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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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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