# 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)

## Abstract

Consider the second order hyperbolic equation
$${u_{{tt}}} - div\left( {a\nabla u} \right) + b{u_{t}} = 0{\text{ }}in{\text{ }}Q = \times \left( {0,T} \right)$$
(1.1)
with zero initial conditions
$$u = {u_{t}} = 0{\text{ }}on{\text{ }}\Omega \left\{ 0 \right\}$$
(1.2)
and the lateral Neumann boundary condition
$$a{\partial _{v}}u = h{\text{ }}on{\text{ }}\partial \Omega \times \left( {0,T} \right)$$
(1.3)
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.

### Keywords

Manifold Geophysics Acoustics

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### References

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.
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.
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.
7. [I1]
Isakov, V., A Nonhyperbolic Cauchy problem forbc and its Applications to Elasticity Theory, Comm. Pure Appl. Math., 39 (1986), 474–469.
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.
10. [I4]
Isakov, V., Completeness of products of solutions and some inverse problems for PDE, J. of Diff. Equat., 92 (1991), 305–317.
11. [I5]
Isakov, V., An Inverse Hyperbolic Problem with Many Boundary Measurements, Comm. in Part. Diff. Equat., 16 (1991), 1183–1197.
12. [I6]
Isakov, V., Uniqueness and Stability in Many dimensional Inverse Problems, Inverse Problems, 9 (1993), 579–621.
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.
15. [J]
John, F., Collected papers, vol. 1, Birkhauser-Verlag, Basel Boston, 1985.
16. [K]
Klibanov, M.V., Inverse Problems and Carleman Estimates, Inverse Problems, 7 (1991), 577–596.
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.
19. [P]
Powell, J., An unconditional estimate for solutions of the wave equation, J. of Math. Anal. Appl., 179 (1993), 179–187.
20. [R]
Rakesh, An inverse impedance transmission problem for the wave equation, Comm. in Part. Diff. Equat., 18 (1993), 583–600.
21. [RS]
Rakesh, Symes, W.W., Uniqueness for an inverse problem for the wave equation, Comm. in Part. Diff. Equat., 16 (1991), 789–801.
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.
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.