Identification problems in linear elasticity
Assume that \({\mathcal E}\) is an elastic isotropic body consisting of two homogeneous parts \({\mathcal E}_1\) and \({\mathcal E}_2\) with Lamé coefficients \(({\widetilde \mu},{\widetilde \lambda})\) and \((\mu,\lambda)\), respectively, such that \(({\widetilde \mu},{\widetilde \lambda})\neq (\mu,\lambda)\). We denote by \(\Omega\) and \(D\) the reference open sets related to \({\mathcal E}\) and \({\mathcal E}_1\) and assume that \(\Omega\) and \(D\) have boundaries which are, respectively, of class \(C^2\) and Lipschitz continuous. As a consequence, if we assume to know the boundary displacement \({\varphi}\) of \({\mathcal E}\), the displacement vector \({\mathbf u}\) solves the Dirichlet problem \[ \sum_{j=1}^n D_i\big[\mu(x)(D_ju_i+D_iu_j) + \lambda(x)D_ju_j\big] = 0 \qquad \text{in } \Omega,\quad i=1,\ldots,n \tag{1} \] \[ u_i=\varphi_i \qquad \text{on } \partial \Omega,\quad i=1,\ldots,n \tag{2} \] where \(\mu(x)={\widetilde \mu}\chi_D(x) + \mu \chi_{{\mathbb{R}}^n\setminus D}(x)\) and \(\lambda(x)={\widetilde \lambda}\chi_D(x)+\lambda \chi_{{\mathbb{R}}^n\setminus D}(x)\), \(\chi_F\) denoting the characteristic function of a set \(F\). Denote by \(\Lambda_D\) the Dirichlet to Neumann map defined on \(\partial \Omega\) by \(\Lambda_D\phi=\mu [\nabla {\mathbf u}+(\nabla {\mathbf u})^t] {\nu} + \lambda (\text{div}){\nu}\), \({\mathbf u}\) and \({\nu}\) being, respectively, the solution to problem (1)--(2) and the outward unit vector normal to \(\partial \Omega\). Concerning the unique determination of \(D\) the authors prove the following result. Theorem. Let \((\mu-{\widetilde \mu})(\lambda-{\widetilde \lambda})\geq 0\) and let \(D_i\) \((i=1,2)\) be two Lipschitz domains such that \({\overline D}_i \subset \Omega\) and \(\Omega \setminus {\overline D}_i\) is connected \((i=1,2)\). Then \(\Lambda_{D_1}=\Lambda_{D_2}\) implies \(D_1=D_2\). In the case when it is a priori known that the one of the two domains is contained in the latter, the authors show that it is possible to require, as an additional information, the single measurement \(\Lambda_D{\varphi}\), provided \({\varphi}\) satisfies some (non strict) restrictions.
determination of the separating boundaries
scientific article; zbMATH DE number 1115430
scientific article