RECONSTRUCTION FOR AN INVERSE PROBLEM FOR THE WAVE EQUATION WITH CONSTANT VELOCITY Rakesh

Let Ω ⊂ R, n > 1 be a bounded domain with smooth boundary. Consider utt −4xu + q(x)u = 0 in Ω× [0, T ] u(x, 0) = 0, ut(x, 0) = 0 if x ∈ Ω u(x, t) = f(x, t) on ∂Ω× [0, T ] Define the Dirichlet to Neumann map Λq : H (∂Ω× [0, T ] ) −→ L(∂Ω× [0, T ] ) f(x, t) 7−→ ∂u ∂n where ~n is the unit outward normal to ∂Ω. If T > diam(Ω), we express q(x) in terms of Λq, and show that if q(x) is piecewise constant then q(x) is uniquely determined by Λq(fi), i = 1 . . . N for some integer N . Our problem is motivated by a desire to obtain properties e.g. density of an inhomogeneous medium by probing it with disturbances generated on the boundary. The data is the response of the medium to these disturbances, measured on the boundary, and the goal is to recover the function which measures the property of the medium. We consider a very simple model. Let Ω ⊂ R, n > 1 be a bounded domain with smooth boundary, which is the region occupied by our medium. Suppose q(x) is a function on Ω which represents some property of the medium. If u(x, t) is a measure of the magnitude of the disturbance at the point x at time t, then the interaction of the medium with the disturbance is assumed to be modelled by 2u + q(x)u ≡ utt −4xu + q(x)u = 0 in Ω× [0, T ] u(x, 0) = 0, ut(x, 0) = 0 if x ∈ Ω (1) u(x, t) = f(x, t) on ∂Ω× [0, T ] Here we have assumed that the medium is quiet initially, and f(x, t) is the disturbance which is used to probe the medium. Roughly speaking, the data is ∂u ∂n measured on ∂Ω× [0, T ] , for different choices of f(x, t), and the goal is to recover q(x). Here ~n is the outward pointing normal to ∂Ω. For a fixed q(x), (1) is a well posed initial boundary value problem, hence one may define the Dirichlet to Neumann map Λq : H (∂Ω× [0, T ] ) −→ L(∂Ω× [0, T ] ) f(x, t) 7−→ ∂u ∂n So knowing Λq is equivalent to knowing the response of the medium, on the boundary, to all possible input disturbances. In [3] Rakesh and Symes showed that the map L∞(Ω) −→ B( H(∂Ω× [0, T ] ) , L(∂Ω× [0, T ] ) ) q 7−→ Λq is injective provided T > diameter(Ω). In [7] Sun showed that that the above map restricted to H(Ω), s > n/2, is an open map for T > diameter(Ω). In this paper we prove two results. First we show that if q(x) is piecewise constant on a known grid on Ω, then q(x) is uniquely determined given Λq(f) for a finite number of f (only a Uniqueness Theorem). Note that the uniqueness result in the general case needed a knowledge of Λq(f) for an infinite