Uniqueness in the Inverse Boundary Value Problem for Piecewise Homogeneous Anisotropic Elasticity

Abstract

Consider a three dimensional piecewise homogeneous anisotropic elastic medium $\Omega$ which is a bounded domain consisting of a finite number of bounded subdomains $D_\alpha$, with each $D_\alpha$ a homogeneous elastic medium. One typical example is a finite element model with elements with curvilinear interfaces for an anisotropic elastic medium. Assuming the $D_\alpha$ are known and Lipschitz, we are concerned with the uniqueness in the inverse boundary value problem of identifying the anisotropic elasticity tensor on $\Omega$ from a localized Dirichlet-to-Neumann map given on a part of the boundary $\partial D_{\alpha_0}\cap\partial\Omega$ of $\partial\Omega$ for a single $\alpha_0$, where $\partial D_{\alpha_0}$ denotes the boundary of $ D_{\alpha_0}$. If we can connect each $D_\alpha$ to $D_{\alpha_0}$ by a chain of $\{D_{\alpha_i}\}_{i=1}^n$ such that interfaces between adjacent regions contain a curved portion, we obtain global uniqueness for this inverse boundary value problem. If the $D_\alpha$ are not known but are subanalytic subsets of $\mathbb{R}^3$ with curved boundaries, then we also obtain global uniqueness.