Mixed formulation in 3D for a nonlinear eddy current problem from applied superconductivity

The purpose of this paper is to demonstrate the well-posedness of a mixed formulation in $ 3 $D for a nonlinear eddy current problem encountered in the modelling of Type Ⅱ superconductors. The eddy current problem models the magnetic field in a material with a power law magnetic resistivity which makes it an analogue of the $ p $-Laplacian. Yet in contrast to the $ p $-Laplacian, the resulting degenerate elliptic problem includes a divergence constraint. The identification of the mixed formulation follows from a Hellinger-Reissner functional thereby leading to mixed, primal and dual formulations and a consistent treatment of mixed boundary conditions on the electric field and the magnetic field density. The proof of well-posedness relies on a novel inf-sup condition in $ L^p $ spaces, an additional analysis of the Schur complement, before nonlinear techniques can be adapted to the usual framework of the Babuška-Brezzi Theorem. This work extends earlier work in $ 2 $D by Prigozhin and Barrett on the $ p $-curl and analysis by Farhloul on the $ p $-Laplacian.