Detecting anisotropic inclusions through EIT

TL;DR

This paper develops a method to detect anisotropic inclusions in a manifold using evolution equations related to Dirichlet-Neumann maps, providing quantitative estimates for inverse boundary value problems.

Contribution

It introduces a new lower bound for solutions of evolution equations involving Dirichlet-Neumann operators, enabling improved detection of inclusions in anisotropic media.

Findings

Derived a lower bound for solutions of the evolution equation involving Dirichlet-Neumann operators.

Established a quantitative density estimate for harmonic functions on manifolds.

Provided a lower bound for the difference of Dirichlet-Neumann maps in terms of metric differences.

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 $\mathcal{M}$.