The Inverse Conductivity Problem with One Measurement: Stability and Estimation of Size

We consider the inverse problem to the refraction problem div$((1 + (k -1)\chi_D)\nabla u)=0 in $\Omega$ and $\pd{u}{\nu}=g$ on $\partial\Omega$. The inverse problem is to determine the size and the location of an unknown object $D$ from the boundary measurement $\Lambda_D(g)=u|_{\bO}$. The results of this paper are twofold: stability and estimation of size of $D$. We first obtain upper and lower bounds of the size of $D$ by comparing $\Lambda_D(g)$ with the Dirichlet data corresponding to the harmonic equation with the same Neumann data $g$. We then obtain logarithmic stability in the case of the disks. In the course of deriving the stability, we are able to compute a positive lower bound (independent of $D$) of the gradient of the solution $u$ to the refraction problem with the Neumann data $g$ satisfying some mild conditions.