Inverse Problems and Imaging (IPI)

Recovering two Lamé kernels in a viscoelastic system

Pages: 431 - 464, Volume 5, Issue 2, May 2011      doi:10.3934/ipi.2011.5.431

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Alfredo Lorenzi - Dipartimento di Matematica “F. Enriques”, Universitá di Milano, via C. Saldini 50, 20133 Milano, Italy (email)
Vladimir G. Romanov - Sobolev Institute of Mathematics, Siberian branch of Russian Academy of Sciences, Acad. Koptyug prosp., 4, Novosibirsk, 630090, Russian Federation (email)

Abstract: 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.

Keywords:  Identification problems, linear viscoelastic materials, hyperbolic second-order integrodifferential systems, recovering relaxation kernels, uniqueness, continuous dependence.
Mathematics Subject Classification:  Primary: 45Q05, 45K05; Secondary: 35L20, 74H05, 74H45, 74J25.

Received: September 2009;      Revised: February 2011;      Available Online: May 2011.