On a nonlocal stationary free-boundary problem arising in the confinement of a plasma in a Stellarator geometry

We prove the existence and some qualitative properties of the solution to a two-dimensional free-boundary problem modeling the magnetic confinement of a plasma in a Stellarator configuration. The nonlinear elliptic partial differential equation on the plasma region was obtained from the three-dimensional magnetohydrodynamic system by Hender & Carreras in 1984 by using averaging arguments and a suitable system of coordinates (Boozer's vacuum coordinates). The free boundary represents the separation between the plasma and vacuum regions, and the model is described by an inverse-type problem (some nonlinear terms of the equation are unknown). Using the zero net current condition for the Stellarator configurations, we reformulate the problem with the help of the notion of relative rearrangement, leading to a new problem involving nonlocal terms in the equation. We use an iterative algorithm and establish some new properties on the relative rearrangement in order to prove the convergence of the algorithm and then the existence of a solution.