On the scattered field generated by a ball inhomogeneity of constant index in dimension three

TL;DR

This paper extends two-dimensional results on the scattered field caused by a small inhomogeneity to three dimensions, providing sharp estimates and broadband bounds for the Helmholtz equation with variable index.

Contribution

It generalizes previous 2D findings to 3D, offering sharp and broadband estimates of the scattered field for small inhomogeneities in the Helmholtz equation.

Findings

Sharp estimates of the scattered field size in 3D

Broadband uniform bounds for the scattered field

Extension of 2D results to three dimensions

Abstract

We consider the solution of a scalar Helmholtz equation where the potential (or index) takes two positive values, one inside a ball of radius $\eps$ and another one outside. In this short paper, we report that the results recently obtained in the two dimensional case in [1] can be easily extended to three dimensions. In particular, we provide sharp estimates of the size of the scattered field caused by this ball inhomogeneity, for any frequencies and any contrast. We also provide a broadband estimate, that is, a uniform bound for the scattered field for any contrast, and any frequencies outside of a set which tends to zero with $\eps$.