Reconstructions from boundary measurements: complex conductivities

TL;DR

This paper extends Nachman's boundary measurement reconstruction method to complex conductivities in higher dimensions, providing new theoretical insights and low frequency estimates for such inverse problems.

Contribution

It demonstrates that Nachman's method can be adapted for complex conductivities in three or more dimensions, filling a gap in inverse boundary value problem literature.

Findings

Reconstruction of complex conductivities from boundary data in higher dimensions.

Extension of Nachman's results to complex-valued conductivities.

Establishment of low frequency estimates for $C^{1,1}$-boundaries.

Abstract

In this paper we show that following Nachman's method we can still reconstruct complex conductivities in $C^{1,1}$ from its Dirichlet-to-Neumann map in three and higher dimensions. For such, we analyze all of the results in Nachman and pinpoint what really needs to be shown for complex conductivities. Moreover, we also obtain low frequency estimates for $C^{1,1}$-boundaries following the approach established by Cornean, Knudsen and Siltanen. As far as we aware, this is the first reconstruction procedure for complex conductivities, even though the proof follows trivially by extending some of Nachman's theorems to the complex case.