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

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We consider a second-order hyperbolic equation on an open bounded domain Ω in Rⁿ for n≥2, with C²-boundary Γ=Ω=Γ₀∪Γ₁, Γ₀∩Γ₁=, subject to non-homogeneous 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 sub-portion Γ₁ of the boundary Γ, and over a computable time interval T>0. Under sharp conditions on the complementary part Γ₀= ΓΓ₁, T>0, and under weak regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and (ii) stability (at the L²-level). The latter (ii) is the main result of the paper. Our proof relies on three main ingredients: (a) sharp Carleman estimates at the H¹ × L₂-level for second-order hyperbolic equations L-T-Z.1; (b) a correspondingly implied continuous observability inequality at the same energy level L-T-Z.1; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Neumann boundary data L-T.4, L-T.5, L-T.6, Ta.3. The proof of the linear uniqueness result (Section 4, step 5) also takes advantage of a convenient tactical route "post-Carleman estimates" suggested by V.Isakov in [Thm.8.2.2, p.231]Is.2.