{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29239",
  "author_total": 2,
  "authors": [
    {
      "name": "Doosung Choi"
    },
    {
      "name": "Mikyoung Lim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1acerscntzj1f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29239",
  "fulltext_url": null,
  "id": "1acerscntzj1f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29239",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Numerical methods in inverse problems"
    ]
  ],
  "openalex": "W7214396906",
  "pdf_url": "https://arxiv.org/pdf/2609.29239",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1acerscntzj1f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29239",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29239"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Faber-Walsh field expansions for the planar conductivity problem with multiple inclusions",
  "title_text": "Faber-Walsh field expansions for the planar conductivity problem with multiple inclusions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/1acerscntzj1f",
  "venue": "arXiv",
  "versions": []
}