Stability Estimates for Coefficients of Magnetic Schr\"odinger Equation From Full and Partial Boundary Measurements

In this paper we establish a $log log$-type estimate which shows that in dimension $n\geq 3$ the magnetic field and the electric potential of the magnetic Schr\"odinger equation depends stably on the Dirichlet to Neumann (DN) map even when the boundary measurement is taken only on a subset that is slightly larger than half of the boundary $\partial\Omega$. Furthermore, we prove that in the case when the measurement is taken on all of $\partial\Omega$ one can establish a better estimate that is of $log$-type. The proofs involve the use of the complex geometric optics (CGO) solutions of the magnetic Schr\"odinger equation constructed in \cite{sun uhlmann} then follow a similar line of argument as in \cite{alessandrini}. In the partial data estimate we follow the general strategy of \cite{hw} by using the Carleman estimate established in \cite{FKSU} and a continuous dependence result for analytic continuation developed in \cite{vessella}.