Abstract
A free boundary value problem arising in plasma physics is reduced to a non-linear eigenvalue problem of a non-classical type. We establish the existence of solutions of the non-linear eigenvalue problem; these solutions are critical points of appropriate functionals.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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).
Auchmuty, G., & R. Beals, Arch. Rational Mech. Anal. 43, 255 (1971).
Baiocchi, C., Free boundary problems in the theory of fluid through porous media. Proceedings of the International Congress of Mathematicians, Vancouver, August 1974.
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).
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.
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.
Boujot, J.P., J. Laminie, & R. Temam, to appear.
Brezis, H., & C. Stampacchia, Une nouvelle méthode pour l'étude d'écoulements stationnaires. C.R.A.S., série A, 276, 129–132 (1973).
Courant, R., & D. Hilbert, Methods of Mathematical Physics, Vol. II. New York: Interscience Publishers 1953.
Duvaut, G., & J.L. Lions, Les Inéquations en Mécanique et en Physique. Paris: Dunod 1973.
Ekeland, I., & R. Temam, Analyse Convexe et Problèmes Variationnels. Paris: Dunod-Gauthier-Villars 1974. English translation to appear. Amsterdam: North Holland 1975.
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.
Kinderlehrer, D., The coincidence set of solutions of certain variational inequalities. Arch. Rational Mech. Anal. 40, 231–250 (1971).
Kinderlehrer, D., & R. Temam, to appear.
Krasnosel'skii, M. A., Topological Methods in the Theory of Non-Linear Integral Equations. New York: Pergamon Press 1964.
Lions, J.L., Problèmes aux limites dans les équations aux dérivées partielles. Presses de l'Université de Montréal, 1962.
Lions, J.L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Paris: Dunod-Gauthier-Villars 1969.
Lions, J.L., & E. Magènes, Problèmes aux Limites Non Homogènes. Paris: Dunod 1968.
Mercier, C., The magnetohydrodynamic approach to the problem of plasma confinment in closed magnetic configurations. Publication of EURATOM C.E.A., Luxembourg 1974.
Morrey, C.B., Multiple Integrals in the Calculus of Variations. Berlin, Heidelberg, New-York: Springer 1966.
Nečas, J., Les Méthodes Directes en Théorie des Equations Elliptiques. Paris: Masson 1967.
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).
Stampacchia, G., Equations elliptiques du second ordre à coefficients discontinus. Presses de l'Université de Montréal.
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).
Grad, H., A. Kadish, & D.C. Stevens, A free boundary Tokamak Equilibrium. Comm. Pure Appl. Math. XXVII, 39–57 (1974).
Fraenkel, L.E., & M.S. Berger, A global theory of steady vortex rings in an ideal fluid. Acta Math. 132, 13–51 (1974).
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.
Author information
Authors and Affiliations
Additional information
Communicated by S. Antman
Rights and permissions
About this article
Cite this article
Temam, R. A non-linear eigenvalue problem: The shape at equilibrium of a confined plasma. Arch. Rational Mech. Anal. 60, 51–73 (1975). https://doi.org/10.1007/BF00281469
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00281469