A non-linear eigenvalue problem: the shape at equilibrium of a confined plasma
type: Item
instance of: scholarly article
title: A non-linear eigenvalue problem: the shape at equilibrium of a confined plasma
zbMATH Open document ID: 0328.35069
DOI: 10.1007/BF00281469
published in: Archive for Rational Mechanics and Analysis
publication date: 1975-01-01T00:00:00Z
review text: 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.
reviewed by: Hansgeorg Jeggle
Mathematics Subject Classification ID: 35Q61
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 58E15
zbMATH DE Number: 3514352
zbMATH Keywords: nonlinear eigenvalue problem
zbMATH Keywords: confined plasma
zbMATH Keywords: free boundary value problem
zbMATH Keywords: Maxwell equations
zbMATH Keywords: critical points of some functional
author: R. Temam
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00281469
OpenAlex ID: W2075328988
description: scientific article; zbMATH DE number 3514352