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

 Consider the solution of a scalar Helmholtz equation where the potential (or index) takes two positive values, one inside a disk or a ball (when d=2 or 3) of radius epsilon and another one outside. For this classical problem, it is possible to derive sharp explicit estimates of the size of the scattered field caused by this inhomogeneity, for any frequencies and any contrast. We will see that uniform estimates with respect to frequency and contrast do not tend to zero with epsilon, because of a quasi-resonance phenomenon. However, broadband estimates can be derived: uniform bounds for the scattered field for any contrast, and any frequencies outside of a set which tends to zero with epsilon.