On Modifled Lagrange Multiplier Rule and its Application to the Regularity of Magnetic Flux Function in Nuclear Fusion

A state of equilibrium of a plasma flow in a nuclear fusion reactor is achieved as a minimizer of the magnetic flux energy over a class of rearrangements of a prescribed profile magnetic flux function. It is known that a Lipschitz continuous minimizer exists when the cross section of the reactor is convex. The classical Lagrange Multiplier Rule with a functional constraint is modified and is applied to the Kruskal-Kulsrud Principle to derive a non-degenerate Euler-Lagrange equation satisfied by a minimizer of the energy functional that arises in magnetohydrodynamics (MHD). A sufficient condition for the existence of a smooth minimizer is presented in this work. Under this sufficient condition, an explicit Euler-Lagrange equation that is compatible with numerical algorithms is obtained within the framework of the Lagrange Multiplier Rule. As a corollary to the main result, the Grad-Shafranov equation is derived rigorously within the framework of the Kruskal-Kulsrud Principle.