Global Uniqueness and Stability in Determining the Damping Coefficient of an Inverse Hyperbolic Problem with NonHomogeneous Neumann B.C. through an Additional Dirichlet Boundary Trace

• Published in: SIAM journal on mathematical analysis Vol. 43; no. 4; pp. 1631 - 1666
• Main Authors: Liu, Shitao, Triggiani, Roberto
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
• Subjects: Applied mathematics; Boundary conditions; Estimates; Exact sciences and technology; Global analysis, analysis on manifolds; Inverse problems; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis in abstract spaces; Numerical analysis. Scientific computation; Ordinary differential equations; Partial differential equations; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Damping; Second order equation; Neumann problem; inverse PDE-problems; Differential equation; 35R30; Dynamical system; Boundary condition; Partial differential equation; 35L10; Inverse problem; Carleman estimate; 49K20; Energy; Mathematical analysis; uniqueness; Hyperbolic equation; Time interval; Observability; Regularity; Boundary theory; Numerical stability; stability
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/100808988

We consider a second-order hyperbolic equation on an open bounded domain $\Omega$ in $\mathbb{R}^n$ for $n\geq2$, with $C^2$-boundary $\Gamma=\partial\Omega=\overline{\Gamma_0\cup\Gamma_1}$, $\Gamma_0\cap\Gamma_1=\emptyset$, subject to nonhomogeneous Neumann boundary conditions on the entire boundary $\Gamma$. We then study the inverse problem of determining the interior damping coefficient of the equation by means of an additional measurement of the Dirichlet boundary trace of the solution, in a suitable, explicit subportion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T>0$. Under sharp conditions on the complementary part $\Gamma_0= \Gamma\backslash\Gamma_1$, and $T>0$, and under weak regularity requirements on the data, we establish the two canonical results in inverse problems: (i) global uniqueness and (ii) Lipschitz stability (at the $L^2$-level). The latter is the main result of this paper. Our proof relies on three main ingredients: (a) sharp Carleman estimates at the $H^1 \times L_2$-level for second-order hyperbolic equations [I. Lasiecka, R. Triggiani, and X. Zhang, Contemp. Math., 268 (2000), pp. 227-325]; (b) a correspondingly implied continuous observability inequality at the same energy level [I. Lasiecka, R. Triggiani, and X. Zhang, Contemp. Math., 268 (2000), pp. 227-325]; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Neumann boundary data [I. Lasiecka and R. Triggiani, Ann. Mat. Pura. Appl. (4), 157 (1990), pp. 285-367], [I. Lasiecka and R. Triggiani, J. Differential Equations, 94 (1991), pp. 112-164], [I. Lasiecka and R. Triggiani, Recent advances in regularity of second-order hyperbolic mixed problems, and applications, in Dynamics Reported: Expositions in Dynamical Systems, Vol. 3, Springer-Verlag, Berlin, 1994, pp. 104-158], [D. Tataru, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), pp. 185-206]. 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, Inverse Problems for Partial Differential Equations, 2nd ed., Springer, New York, 2006, Thm.,8.2.2, p.,231].