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

TL;DR

This paper derives stability estimates for the magnetic field and electric potential in the magnetic Schrödinger equation based on boundary measurements, with different estimates depending on whether measurements are full or partial.

Contribution

It establishes new log-log and log-type stability estimates for magnetic Schrödinger coefficients from boundary data, extending previous results to partial boundary measurements.

Findings

Log-log stability estimate for partial boundary data

Log stability estimate for full boundary data

Use of complex geometric optics solutions and Carleman estimates

Abstract

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}.