# Characterization for stability in planar conductivities

## TL;DR

This paper provides a complete characterization of isotropic conductivities that can be stably recovered in the Calderón problem on a disk, using minimal assumptions related to their continuity and ellipticity.

## Contribution

It introduces a new, minimal set of conditions based on modulus of continuity and ellipticity that guarantees stable recovery of conductivities in a bounded domain.

## Key findings

- Complete characterization for stable recovery in a disk
- Minimal assumptions on conductivities for stability
- Conditions expressed via modulus of continuity and ellipticity

## Abstract

We find a complete characterization for sets of isotropic conductivities with stable recovery in the $L^2$ norm when the data of the Calder\'on Inverse Conductivity Problem is obtained in the boundary of a disk and the conductivities are constant in a neighborhood of its boundary. To obtain this result, we present minimal a priori assumptions which turn to be sufficient for sets of conductivities to have stable recovery in a bounded and rough domain. The condition is presented in terms of the modulus of continuity of the coefficients involved and their ellipticity bound.