Faber-Walsh 域展开用于多夹杂物平面导电问题
Faber-Walsh field expansions for the planar conductivity problem with multiple inclusions
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- Case Western Reserve University（凯斯西储大学）
- Korea Advanced Institute of Science and Technology（韩国科学技术院）
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中文总结 AI 辅助
针对多夹杂物平面导电问题，推导显式 Faber-Walsh 展开，通过极化张量分解分离相互作用与变形，并证明收敛性，数值验证了其有效性。
中文摘要 AI 辅助
我们推导了平面导电问题的一个显式 Faber-Walsh 表示，该问题涉及有限多个一般形状的不相交夹杂物，且各夹杂物具有独立选择的正电导率。Walsh 的 lemniscatic 映射提供了一个共形坐标，无需小性假设或宽分离假设。场展开通过 Faber-Walsh 极化张量表达，我们通过三角基变换将其与复收缩广义极化张量联系起来。对于解析界面，我们证明了共形延拓以及 Walsh-Grunsky 级数在边界附近的绝对且一致收敛，从而获得 Neumann-Poincaré 算子和 Faber-Walsh 极化张量的显式分解。这些分解将典型相互作用（即使在所有 Grunsky 系数消失时也可能非零）与共形变形分离，同时在一个耦合预解式中保留逐分量的电导率依赖性。对于调和多项式入射场，我们证明了在固定外部分支域的紧子集上，截断项以精确系数几何收敛。与独立边界积分解的数值比较说明了该表示在异质、非对称和近接触构型中的有效性。
英文摘要
We derive an explicit Faber-Walsh representation for the planar conductivity problem with finitely many disjoint inclusions of general shape and independently chosen positive conductivities. Walsh's lemniscatic map gives a conformal coordinate without smallness or wide-separation assumptions. The field expansion is expressed through Faber-Walsh polarization tensors, which we relate to the complex contracted generalized polarization tensors by a triangular change of basis. For analytic interfaces, we prove conformal continuation and absolute and uniform convergence of the Walsh-Grunsky series near the boundary to obtain explicit factorizations of the Neumann-Poincaré operator and the Faber-Walsh polarization tensors. These factorizations separate canonical interaction, which can remain nonzero even when all Grunsky coefficients vanish, from conformal deformation, while retaining the componentwise conductivity dependence in a coupled resolvent. For harmonic polynomial incident fields, we prove geometric convergence of outgoing truncations with exact coefficients on compact subsets of fixed exterior branch domains. Numerical comparisons with independent boundary-integral solutions illustrate the representation for heterogeneous, asymmetric, and nearly touching configurations.
