Reconstruction from boundary measurements for less regular conductivities

TL;DR

This paper advances the reconstruction of less regular conductivities, specifically $C^1$ and Lipschitz types, from boundary measurements using ideas inspired by Nachman, Haberman, and Tataru, with new boundary gradient recovery results.

Contribution

It extends boundary reconstruction methods to less regular conductivities and includes boundary gradient recovery techniques for Lipschitz domains.

Findings

Reconstruction of $C^1$ conductivities from Dirichlet-to-Neumann map.

Reconstruction of Lipschitz conductivities with small gradient of log-conductivity.

Boundary gradient recovery of $C^1$ conductivities in Lipschitz domains.

Abstract

In this paper, following Nachman's idea and Haberman and Tataru's idea, we reconstruct $C^1$ conductivity $\gamma$ or Lipchitz conductivity $\gamma$ with small enough value of $|\nabla log\gamma|$ in a Lipschitz domain $\Omega$ from the Dirichlet-to-Neumann map $\Lambda_{\gamma}$. In the appendix the authors and R. M. Brown recover the gradient of a $C^1$-conductivity at the boundary of a Lipschitz domain from the Dirichlet-to-Neumann map $\Lambda_{\gamma}$.