Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace

Abstract

We consider a second-order hyperbolic equation defined on an open bounded domain $\Omega$ in $\mathbb{R}^n$ for $n \geq 2$, with $C^2$-boundary $\Gamma = \partial \Omega = \overline{\Gamma_0 \cup \Gamma_1}$, $\Gamma_0 \cap \Gamma_1 = \emptyset$, subject to non-homogeneous Dirichlet boundary conditions for the entire boundary $\Gamma$. We then study the inverse problem of determining both the damping and the potential (source) coefficients simultaneously, in one shot, by means of an additional measurement of the Neumann 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 sharp regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and (ii) Lipschitz-stability. The latter (ii) is the main result of the paper. Our proof relies on a few main ingredients: (a) sharp Carleman at the $H^1(\Omega) \times L^2(\Omega)$-level for second-order hyperbolic equations [23], originally introduced for control theory issues; (b) a correspondingly implied continuous observability inequality at the same energy level [23]; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Dirichlet boundary data [14,15,16]. The proof of the linear uniqueness result (Section 3) also takes advantage of a convenient tactical route ``post-Carleman estimates proposed by V. Isakov in [8, Thm. 8.2.2, p. 231]. Expressing the final results for the nonlinear inverse problem directly in terms of the data offers an additional challenge.