# Bounds on complex polarizabilities and a new perspective on scattering by a lossy inclusion

## TL;DR

This paper derives explicit bounds on the complex polarizabilities of inclusions and introduces a new variational framework for acoustic scattering, connecting scattering problems to Y-problems and avoiding explicit radiation conditions.

## Contribution

It provides novel explicit bounds on complex polarizabilities using quasistatic bounds and reformulates scattering problems as Y-problems with variational principles, offering new analytical tools.

## Key findings

- Explicit bounds on complex polarizabilities derived from quasistatic bounds.
- Reformulation of scattering problems as Y-problems with auxiliary fields.
- Variational principles for acoustic scattering that avoid explicit radiation conditions.

## Abstract

Here we obtain explicit formulae for bounds on the complex electrical polarizability at a given frequency of an inclusion with known volume that follow directly from the quasistatic bounds of Bergman and Milton on the effective complex dielectric constant of a two-phase medium. We also describe how analogous bounds on the orientationally averaged bulk and shear polarizabilities at a given frequency can be obtained from bounds on the effective complex bulk and shear moduli of a two-phase medium obtained by Milton, Gibiansky and Berryman, using the quasistatic variational principles of Cherkaev and Gibiansky. We also show how the polarizability problem and the acoustic scattering problem can both be reformulated in an abstract setting as "Y -problems". In the acoustic scattering context, to avoid explicit introduction of the Sommerfeld radiation condition, we introduce auxilliary fields at infinity and an appropriate "constitutive law" there, which forces the Sommerfeld radiation condition to hold. As a consequence we obtain minimization variational principles for acoustic scattering that can be used to obtain bounds on the complex backwards scattering amplitude. Some explicit elementary bounds are given.