Recovering two Lamé kernels in a viscoelastic system

Alfredo Lorenzi; Vladimir G. Romanov

doi:10.3934/ipi.2011.5.431

Article Contents

2011, Volume 5, Issue 2: 431-464. Doi: 10.3934/ipi.2011.5.431

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Recovering two Lamé kernels in a viscoelastic system

Alfredo Lorenzi^1, and
Vladimir G. Romanov^2,

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

Received: September 2009

Revised: February 2011

Published: May 2011

Abstract / Introduction Related Papers Cited by

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.

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