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 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 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.

中文翻译：

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复极化率的界线和有损包含的散射新观点

在这里，我们获得了在已知体积的夹杂物给定频率下复电极化率界限的明确公式，该公式直接从Bergman和Milton关于两相介质有效复数介电常数的准静态界限得出。我们还描述了如何通过准静态变分方法从Milton，Gibiansky和Berryman获得的两相介质的有效复数体积和剪切模量的界限上，获得给定频率下定向平均体积和剪切极化率的相似界限。 Cherkaev和Gibiansky的原则。我们还展示了如何在一个抽象的环境中将极化率问题和声散射问题都重新表述为“$ÿ$问题。” 在声散射环境中，为避免明确引入Sommerfeld辐射条件，我们在无穷大处引入了辅助场并在此处引入了适当的“本构定律”，这迫使Sommerfeld辐射条件得以保持。结果，我们获得了声散射的最小化变分原理，该原理可以用来获得复杂的向后散射振幅的界限。给出了一些明确的基本范围。