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.

We consider a second-order hyperbolic equation on an open bounded domain Ω in ℝn for n ≥ 2, with C²-boundary Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. 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 Γ1 of the boundary Γ, and over a computable time interval T > 0. Under sharp conditions on the complementary part Γ0 = Γ\Γ1, 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 L²-level). The latter is the main result of this paper. Our proof relies on three main ingredients: (a) sharp Carleman estimates at the H¹ x L2-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]. [ABSTRACT FROM AUTHOR]