Resonances for transparent obstacles

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

“…In the case a 1 > a 2 and if the inner domain is strictly convex, it has been shown using micro-local analysis [34,35] that the speed of the energy decay is exponential if the dimension of space is odd and polynomial otherwise. In the general case, including the cases when a 1 < a 2 or the inner domain is not strictly convex, it has been proved using micro-local analysis and global Carleman estimates for the spectral problem [3] that the energy decays as the inverse of the logarithm of the time.…”

“…Secondly, there exist several results about the growth of the resolvent for the spectral stationary transmission problem, from where it is possible to derive the speed of local energy decay for the evolution wave equation with transmission conditions [10]. In the case a 1 > a 2 and if the inner domain is strictly convex, it has been shown using micro-local analysis [34,35] that the speed of the energy decay is exponential if the dimension of space is odd and polynomial otherwise. In the general case, including the cases when a 1 < a 2 or the inner domain is not strictly convex, it has been proved using micro-local analysis and global Carleman estimates for the spectral problem [3] that the energy decays as the inverse of the logarithm of the time.…”

Boundary Control for a Generalized Wave Equation -- Revisiting Russell's Method of Control

“…This is a question studied by several authors, as for example [7], [8], [27], [28] and [32]. Finally, using the concept of generalized bicharacteristics, introduced by R. B. Melrose an J. Sjöstrand in [21] and [22], it was proved in [8] and [29] that the condition that every generalized geodesic leaves any compact in a finite time is sufficient for ii) to be fulfilled, that is, the metric associated with the equation (1.4) must be non-trapping. For this reason, we assume this condition, that is: Assumption 1.1.…”