A gradient bound for the Grad-Kruskal-Kulsrud functional

We establish the existence of minimizers with a bounded gradient for a variational problem arising in the Grad-Kruskal-Kulsrud model for the equilibrium of a confined plasma. The variational problem involves derivatives of the nondecreasing rearrangement of minimizers. Results are derived for bounded convex domains in R ². Our results answer in the negative (in the case of convex domains) a question raised by Grad concerning the possibility of singular behavior of the magnetic field at the point of maximum flux. The main approach is to use an approximating free boundary problem to handle the nonlinear nonlocal nature of the variational functional. Limiting minimizers of the variational problem are shown to have bounded gradient and to satisfy a weak equation that for one model problem takes the form where ψ*, μ_ψ are, respectively, the nondecreasing rearrangement and the distribution function of ψ. © 1996 John Wiley & Sons, Inc.