Recovering damping and potential coefficients for an inverse nonhomogeneous secondorder hyperbolic problem via a localized Neumann boundary trace
Shitao Liu  Department of Mathematics and Statistics, University of Helsinki, FI00014 Helsinki, Finland (email) Abstract: We consider a secondorder 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 nonhomogeneous 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 subportion $\Gamma_1$ 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) Lipschitzstability. The latter (ii) is the main result of the paper. Our proof relies on a few main ingredients: (a) sharp Carleman estimates at the $H^1(\Omega) \times L^2(\Omega)$level for secondorder 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 secondorder 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 ``postCarleman 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.
Keywords: Inverse hyperbolic problems, uniqueness, stability, Carleman estimate.
Received: February 2012; Revised: September 2012; Available Online: May 2013. 
2016 Impact Factor1.099
