Single-logarithmic stability for the Calder\'on problem with local data

TL;DR

This paper establishes an optimal stability estimate for Electrical Impedance Tomography with local boundary data, demonstrating how local information can determine global conductivity with Hölder dependence when boundary conductivities are known.

Contribution

It introduces a general method to derive global stability estimates from local data in the Calderón problem, assuming known conductivities near the boundary.

Findings

Proves optimal stability estimate for local data in EIT

Demonstrates Hölder dependence of global map on local data

Provides a new approach for boundary and interior conductivity recovery

Abstract

We prove an optimal stability estimate for Electrical Impedance Tomography with local data, in the case when the conductivity is precisely known on a neighborhood of the boundary. The main novelty here is that we provide a rather general method which enables to obtain the H\"older dependence of a global Dirichlet to Neumann map from a local one on a larger domain when, in the layer between the two boundaries, the coefficient is known.