Faber-Walsh field expansions for the planar conductivity problem with multiple inclusions
Abstract
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.
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BibTeX
@misc{choi2026faberwalsh,
title = {{Faber-Walsh field expansions for the planar conductivity problem with multiple inclusions}},
author = {Doosung Choi and Mikyoung Lim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29239},
eprint = {2609.29239},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29239},
}
RIS
TY - UNPB
TI - Faber-Walsh field expansions for the planar conductivity problem with multiple inclusions
AU - Doosung Choi
AU - Mikyoung Lim
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29239
UR - https://arxiv.org/abs/2609.29239
AB - 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.
ER -