任意尺寸小导体电磁散射的Foldy-Lax近似
The Foldy-Lax Approximation for the Full Electromagnetic Scattering by Small Conductive Bodies of Arbitrary Shapes
摘要：
我们处理的电磁波传播的谐波制度。我们推导了由任意分布在$\mathbb{R}^3$的有界区域$Ω$中的小导电不均匀体簇所产生的散射场的Foldy-Lax近似。这种近似是有效的，在一个充分的，但一般的条件下，这种不均匀性$m$的数量，它们的最大半径$ε$和它们之间的最小距离$δ$，形式$$(\ln m)^{\frac{1}{3}}\fracεδ \leq C,$$，其中$C$是一个常数，只取决于Lipschitz字符的缩放不均匀性。此外，我们提供了明确的误差估计，这种近似在上述参数，$m, ε, δ$，但也使用的频率$k$下的瑞利政权。估计远场和近场（以距群集的距离$δ$表示）。特别是，对于一个中等数量的小的不均匀性$m$，派生的扩展是有效的中尺度制度，其中$δ\sim ε$。在数学分析的水平上，基于积分方程方法，我们证明了在$L^{2,Div}_t$空间，而不是通常的$L^2$空间（这是不够的）的密度的先验估计。在这样一个证明的关键点是一个特定的亥姆霍兹型分解的密度推导。这些估计数可以在改进近似值的同时获得所需的定性和定量估计数。最后，为了证明Foldy-Lax线性代数系统的可逆性，我们将可逆性不等式化为与标量Helmholtz模型相关的不等式。由于这个线性代数系统来自边界条件，这样的约化不是直接的。
英文摘要：
We deal with the electromagnetic waves propagation in the harmonic regime. We derive the Foldy-Lax approximation of the scattered fields generated by a cluster of small conductive inhomogeneities arbitrarily distributed in a bounded domain $Ω$ of $\mathbb{R}^3$. This approximation is valid under a sufficient but general condition on the number of such inhomogeneities $m$, their maximum radii $ε$ and the minimum distances between them $δ$, the form $$(\ln m)^{\frac{1}{3}}\fracεδ \leq C,$$ where $C$ is a constant depending only on the Lipschitz characters of the scaled inhomogeneities. In addition, we provide explicit error estimates of this approximation in terms of aforementioned parameters, $m, ε, δ$ but also the used frequencies $k$ under the Rayleigh regime. Both the far-fields and the near-fields (stated at a distance $δ$ to the cluster) are estimated. In particular, for a moderate number of small inhomogeneities $m$, the derived expansions are valid in the mesoscale regime where $δ\sim ε$. At the mathematical analysis level and based on integral equation methods, we prove a priori estimates of the densities in the $L^{2,Div}_t$ spaces instead of the usual $L^2$ spaces (which are not enough). A key point in such a proof is a derivation of a particular Helmholtz type decomposition of the densities. Those estimates allow to obtain the needed qualitative as well as quantitative estimates while refining the approximation. Finally, to prove the invertibility of the Foldy-Lax linear algebraic system, we reduce the coercivity inequality to the one related to the scalar Helmholtz model. As this linear algebraic system comes from the boundary conditions, such a reduction is not straightforward.
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