Inverse Conductivity Problem with One Measurement: Error Estimates and Approximate Identification for Perturbed Disks

Source abstract

This paper studies the global uniqueness and stability questions of the inverse\linebreak conductivity problem to determine the unknown object D entering $\mbox{div}((1+(k-1)\chi_D)\nabla u)=0$\linebreak in $\Omega$ and $\pd{u}{\nu}=g$ on $\partial\Omega$ from the boundary measurement $\Lambda_D(g)=u|_{\bO}$. The results of this paper\linebreak are fourfold. We first obtain a Hölder stability estimate for disks. Second, a uniform stability estimate for the direct problem is obtained. Third, we obtain the stability estimates $|D_1 \setminus \bar D_2| +|D_2 \setminus \bar D_1| \le C ( \| \Lambda_{D_1} (g) - \Lambda_{D_2} (g) \|_{L^\infty(\bO)}^{\alpha} + \ep )$ for some $\alpha >0$ when g satisfies some condition if D 1 and D 2 are $\ep$-perturbations of two disks. We then drop the condition on g and show that if $\Lambda_{D_1} (g) = \Lambda_{D_2} (g)$ on $\bO$, then the two domains must be very close.