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

Abstract

We consider a second-order hyperbolic equation on an open bounded domain $\Omega$ in $\mathbb{R}^n$ for $n\geq2$, with $C^2$-boundary $\Gamma=\pa\Omega=\bar{\Gamma_0\cup\Gamma_1}$, $\Gamma_0\cap\Gamma_1=\emptyset$, subject to non-homogeneous 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 sub-portion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T>0$. Under sharp conditions on the complementary part $\Gamma_0= \Gamma\backslash\Gamma_1$, $T>0$, and under weak regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and (ii) stability (at the $L^2$-level). The latter (ii) is the main result of the 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 \cite{L-T-Z.1}; (b) a correspondingly implied continuous observability inequality at the same energy level \cite{L-T-Z.1}; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Neumann boundary data \cite{L-T.4}, \cite{L-T.5}, \cite{L-T.6}, \cite{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 \cite[Thm.\,8.2.2, p.\,231]{Is.2}.