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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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[1] Agmon, S., A. Douglis, & L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Appl. Math. 12, 623–727 (1959). · Zbl 0093.10401
[2] Auchmuty, G., & R. Beals, Arch. Rational Mech. Anal. 43, 255 (1971). · Zbl 0225.49013
[3] Baiocchi, C., Free boundary problems in the theory of fluid through porous media. Proceedings of the International Congress of Mathematicians, Vancouver, August 1974. · Zbl 0347.76067
[4] Bernstein, J., E.A. Frieman, M.D. Kruskal, & R.M. Kulsrud, An energy principle for hydromagnetic stability problems. Proc. Royal Soc., A 244, 17–40 (1958). · Zbl 0081.21704
[5] Boujot, J.P., J.P. Morera, & R. Temam, Problèmes de controle optimal en physique des plasmas. Proceedings of the Symposium ”Colloque International” sur les méthodes de Calcul Scientifique et Technique, IRIA – Décembre 1973. Lectures Notes in Computer Science (vol. 11). Berlin, Heidelberg, New York: Springer 1973.
[6] Boujot, J.P., J.P. Morera, & R. Temam, An optimal control problem related to the equilibrium of a plasma in a cavity. Applied Mathematics and Optimisation Journal, to appear. · Zbl 0324.49003
[7] Boujot, J.P., J. Laminie, & R. Temam, to appear.
[8] Brezis, H., & C. Stampacchia, Une nouvelle méthode pour l’étude d’écoulements stationnaires. C.R.A.S., série A, 276, 129–132 (1973). · Zbl 0246.35021
[9] Courant, R., & D. Hilbert, Methods of Mathematical Physics, Vol. II. New York: Interscience Publishers 1953. · Zbl 0051.28802
[10] Duvaut, G., & J.L. Lions, Les Inéquations en Mécanique et en Physique. Paris: Dunod 1973.
[11] Ekeland, I., & R. Temam, Analyse Convexe et Problèmes Variationnels. Paris: Dunod-Gauthier-Villars 1974. English translation to appear. Amsterdam: North Holland 1975.
[12] Fučik, S., J. Nečas, J. Souček, & V. Souček, Spectral Analysis of Non-Linear Operators. Lecture Notes in Mathematics, Vol. 346. Berlin Heidelberg New York: Springer 1973.
[13] Kinderlehrer, D., The coincidence set of solutions of certain variational inequalities. Arch. Rational Mech. Anal. 40, 231–250 (1971). · Zbl 0219.49014
[14] Kinderlehrer, D., & R. Temam, to appear.
[15] Krasnosel’skii, M. A., Topological Methods in the Theory of Non-Linear Integral Equations. New York: Pergamon Press 1964.
[16] Lions, J.L., Problèmes aux limites dans les équations aux dérivées partielles. Presses de l’Université de Montréal, 1962. · Zbl 0168.08602
[17] Lions, J.L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Paris: Dunod-Gauthier-Villars 1969.
[18] Lions, J.L., & E. Magènes, Problèmes aux Limites Non Homogènes. Paris: Dunod 1968. · Zbl 0235.65074
[19] Mercier, C., The magnetohydrodynamic approach to the problem of plasma confinment in closed magnetic configurations. Publication of EURATOM C.E.A., Luxembourg 1974.
[20] Morrey, C.B., Multiple Integrals in the Calculus of Variations. Berlin, Heidelberg, New-York: Springer 1966. · Zbl 0142.38701
[21] Nečas, J., Les Méthodes Directes en Théorie des Equations Elliptiques. Paris: Masson 1967.
[22] Nitsche, J.C.C., Variational problems with inequalities as boundary conditions or how to fashion a cheap hat for Giacometti’s brother. Arch. Rational Mech. Anal. 35, 83–113 (1969). · Zbl 0209.41601
[23] Stampacchia, G., Equations elliptiques du second ordre à coefficients discontinus. Presses de l’Université de Montréal. · Zbl 0151.15501
[24] Temam, R., Configuration d’équilibre d’un plasma: un problème de valeur propre non linéaire. C.R.A.S., série A, 280, 419–421 (1975). · Zbl 0296.35028
[25] Grad, H., A. Kadish, & D.C. Stevens, A free boundary Tokamak Equilibrium. Comm. Pure Appl. Math. XXVII, 39–57 (1974). · Zbl 0283.76076
[26] Fraenkel, L.E., & M.S. Berger, A global theory of steady vortex rings in an ideal fluid. Acta Math. 132, 13–51 (1974). · Zbl 0282.76014
[27] Temam, R., Applications de l’analyse convexe au calcul des variations. Proceedings of the Conference on Non-linear Operators and the Calculus of Variations, Brussels, Sept. 1972. Lecture Notes in Mathematics, Berlin-Heidelberg-New York: Springer 1976. · Zbl 0337.49022
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