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

We obtain sharp $L^p$ and H"older estimates for the Neumann function of the operator $\mathrm{\nabla }\cdot \gamma \mathrm{\nabla }-ik$ 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.