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A non-linear eigenvalue problem: the shape at equilibrium of a confined plasma. (English) Zbl 0328.35069
The equilibrium of a plasma in a cavity (the tokamak machine) is described by a free boundary value problem related to the Maxwell equations. The paper considers two versions of the problem, a simplified one and a more general one. The main ideas behind the equilibrium equations in the tokamak and the derivation of the two models mentioned above are indicated in an appendix of the paper. The unknowns of the problems are the shape at equilibrium of the plasma and the values of the flux function. If the shape is determined, the latter ones can be evaluated from certain elliptic problems. Using auxiliary formulations of the problems leading to variational problems, in both cases the original problems are shown to be equivalent to finding critical points of some functional $$k_1$$ with respect to some other functional $$k_2$$. In the first case, the critical points are obtained as those where the minimum of $$k_1$$ is attained on a set where $$k_2$$ is constant. In the second case, it turns out that the functional $$k_1$$ is not bounded from below on the whole function space involved in the problem. In order to overcome this serious difficulty a nonclassical inequality is proved that relates the positive part of a function to its gradient and its negative part which must be assumed to be different from zero. Applying this result one can show that $$k_1$$ is bounded from below on the sets where $$k_2$$ is constant and that it attains its minimum there.

##### MSC:
 35Q61 Maxwell equations 35R35 Free boundary problems for PDEs 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J40 Boundary value problems for higher-order elliptic equations 35A15 Variational methods applied to PDEs 35J60 Nonlinear elliptic equations 35B45 A priori estimates in context of PDEs 58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
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