Sharp estimates for the Neumann functions and applications to quantitative photo-acoustic imaging in inhomogeneous media

TL;DR

This paper derives precise mathematical estimates for Neumann functions related to a differential operator and applies these results to improve quantitative photo-acoustic imaging techniques in media with variable properties.

Contribution

The paper provides sharp $L^p$ and H"older estimates for Neumann functions and demonstrates their application in reconstructing optical properties in inhomogeneous media.

Findings

Sharp $L^p$ and H"older estimates for Neumann functions.

Quantitative description of the singularity of Neumann functions.

Improved reconstruction methods in photo-acoustic imaging.

Abstract

We obtain sharp $L^p$ and H\"older estimates for the Neumann function of the operator $\nabla \cdot \gamma \nabla - ik$ on a bounded domain. We also obtain quantitative description of its singularity. We then apply these estimates to quantitative photo-acoustic imaging in inhomogeneous media. The problem is to reconstruct the optical absorption coefficient of a diametrically small anomaly from the absorbed energy density.