Dynamical inverse problem for a Lamé type system

Abstract

The system under consideration is governed by the equation utt = ∇ div u – rot µ rot u in Ω × (0, T); its response operator ("input output" map) RT plays the role of the inverse data. As in the case of the proper Lamé system, such a model describes a dynamical system with two wave modes (p-waves and s-waves) propagating with different velocities cp = and cs = correspondingly. We show that R2T determines and , where and are the subdomains of Ω filled (at the moment T) with p- and s-waves propagating from ∂Ω. Due to the wave splitting u = ∇p + rot s the problem is reduced to the inverse problems for the acoustical and Maxwell subsystems governed by the equations ptt = Δp and stt = −µ rot rot s with the response operators and determined by R2T . The first problem can be solved by the BC method (Belishev, 1986), the second one is solved by a version of the method based on a blow up effect. This version is the main subject of the paper. In addition, we derive the inequality, which can be used for approximate determination of the shape of .