Stability estimates for the Calder\'on problem with partial data

TL;DR

This paper extends stability estimates for the Calderón inverse problem with partial data from local to global in all dimensions higher than three, using complex geometrical optics solutions.

Contribution

It provides the first global stability estimates for the Calderón problem with partial data in dimensions greater than three, building on complex geometrical optics techniques.

Findings

Established global stability estimates in higher dimensions.

Extended previous local results to a global setting.

Utilized advanced complex geometrical optics constructions.

Abstract

This is a follow-up of a previous article where we proved local stability estimates for a potential in a Schr\"odinger equation on an open bounded set in dimension $n=3$ from the Dirichlet-to-Neumann map with partial data. The region under control was the penumbra delimited by a source of light outside of the convex hull of the open set. These local estimates provided stability of log-log type corresponding to the uniqueness results in Calder\'on's inverse problem with partial data proved by Kenig, Sj\"ostrand and Uhlmann. In this article, we prove the corresponding global estimates in all dimensions higher than three. The estimates are based on the construction of solutions of the Schr\"odinger equation by complex geometrical optics developed in the anisotropic setting by Dos Santos Ferreira, Kenig, Salo and Uhlmann to solve the Calder\'on problem in certain admissible geometries.