The local Calderon problem and the determination at the boundary of the conductivity

TL;DR

This paper extends the uniqueness and stability results for the inverse boundary value problem of determining anisotropic conductivities, using local boundary data instead of global, with applications to both Dirichlet-to-Neumann and Neumann-to-Dirichlet maps.

Contribution

It generalizes previous global boundary results to local boundary data, providing new uniqueness and stability theorems for anisotropic conductivities at the boundary.

Findings

Extended boundary uniqueness and stability results to local data

Established pointwise stability among continuous conductivities at boundary points

Applicable to both Dirichlet-to-Neumann and Neumann-to-Dirichlet maps

Abstract

We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $\Omega\subset\mathbb{R}^{n}$ when the so--called Dirichlet-to-Neumann map is locally given on a non empty portion $\Gamma$ of the boundary $\partial\Omega$. We extend results of uniqueness and stability at the boundary, obtained by the same authors in SIAM J. Math. Anal. 33 (2001), no. 1, 153--171, where the Dirichlet-to-Neumann map was given on all of $\partial\Omega$ instead. We also obtain a pointwise stability result at the boundary among the class of conductivities which are continuous at some point $y\in\Gamma$. Our arguments also apply when the local Neumann-to-Dirichlet map is available.