Journal of the
Korean Mathematical Society JKMS

We investigate the behavior of the gradient of solutions to the refraction equation $\text{div}(( 1+(k-1)\chi_{D})\nabla u)=0$ under perturbation of domain $D$. If $u$ and $u_h$ are solutions to the refraction equation corresponding to subdomains $D$ and $D_h$ of a domain $\Omega$ in 2 dimensional plane with the same Neumann data on $\partial \Omega$, respectively, we prove that $\| \nabla (u-u_h) \|_{L^2(\Omega)} \le C \sqrt{\text{dist}(D,D_h)}$ where $\text{dist}(D,D_h)$ is the Hausdorff distance between $D$ and $D_h$. We also show that this is the best possible result.

Keywords: conductivity problem, stability, layer potential

MSC numbers: 35B35