On the determination of wave speed and potential in a hyperbolic equation by two measurements

We discuss a problem of finding a speed of sound c(x) and a potential q(x) in a second-order hyperbolic equation from two boundary observations. The coefficients are assumed to be unknown inside a disc in R. On a suitable bounded part of the cylindrical surface, we are given Cauchy data for solutions to a hyperbolic equation with zero initial data and sources located on the lines {(x, t) ∈ R3|x · ν = 0, t = 0} for two distinct unit vectors ν = ν, k = 1, 2. We obtain a conditional stability estimate under a priori assumptions on smallness of c(x)− 1 and q(x). §1. Statement of the inverse problem and main results In the papers [2], [6] [9], a new method for obtaining conditional stability estimates for problems related to determination of coefficients for linear hyperbolic equations has been proposed. This method uses a single observation for finding one unknown coefficient. By our method, we can prove the stability in determining coefficients by means of a finite number of measurements where initial data are zero and impulsive inputs are added. As other methodology for inverse problems with a finite number of measurements, we refer to [1], [4], [5] and the references therein. However in those papers, we have to assume some positivity or non-degeneracy of initial values, which is not practical. For our method, we need not such restrictions on initial data, which is very practical. On the other hand, we have to assume that unknown coefficients should be close to fixed reference coefficients which are constant. An analysis shows that the problem with several unknown coefficients under the derivatives of the first order can also be successfully studied by this method (see [7], [8]). However its application to determination of coefficients under derivatives of different orders meets some Sobolev Institute of Mathematics of Siberian Division of Russian Academy of Sciences, Acad. Koptyug prospekt 4, 630090 Novosibirsk Russia; e-mail: romanov@math.nsc. ru Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153 Japan; e-mail: myama@ms.u-tokyo.ac. jp 1 difficulties. Recently the problems of finding a damping coefficient and a potential from two measurements, and the speed of sound and damping, were considered in papers [3] and [10], respectively. In this paper, by two measurements, we consider the inverse problem where coefficients of the leading term and the lowest term are unknown. The technique of this paper differs from [3] and [10], but keeps some common features with them. Let u = u(x, t), x ∈ R, satisfy the equation utt − c(∆u+ qu) = 2δ(t) δ(x · ν), (x, t) ∈ R, (1.1) and the zero initial condition u|t<0 = 0. (1.2) Here ν is a unit vector and the symbol x · ν means the scalar product of the vectors x and ν. The solution to problem (1.1) (1.2) depends on the parameter ν, i.e., u = u(x, t, ν). Assume that supports of the coefficients q(x) and c(x) − 1 are located strictly inside the disc B := {x ∈ R2| |x− x0| < r} and B belongs to the half-plane x · ν > 0. Suppose also that q(x) and c(x) > 0 are smooth functions in R (see below). Introduce the function τ(x, ν) as the solution to the following problem for the eikonal equation: |∇τ |2 = c−2(x), τ |x·ν=0 = 0. (1.3) Let G(ν) be the cylindrical domain G(ν) := {(x, t)| x ∈ B, τ(x, ν) < t < T + τ(x, ν)} where T is a positive number. Denote by S(ν) the lateral boundary of this domain and by Σ0(ν) and ΣT (ν) the lower and upper basements, respectively. That is, S(ν) := {(x, t)| x ∈ ∂B, τ(x, ν) ≤ t ≤ T + τ(x, ν)}, Σ0(ν) := {(x, t)| x ∈ B, t = τ(x, ν)}, ΣT (ν) := {(x, t)| x ∈ B, t = T + τ(x, ν)}, ∂B := {x ∈ R2| |x− x0| = r}. Consider the problem of determination of q(x) and c(x). Let the following information be known. We take distinct unit vectors ν and ν such that B belongs to the half plane x · ν > 0 for k = 1, 2. Then we are given the traces of the functions τ(x, ν) on ∂B, and 2 the traces on S(ν) := Sk of solutions and its normal derivatives to problem (1.1) (1.2) with ν = ν, that is, u(x, t, ν) = f (x, t), ∂ ∂n u(x, t, ν) = g(x, t), (x, t) ∈ Sk; τ(x, ν) = τ (x), x ∈ ∂B; k = 1, 2. (1.4) The problem is: find q(x) and c(x) from given data, i.e., from f , g, τ , k = 1, 2. For fixed constants q0 > 0 and d > 0, let Λ(q0, d) be the set of functions (q, c) satisfying the following two conditions: 1) supp q(x), supp (c(x)− 1) ⊂ Ω ⊂ B, dist(∂B,Ω) ≥ d, 2) ‖q‖C17(Rn) ≤ q0, ‖c− 1‖C19(Rn) ≤ q0. In particular, we note that ν and ν are linearly independent. We prove here the following stability and uniqueness theorems. Theorem 1.1. Let (qj, cj) ∈ Λ(q0, d), and let {f (k) j , g j , τ (k) j } be the data corresponding to the solution to (1.1) (1.2) with q = qj(x), c = cj(x) and ν = ν , k, j = 1, 2. Moreover let the condition 4r/T < 1 be satisfied. Then there exist positive numbers q∗ and C depending on T , r, d and |ν (1) − ν (2)| such that for all q0 ≤ q∗ the following inequality holds: ‖q1 − q2‖L2(B) + ‖c1 − c2‖H2(B) ≤ C 2 ∑ k=1 ( ‖f̂ (k) 1 − f̂ (k) 2 ‖H3(∂B×{0}) + ‖(f̂ (k) 1 − f̂ (k) 2 )t‖H2(∂B×(0,T )) (1.5) +‖(ĝ 1 − ĝ 2 )t‖H1(∂B×(0,T )) + ‖τ (k) 1 − τ (k) 2 ‖H5(∂B) ) , where f̂ (k) j (x, t) = f (k) j (x, t− τ (k) j (x)) and ĝ j (x, t) = g j (x, t− τ (k) j (x)). Theorem 1.2. Let the conditions the Theorem 1.1 be fulfilled. Then one can find a number q∗ > 0 such that if (qj , cj) ∈ Λ(q∗, d), j = 1, 2, and the corresponding data partly coincide, namely, f (k) 1 (x, t) = f (k) 2 (x, t), (x, t) ∈ Sk; τ (k) 1 (x) = τ (k) 2 (x), x ∈ ∂B; k = 1, 2, (1.6)