Stability Estimates for Coefficients of Magnetic Schrödinger Equation from Full and Partial Boundary Measurements

Abstract

In this paper we establish a log log-type estimate which shows that in dimension n ≥ 3 the magnetic field and the electric potential of the magnetic Schrödinger 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 ∂Ω. Furthermore, we prove that in the case when the measurement is taken on all of ∂Ω 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 Schrödinger equation constructed in [8] then follow a similar line of argument as in [1]. In the partial data estimate we follow the general strategy of [5] by using the Carleman estimate established in [4] and a continuous dependence result for analytic continuation developed in [14].

The Calderón problem with partial data in two dimensions

“…The case of the magnetic Schrödinger equation was considered in [11] and an improvement on the regularity of the coefficients is done in [21]. Stability estimates for the magnetic Schrödinger equation with partial data were proven in [30].…”

Inverse Problems: Visibility and Invisibility

“…Stability estimates for the uniqueness result of [36] were given in [77]. Stability estimates for the XI-11 magnetic Schrödinger operator with partial data in the setting of [36] can be found in [201]. The result [36] was substantially improved in [108].…”

“…The case of partial data on a slab was studied in [127]. The DN map with partial data for the magnetic Schrödinger operator was studied in [52], [110], [201], [118]. The case of the polyharmonic operator was considered in [119].…”

Inverse problems: seeing the unseen

“…Stability estimates for the uniqueness result of [35] were given in [76]. Stability estimates for the magnetic Schrödinger operator with partial data in the setting of [35] can be found in [198]. The result [35] was substantially improved in [107].…”

“…The case of partial data on a slab was studied in [122]. The DN map with partial data for the magnetic Schrödinger operator was studied in [51,109,117,198]. The case of the polyharmonic operator was considered in [118].…”