A Negative Answer to Harrach's Question on the Potential Map in the Unified Eddy-Current Formulation

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. Version 1.1 (30 September 2026) revises version 1.0 after an independent referee round. Main changes: a sentence of the abstract is corrected (the conductivity in the counterexample vanishes at least linearly across one cross-section, σ ≤ C|sin(θ/2)|); the solvability statement is qualified by the integrability condition of Theorem 1.4; the doctoral thesis of L. Simon (née Arnold, Mainz 2014) is cited; and several smaller corrections are made to the text, the bibliography and the verification record. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-11578-001 (Oberwolfach Report 11/2012, p. 631, open problem stated by B. Harrach).