Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Dirichlet B.C. through an additional localized Neumann boundary trace

Abstract

We consider a second-order hyperbolic equation on an open bounded domain Ω in ℝ ⁿ for n ≥ 2, with C ²-boundary , Γ₀ ∩ Γ₁ = ∅, subject to non-homogeneous Dirichlet boundary conditions on the entire boundary Γ. We then study the inverse problem of determining the interior damping coefficient of the equation by means of an additional measurement of the Neumann boundary trace of the solution, in a suitable, explicit sub-portion Γ₁ of the boundary Γ, and over a computable time interval T > 0. Under sharp conditions on the complementary part Γ₀ = Γ\Γ₁, T > 0, and under sharp regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and Lipschitz stability (at the H ^θ-level, 0 < θ ≤ 1, ). The latter (ii) is the main result of this article. Our proof relies on three main ingredients: (a) sharp Carleman estimates at the H ¹ × L ²-level for second-order hyperbolic equations [I. Lasiecka, R. Triggiani, and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot, Contemp. Math. 268 (2000), pp. 227–325] (b) a correspondingly implied continuous observability inequality at the same energy level Lasiecka et al.; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Dirichlet boundary data [I. Lasiecka, J.L. Lions, and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), pp. 149–192; I. Lasiecka and R. Triggiani, A cosine operator approach to modelling L ₂(0, T; L ₂(Ω)) boundary input hyperbolic equations, Appl. Math. Optimiz., 7 (1981), 35–83; I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under L ₂(0, T; L ₂(Ω))-Dirichlet boundary terms, Appl. Math. Optimiz., 10 (1983), pp. 243–290]. The proof of the linear uniqueness result (Section 4, Step 5) also takes advantage of a convenient tactical route ‘post-Carleman estimates’ suggested by Isakov in [V. Isakov and M. Yamamoto, Carleman estimate with Neumann B.C. and its application to the observability inequality and inverse hyperbolic problems, Contemp. Math. 268 (2000), pp. 191–225, Theorem 8.2.2, p. 231].

Keywords:

• inverse problem
• uniqueness
• stability
• Carleman estimate

AMS Subject Classifications::

• 35R30
• 35L10
• 49K20

Acknowledgements

We warmly thank two referees for their useful comments and careful reading. The second author's research was partially supported by the National Science Foundation under grant DMS-0104305 and by the Air Force Office of Scientific Research under grant FA 9550-09-1-0459.