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 $\mathcal{ M}$.
Categories: math.AP