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.