Detecting anisotropic inclusions through EIT

Abstract

We study the evolution equation $\partial_{t}u=-\Lambda_{t}u$ where $\Lambda_ {t}$ is the Dirichlet-Neumann operator of a decreasing family of Riemannian manifolds with boundary $\Sigma_{t}$. We derive a lower bound for the solution of such an equation, and apply it to a quantitative density estimate for the restriction of harmonic functions on $\mathcal{M}=\Sigma_{0}$ to the boundaries of $\partial\Sigma_{t}$. Consequently we are able to derive a lower bound for the difference of the Dirichlet-Neumann maps in terms of the difference of a background metrics $g$ and an inclusion metric $g+\chi_{\Sigma}(h-g)$ on a manifold $\mathcal{M}$.