Determining of the parameters of an elastic isotropic medium in a infinite cylinder

T. V. Bugueva

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 9 (2012), pp. 568-617.

Résumé

We consider an inverse problem for a system of an elastic isotropic equations in a cylinder infinite with respect to the variable $z$. The linearized problem of identification of three characteristics of elastic isotropic medium is investigated. It is supposed that the medium density $\rho(r,\theta,\varphi)$, the propagation velocities of longitudinal $c(r,\theta,\varphi)$ and transverse $a(r,\theta,\varphi)$ waves can be represented as $\rho(r,\theta,\varphi)\!=\!\rho_{0}+\rho_{1}(r,\theta,\!\varphi)$, $a^{2}(r,\theta,\varphi)=a_{0}^{2}+a_{1}(r,\theta,\varphi)$, $c^{2}(r,\theta,\varphi)=c_{0}^{2}+c_{1}(r,\theta,\varphi)$, where $\rho_{0}$, $a_{0}^{2}$, $c_{0}^{2}$ are some unknown constants, and unknown functions $\rho_{1}(r,\theta,\varphi)$, $a_{1}(r,\theta,\varphi)$, $c_{1}(r,\theta,\varphi)$ are small in comparison with the constants $\rho_{0}$, $a_{0}^{2}$ и $c_{0}^{2}$, correspondingly. The estimates of conditional stability of the inverse problem solution are obtained.

Détail

Keywords: inverse problems, isotropic elasticity, conditional stability estimate.

Bibliographie

[1] T. V. Bugueva, “Determining of density function in the multi-dimensional inverse problem of isotropic elasticity in a sphere”, Journal of Inverse and Ill-Posed Problems, 6:3 (1998), 173–203 | MR | Zbl

[2] T. V. Bugueva, “Multidimensional inverse problem for isotropic elasticity problem in a sphere”, Journal of Inverse and Ill-Posed Problems, 15:9 (2007), 893–934 | MR

[3] V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Sciences Press, Utrecht, 1987 | MR

[4] T. V. Bugueva, “A linearized inverse problem for the wave equation in a sphere”, Journal of Inverse and Ill-Posed Problems, 4:3 (1996), 171–189 | MR | Zbl