247.Google Scholar
  4. [FI]
    Friedman, A., Isakov, V., On the Uniqueness in the Inverse Conductivity Problem with One Measurement, Indiana Univ. Math. J., 38 (1989), 553–580.MathSciNetGoogle Scholar
  5. [H]
    Hadamard, J., Lectures on Cauchy’s Problem in linear partial differential equations, Dover, New York, 1953.Google Scholar
  6. [Ha]
    Hansen, S., Solution of a Hyperbolic Inverse Problem by Linearization, Comm. in Part. Diff. Equat., 16 (1991), 291–309.MATHCrossRefGoogle Scholar
  7. [I1]
    Isakov, V., A Nonhyperbolic Cauchy problem forbc and its Applications to Elasticity Theory, Comm. Pure Appl. Math., 39 (1986), 474–469.MathSciNetCrossRefGoogle Scholar
  8. [I2]
    Isakov, V., Inverse Source Problems, Math. Surveys and Monographs Series, Vol. 34, AMS, Providence, R.I., 1990.Google Scholar
  9. [I3]
    Isakov, V., On uniqueness in the inverse scattering problem, Comm. in Part. Diff. Equat., 15 (1990), 1565–1581.MathSciNetMATHCrossRefGoogle Scholar
  10. [I4]
    Isakov, V., Completeness of products of solutions and some inverse problems for PDE, J. of Diff. Equat., 92 (1991), 305–317.MathSciNetMATHCrossRefGoogle Scholar
  11. [I5]
    Isakov, V., An Inverse Hyperbolic Problem with Many Boundary Measurements, Comm. in Part. Diff. Equat., 16 (1991), 1183–1197.MathSciNetMATHCrossRefGoogle Scholar
  12. [I6]
    Isakov, V., Uniqueness and Stability in Many dimensional Inverse Problems, Inverse Problems, 9 (1993), 579–621.MathSciNetMATHCrossRefGoogle Scholar
  13. [I7]
    Isakov, V., On uniqueness of discontinuity surface of the speed of propagation, J. of Inverse and I11-Posed Problems (to appear).Google Scholar
  14. [IS]
    Isakov, V., and Sun, Z., Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems, 8 (1992), 193–206.MathSciNetMATHCrossRefGoogle Scholar
  15. [J]
    John, F., Collected papers, vol. 1, Birkhauser-Verlag, Basel Boston, 1985.MATHGoogle Scholar
  16. [K]
    Klibanov, M.V., Inverse Problems and Carleman Estimates, Inverse Problems, 7 (1991), 577–596.MathSciNetMATHCrossRefGoogle Scholar
  17. [L]
    Lop Fat Ho, Observabilite frontiere de’lequation des ondes, C.R. Acad. Sc. Paris, t.302, ser. I, # 12 (1986), 443–446.Google Scholar
  18. [N]
    Natterer, F., The Mathematics of Computerized Tomography, Wiley, New York, 1986.MATHGoogle Scholar
  19. [P]
    Powell, J., An unconditional estimate for solutions of the wave equation, J. of Math. Anal. Appl., 179 (1993), 179–187.MATHCrossRefGoogle Scholar
  20. [R]
    Rakesh, An inverse impedance transmission problem for the wave equation, Comm. in Part. Diff. Equat., 18 (1993), 583–600.MathSciNetMATHCrossRefGoogle Scholar
  21. [RS]
    Rakesh, Symes, W.W., Uniqueness for an inverse problem for the wave equation, Comm. in Part. Diff. Equat., 16 (1991), 789–801.CrossRefGoogle Scholar
  22. [Ro]
    Romanov, V.G., Inverse Problems of Mathematical Physics, VNU Sc. Press, Utrecht, 1987.Google Scholar
  23. [SU]
    Sylvester, J., Uhlmann, G., Inverse Problems in Anisotropic Media, Contemporary Mathematics, 122 (1991), 105–117.MathSciNetCrossRefGoogle Scholar
  24. [T]
    Tataru, D., Unique continuation for solutions to PDE’s: between Hormander’s Theorem and Holmgren’s Theorem, Comm. in Part. Diff. Equat., 20 (1995), 855–884.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Victor Isakov
    • 1
  1. 1.Department of Mathematics and StatisticsWichita State UniversityWichitaUSA

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