A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem

Abstract

We consider a transmission wave equation in two embedded domains in R 2 , where the speed is a1 > 0 in the inner domain and a2 > 0 in the outer domain. We prove a global Carleman inequality for this problem under the hypothesis that the inner domain is strictly convex and a1 > a2. As a consequence of this inequality, uniqueness and Lipschitz stability are obtained for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient from a single time-dependent Neumann boundary measurement.

Inverse problem on a tree-shaped network: unified approach for uniqueness

“…[8] and [5]) for networks, or also in [16] for the related case of a hyperbolic transmission equation. We should underline that this "global" method goes back to [17,18] for the wave equation.…”

On the determination of the principal coefficient from boundary measurements in a KdV equation

“…Other related papers on this type of inverse problem for pde's can be listed: [1,26] (one dimensional fourth order parabolic equation), [10,4,3] (discontinuous coefficients), [2] (network of one dimensional waves) [9,8] (logarithmic stability estimates), [21] (parabolic system), [41] (unknown coefficient in the nonlinearity), [46,15,21] (two unknown coefficients). We also mention some important books which can be the starting point to study inverse problems [30], Carleman estimates [22], and control theory for partial differential equations [18].…”

“…Inverse problem: Retrieve the principal coefficient a = a(x) of equation (2) from the measurement of y x (0, t), y xx (0, t) and y xx (L, t) on (0, T ), where y is the solution of equation (2).…”

Stability estimate in determination of a coefficient in transmission wave equation by boundary observation

“…Here + = + × (−T, T ) and T > 0 are fixed. Baudouin et al [4] prove global Carleman inequality for a transmission wave equation, uniqueness and Lipschitz stability for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient which is constant on each subdomain (i.e. a 1 and a 2 constants) from a single time-dependent Neumann boundary measurement.…”