A Gradient Bound for the Grad-Kruskal-Kulsrud Functional

We announce a gradient bound for minimizers of 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. In the case of a convex domain our results answer in the negative a question raised by Grad concerning the possibly singular behavior of the magnetic field at the point of maximum flux. It completes a research program begun in 1983 in which an approach using free boundaries was initiated to handle the nonlinear nonlocal nature of the problem. Limiting minimizers of the variational problem are shown to have bounded gradient and satisfy a weak equation which for one model problem takes the form $$ \triangle\psi=-{\psi^*}”(\mu_\psi(\psi)) $$ where $\psi^*, \mu_\psi$ are respectively the nondecreasing rearrangement, and distribution function of $\psi$.

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Published 1 January 1994