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Identification of Lamé coefficients from boundary observations. (English) Zbl 0737.35139

The authors are concerned with the identification of Lamé coefficients \(\lambda\) and \(\mu\) in the following stationary problem related to an inhomogeneous, isotropic, two-dimensional elastic body \(\Omega\): \[ (1) \quad L(\lambda,\mu)u := \hbox{div}[\lambda(tr(e(u))I + 2\mu e(u)]=0 \hbox{ in } \Omega \qquad (2) \quad u=f \hbox { on } \partial\Omega. \] Here \(I\) is the \(2\times2\) identity matrix, \(f\in C^ \infty(\partial\Omega)^ 2\), \(e(u)={1\over 2}(D_ iu_ j+D_ ju_ i)_{i,j=1,2}\) and \(\hbox{div}T=(D_ 1T_{11}+D_ 2T_{12}\), \(D_ 1T_{21}+D_ 2T_{22})\), \(\hbox{tr}T=T_{11}+T_{22}\), \(T=(T_{ij})_{i,j=1,2}\). Coefficients \(\lambda\) and \(\mu\) are assumed to satisfy the properties: \(\lambda>0\), \(\lambda+\mu>0\) in \(\overline\Omega\). As an additional information they suppose to know the Dirichlet-to-Neumann map \(\Pi(\lambda,\mu)f=[\lambda(\hbox{tr}(e(u))I+2\mu e(u)]\cdot n\), \(n\) being the outward unit normal to \(\partial\Omega\).
Using pseudodifferential techniques, the authors show that the derivatives of every order of \(\lambda\) and \(\mu\) at any point \(x_ 0\in\partial\Omega\) can be computed recursively in terms of the asymptotic expansion of the total symbol associated with \(\Pi(\lambda,\mu)\) and the local representation of \(\partial\Omega\) at \(x_ 0\). In the special case where \(\lambda\) and \(\mu\) are real analytic, they derive a uniqueness result for their identification problem.
Stability results are also proved (e.g. in \(L^ \infty(\partial\Omega))\), when \(\lambda,\mu\in C^ \infty(\overline\Omega)\) are assumed to satisfy the following additional assumptions for a fixed pair \(m, M\in\mathbb{R}_ +\): \(m\leq\lambda(x)\leq M\), \(m\leq\mu(x)\leq M\;\forall x\in\overline\Omega\).
Reviewer: A.Lorenzi (Milano)

MSC:

35R30 Inverse problems for PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
58J40 Pseudodifferential and Fourier integral operators on manifolds
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