Weak Solution Theory for Maxwell′s Equations in the Semistatic Limit Case

We consider the quasilinear, semistatic system of the reduced Maxwell′s equations curl H = j, curl E + Ḃ = 0, div B = 0 for the electromagnetic field given by the electric field E and the magnetic induction B in a bounded domain Ω. These equations result from the usual system of Maxwell′s equations by neglecting displacement currents. The domain Ω is permitted to have arbitrary topological genus and a nonsmooth (Lipschitz type) boundary. The constituent relations for the underlying medium occupying Ω are j = σE (linear), H = ζ(B) (nonlinear). Here ζ is a strongly monotone and Lipschitz continuous map in the Hilbert space of vector-valued functions with components in L2(Ω). The electromagnetic field is subject to generalized boundary conditions associated with the classical form n · E = n · B = 0 on ∂Ω. We show global existence and uniqueness, and continuous dependence on the data, of the weak solution of the associated initial-boundary value problem in a suitable space-time Hilbert space setting. Uniqueness is proven directly; the existence result is derived by reduction to an abstract evolution equation of monotone type in an appropriately constructed Gelfand triplet via the introduction of electromagnetic potentials. We solve the evolution by means of a classical Galerkin approximation method and then show that the potentials so obtained do define the desired solution of the original equations. Finally, we prove the continuous dependence of the solutions to the original problem from their initial data.