Consistency of a phase field regularisation for an inverse problem governed by a quasilinear Maxwell system

An inverse problem of reconstructing the magnetic reluctivity in a quasilinear magnetostatic Maxwell system is studied. To overcome the ill-posedness of the inverse problem, we propose and investigate two regularisations posed as constrained minimisation problems. The first uses the total variation (perimeter) regularisation, and the second makes use of the phase field regularisation. Existence of minimisers, sequential stability with respect to data perturbation, and consistency as the regularisation parameters tending to zero are rigorously analysed. Under some regularity assumption, we infer a relation between the regularisation parameters that allows one to recover a solution to the original inverse problem from the phase field regularised problem. The second focus of the paper is set on the first-order analysis of both regularisation approaches. For the phase field approach, two types of optimality systems are derived through a weak directional differentiability result and the domain variation technique of shape calculus. As a final result, we show the convergence of the optimality conditions obtained from shape calculus, leading to a necessary optimality system for the total variation inverse problem.