{
  "schema_version": 1,
  "problem_number": "OWR-11578-001",
  "title": "A Negative Answer to Harrach's Question on the Potential Map in the Unified Eddy-Current Formulation",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In the unified variational formulation of the parabolic-elliptic eddy-current equations due to Arnold and Harrach, the electric field is written as E = A + ∇φ_A, where A is divergence free and φ_A solves div(σ∇φ_A) = −div(σA). To solve this equation, Arnold and Harrach assume a conductivity σ that is bounded below on its support, and a support made of finitely many Lipschitz domains with disjoint closures. Harrach asked whether the map A ↦ φ_A can be defined for general nonnegative σ ∈ L^∞(ℝ³). We show that it cannot. Let σ be the indicator function of a cyclic necklace of n ≥ 3 closed balls in which consecutive balls touch at one point, or a smooth function that is positive exactly on the open balls. Then there is a smooth, compactly supported, divergence-free field A* for which no φ ∈ H¹_loc(ℝ³) solves div(σ(A* + ∇φ)) = 0. A solid torus whose conductivity is positive almost everywhere but vanishes at least linearly across one cross-section (σ ≤ C|sin(θ/2)|, with θ the azimuth) gives the same conclusion. So neither hypothesis can simply be dropped, although we do not claim that either is necessary. In both examples the eddy-current equation with zero initial data and a smooth divergence-free source that vanishes near the conductor has no solution in L²(0,T;W(curl)). On the positive side, the product σ(A + ∇φ_A), which is what the unified formulation uses, can be defined for every σ ≥ 0 by a weighted orthogonal projection. With this definition the unified formulation stays uniquely solvable and uniformly coercive, and it controls every solution of the eddy-current equation. The eddy-current equation is solvable exactly when, for almost every t, the potential equation for the solution A(t) of the unified formulation has a solution φ(t) ∈ H¹_loc(ℝ³), with ∇φ(·) ∈ L²(0,T;L²_ρ). The ingredients are classical: points have zero capacity, and the energy between touching balls diverges logarithmically. This is an unrefereed note.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "keywords": [
    "OWR-11578-001",
    "eddy current equations",
    "degenerate conductivity",
    "Helmholtz decomposition",
    "counterexample",
    "Oberwolfach Reports",
    "math.AP",
    "math.FA"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
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  "date_modified": "2026-09-30",
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  "status": "unrefereed preprint",
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  "doi": "10.5281/zenodo.23049958",
  "zenodo_record_url": "https://zenodo.org/records/23049958",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Scope: the obstruction concerns H¹_loc potentials and the stated variational solution class. The eddy-current equivalence and nonexistence conclusions are proved with zero initial data; this is not a classification of every conductivity. Sketches in Remarks 7.1 and 7.2 are not used in the proved theorems. The ingredients include classical capacity and touching-conductor energy arguments. No absolute priority claim is made. Self-audited, AI-assisted and unrefereed; no independent peer review is claimed.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text.",
  "concept_doi": "10.5281/zenodo.23049210",
  "concept_url": "https://doi.org/10.5281/zenodo.23049210",
  "revision_published_at": "2026-09-30T01:46:04.694459+00:00",
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  "revision_note": "The abstract now specifies conductivity vanishing at least linearly across the cross-section (σ ≤ C|sin(θ/2)|), and the solvability statement includes the required integrability condition. Citations and the verification record are corrected.",
  "review_disclosure": "Internal AI-assisted checks only; unrefereed preprint, no independent human peer review claimed."
}