The Local Calderòn Problem and the Determination at the Boundary of the Conductivity

Abstract

We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body Ω ⊂ ℝ n when the so-called Dirichlet-to-Neumann map is locally given on a non empty portion Γ of the boundary ∂Ω. We extend results of uniqueness and stability at the boundary, obtained by the same authors in SIAM J. Math. Anal. 33:153–171, where the Dirichlet-to-Neumann map was given on all of ∂Ω instead. We also obtain a pointwise stability result at the boundary among the class of conductivities which are continuous at some point y ∈ Γ. Our arguments also apply when the local Neumann-to-Dirichlet map is available.