Strong unique continuation for second-order hyperbolic equations with time-independent coefficients
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Strong unique continuation for second‑order hyperbolic equations with time‑independent coefficients Sergio Vessella1 Received: 16 December 2019 / Accepted: 14 September 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper we prove that )if u is a solution to second-order hyperbolic equation ( 𝜕t2 u + a(x)𝜕t u − ( divx A(x)∇x u + b(x) ⋅ ∇x u + c(x)u) = 0 and u is flat on a segment {x0 } × (−T, T) (T finite), then u vanishes in a neighborhood of {x0 } × (−T, T) . The novelty with respect to earlier papers on the subject is the nonvanishing damping coefficient a(x) in the hyperbolic equation. Keywords Unique continuation property · Stability estimates · Hyperbolic equations · Inverse problems Mathematics Subject Classification 35R25 · 35L10 · 35B60 · 35R30
1 Introduction In this paper we study strong unique continuation property (SUCP) for the equation
𝜕t2 u + a(x)𝜕t u − L(u) = 0,
(1.1)
in B𝜌0 × (−T, T),
where 𝜌0 , T are given positive numbers, B𝜌0 is the ball of ℝn , n ≥ 2 , of radius 𝜌0 and center at 0, a ∈ L∞ (ℝn ) , L is the second-order elliptic operator ( ) L(u) = divx A(x)∇x u + b(x) ⋅ ∇x u + c(x)u, (1.2)
b ∈ L∞ (ℝn ;ℝn ) , c ∈ L∞ (ℝn ) and A(x) is a real-valued symmetric n × n matrix that satisfies a uniform ellipticity condition and entries of A(x) are functions of Lipschitz class. We say that Eq. (1.1) has the SUCP if there exists a neighborhood U of {0} × (−T, T) such that for every solution, u, to Eq. (1.1) we have
* Sergio Vessella [email protected] 1
Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
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‖u‖L2 (Br ×(−T,T)) = O(rN ), ∀N ∈ ℕ, as r → 0,
⟹
u = 0, in U.
(1.3)
Property (1.3) was proved (if the matrix A belongs to C2 ), under the additional condition T = +∞ and u is bounded, by Masuda in 1968, [25]. Later on, in 1978, Baouendi and Zachmanoglou, [5], proved the SUCP whenever the coefficients of equation (1.1) are analytic functions. In 1999, Lebeau, [23], proved the SUCP for solution to (1.1) when a = b = c = 0 . The proof of [23] requires the symmetry of the differential operator, and there seems no obvious extension of the proof to the nonsymmetric case, in particular, to the case of damped wave equation 𝜕t2 u + a(x)𝜕t u − Δu = 0 . We also refer to [29, 32] where the SUCP at the boundary and the quantitative estimate of unique continuation related to property was proved when a = 0. The novelty of the present paper with respect to earlier papers, with finite T, is the nonvanishing damping coefficient a(x) in the hyperbolic Eq. (1.1). It is worth noting that SUCP and the related quantitative estimates, have been extensively studied and today well understood in the context of second-order elliptic and parabolic equation. Among the extensive literature on the subject here we mention, for the elliptic equations, [3, 4, 15, 19], and, for the parabolic equations, [2, 8, 20]. In the contex
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