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 - Liu, Shitao; Triggiani, Roberto (SIAM journal on mathematical analysis, 2011)
We consider a second-order hyperbolic equation on an open bounded domain $Ω$ in $\mathbb{R}^n$ for $n≥2$, with $C^2$-boundary $Γ=\partialΩ=\overline{Γ_0\cupΓ_1}$, $Γ_0\capΓ_1=\emptyset$, subject to nonhomogeneous Neumann 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 Dirichlet boundary trace of the solution, in a suitable, explicit subportion $Γ_1$ of the boundary $Γ$, and over a computable time interval $T>0$. Under sharp conditions on the complementary part $Γ_0= Γ\backslashΓ_1$, 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 $H^1 × L_2$-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].