Nonlocal generalized models for a confined plasma in a Tokamak

The following model appears in plasma physics for a Tokamak configuration: -Delta u + g(u) = 0, u is an element of V = H-0(1)(Omega) + R, integral(par tial derivative Omega) partial derivative u/partial derivative n = I > 0, w here I is a given positive constant, which is equivalent to find a fixed po int u = F(u - g(u)) + phi(o) where F is a compact operator on L-2(Omega). A ccording to Grad and Shafranov the nonlinearity g can depend on u* which is the generalized inverse of the distribution function m(t) = meas{x : u(x) > t} = \{u > t}\ (see [1]). But in these cases the map u --> g(u) cannot be continuous on all the space V but only on a nonlinear nonclosed set V-o. T his implies that the standard direct method for fixed point cannot be appli ed to solve the preceding problem. Nevertheless, using the Galerkin method and a topological argument, we prove that there exists a solution u fixed p oint of u = F(u - g(u)) + phi(o) under suitable assumptions on g. The model we treat covers a large new class of nonlinearities including rel ative rearrangment and monotone rearrangment. The resolution of the concret e model needs an extension of the strong continuity result of the relative rearrangement map made in [2] (see Theorem 1.1 below for the definition and result). (C) 1998 Elsevier Science Ltd. All rights reserved.