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.