# American Institute of Mathematical Sciences

May  2011, 5(2): 431-464. doi: 10.3934/ipi.2011.5.431

## Recovering two Lamé kernels in a viscoelastic system

 1 Dipartimento di Matematica “F. Enriques”, Universitá di Milano, via C. Saldini 50, 20133 Milano, Italy 2 Sobolev Institute of Mathematics, Siberian branch of Russian Academy of Sciences, Acad. Koptyug prosp., 4, Novosibirsk, 630090, Russian Federation

Received  September 2009 Revised  February 2011 Published  May 2011

Let $\mathcal B$ be a viscoelastic body with a (smooth) bounded open reference set $\Omega$ in $\mathbb R^3$, with the equation of motion being described by the Lamé coefficients $\lambda_0$ and $\mu_0$ and the related viscoelastic coefficients $\lambda_1$ and $\mu_1$. The latter are assumed to be factorized with the same temporal part, i.e. $\lambda_1(t,x)=k(t)p(x)$ and $\mu_1(t,x)=k(t)q(x)$. Furthermore, it is assumed that the spatial parts $p$ and $q$ of $\lambda_1$ and $\mu_1$ are unknown and the three additional measurements $\sum_{j=1}^3\sigma_{i,j}^0(t,x)$n$_j(x) = g_i(t,x)$, $i=1,2,3$, are available on $(0,T)\times \partial \Omega$ for some (sufficiently large) subset $\Gamma\subset \partial \Omega$.
The fundamental task of this paper is to show the uniqueness of the pair $(p,q)$ as well as its continuous dependence on the boundary conditions, the initial data being kept fixed and the initial velocity being suitably related to the initial displacement.
Citation: Alfredo Lorenzi, Vladimir G. Romanov. Recovering two Lamé kernels in a viscoelastic system. Inverse Problems & Imaging, 2011, 5 (2) : 431-464. doi: 10.3934/ipi.2011.5.431
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