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

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 $\Gamma=\partial\Omega=\overline{\Gamma_0\cup\Gamma_1}$, $\Gamma_0\cap\Gamma_1=\emptyset$, subject to nonhomogeneous 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 subportion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T>0$. Under sharp conditions on the complementary part $\Gamma_0= \Gamma\backslash\Gamma_1$, 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 $L^2$-level). The latter is the main result of this paper. Our proof relies on three main ingredients: (a) sharp Carleman estimates at the ...