Bounds on complex polarizabilities and a new perspective on scattering by a lossy inclusion
Abstract
Here, we obtain explicit formulas 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 twophase 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 twophase 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 auxiliary 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.
 Publication:

Physical Review B
 Pub Date:
 September 2017
 DOI:
 10.1103/PhysRevB.96.104206
 arXiv:
 arXiv:1704.06832
 Bibcode:
 2017PhRvB..96j4206M
 Keywords:

 Mathematical Physics
 EPrint:
 27 pages, 1 figure