Estimate of solution stability in a~two-dimensional inverse problem for elasticity equations

V. G. Romanov

Trudy Instituta matematiki i mehaniki, Control, stability, and inverse problems of dynamics, Tome 12 (2006) no. 2, pp. 152-161

Résumé

The problem of determining the density of the medium and one of its elasticity moduli is considered. Properties of the elastic medium and external forces are assumed to be independent of the coordinate $x_3$. In this case, the third component of the displacement vector satisfies a scalar equation of the second order, which contains the density $\rho$ of the medium and elasticity modulus $\mu$ as coefficients. The parameters $\rho$ and $\mu$ are known to be positive and constant everywhere outside some compact domain $D\subset\mathbb R^2$, but they are unknown inside $D$. The problem of determining these coefficients in $D$ via information, given on the boundary of the domain $D$ for some finite time interval, about a solution of two direct problems is considered. An estimate of the conditional stability of a solution of the inverse problem under consideration is established.