A Stability Estimate for an Inverse Problem of Determining a Coefficient in a Hyperbolic Equation with a Point Source
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A Stability Estimate for an Inverse Problem of Determining a Coefficient in a Hyperbolic Equation with a Point Source Xue Qin1 · Shumin Li1
Received: 26 March 2016 / Revised: 6 July 2016 / Accepted: 8 July 2016 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2016
Abstract For the solution to ∂t2 u(x, t)−u(x, t)+q(x)u(x, t) = δ(x, t) and u |t 0} is a bounded domain, ST = {(x, t) | x ∈ ∂, |x| < t < T + |x|}, n = n(x) is the outward unit normal n to ∂, and T > 0. For suitable T > 0, prove a Lipschitz stability estimation: q1 − q2 L 2 () ≤ C f 1 − f 2 H 1 (ST ) + g1 − g2 L 2 (ST ) , provided that q1 satisfies a priori uniform boundedness conditions and q2 satisfies a priori uniform smallness conditions, where u k is the solution to problem (1.1) with q = qk , k = 1, 2. Keywords Inverse problem · Stability · Carleman estimate · Hyperbolic equation Mathematics Subject Classification 35R30 · 35R25 · 35L10
1 Introduction and Main Results Consider an inverse problem of determining a coefficient in a hyperbolic equation by an impulsive source located outside the domain where a coefficient is unknown (see, e.g., [19,20,22]). Let u(x, t), x = (x1 , x2 , x3 ) solve the Cauchy problem
B
Xue Qin [email protected] Shumin Li [email protected]
1
School of Mathematical Sciences, University of Science and Technology of China, No. 96, JinZhai Road Baohe District, Hefei, Anhui 230026, People’s Republic of China
123
X. Qin, S. Li
∂t2 u − u + q(x)u = δ(x, t), u |t 0 be suitably given,
|x| = x12 + x22 + x32 . Set
G T = {(x, t) | x ∈ , |x| < t < T + |x| } , 0 = {(x, t) | x ∈ , t = |x| + 0 } , T = {(x, t) | x ∈ , t = T + |x| } , ST = {(x, t) | x ∈ ∂, |x| < t < |x| + T } .
(1.2)
Consider: Inverse problem Let Cauchy data of the solution u to (1.1) be given on ST : u(x, t) = f (x, t),
∂ u(x, t) = g(x, t), (x, t) ∈ ST , ∂n
(1.3)
where n = n(x) is the outward unit normal vector to ∂ at x. Then determine q(x), x ∈ from given data (1.3). In order to state the main result, introduce the notations. Let r =(diam )/2. Assume that x10 > r > 0, and ⊆ B(x 0 , r ) = x ∈ R3 | |x − x 0 | < r , where x 0 = (x10 , 0, 0) ∈ R3+ . Set K = K (x 0 , T0 ) = (x, t) | |x| < t < T0 − |x − x 0 | ,
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(1.4)
A Stability Estimate for an Inverse Problem of Determining…
and choose T0 = T0 T, x 0 , r > T , so that G T ⊂ K . Denote by U = U (x 0 , T0 ) = x | |x| < T0 − |x − x 0 | , the projection of K on the space R3 . For any fixed positive constant Q, set Y (Q) = q(x) ∈ H 3 (U ) | qC(U ) ≤ Q .
(1.5)
Throughout this paper, H 1 (ST ), H 3 (U ), etc., denote usual Sobolev spaces (e.g., Adams [1]). By x10 > r > 0, one can choose a constant ε ∈ (1, 3/2) such that α :=
x10 − εr x10
∈ (0, 1)
(1.6)
and fix it. Then take a small positive constant β such that
2
2 0 < β < 1 and 0 < β rβ + 2x10 + r < (1 − α)x10 − r .
(1.7)
Now state the main result. Theorem 1.1 Let x10 > r > 0 and assume that satisfies (1.4). Let M > 0 and
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