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A semilinear elliptic eigenvalue problem, II. The plasma problem

Published online by Cambridge University Press:  14 November 2011

Grant Keady  and
John Norbury
Grant Keady
Affiliation:
Mathematics Department, University of Western Australia
John Norbury
Affiliation:
St Catherine's College, Oxford
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Synopsis

This paper continues the study of the boundary value problem, for (λ, ψ)

Here δ denotes the Laplacian, k is a given positive constant, and λ1 will denote the first eigenvalue for the Dirichlet problem for −δ on Ω. For λ ≦ λ1, the only solutions are those with ψ = 0. Throughout we will be interested in solutions (λ, ψ) with λ > λ1 and with ψ > 0 in Ω.

In the special case Ω = B(0, R) there is a branch ℱe, of explicit exact solutions which bifurcate from infinity at λ = λ1 and for which the following conclusions are valid, (a) The set Aψ,

is simply-connected, (b) Along ℱe, ψm → k, ‖ψ‖1 → 0 and the diameter of Aψ tends to zero as λ → ∞, where

Here it is shown that the above conclusions hold for other choices of Ω, and in particular, for Ω = (−a, a)×(−b, b). (Existence is settled in Part I, and elsewhere.)

The results of numerical and asymptotic calculations when Ω = (−a, a)×(−b, b) are given to illustrate both the above, and some limitations in the conclusions of our analysis.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 87 , Issue 1-2 , 1980 , pp. 83 - 109
Copyright
Copyright © Royal Society of Edinburgh 1980

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References

1Berestycki, H. and Brezis, H.. Sur certains problèmes de frontière libre. C.R. Acad. Sci. Paris 283 (1976), 1091–1094.Google Scholar
2Berger, M. S. and Fraenkel, L. E.. Nonlinear desingularization in certain free-boundary problems. Comm. Math. Phys. (1980), to appear.Google Scholar
3Brascamp, H. J. and Lieb, E. H.. On extensions of the Brunn-Minkowski and Prékopa-Leindler Theorems, including inequalities for log-concave functions, and with an application to the diffusion equation. J. Functional Analysis 22 (1976), 366–389.Google Scholar
4Budden, P. J. and Norbury, J.. A nonlinear elliptic eigenvalue problem. J. Inst. Math. Appl. 24 (1979), 9–33.Google Scholar
5Courant, R. and Hilbert, D.. Methods of mathematical physics, Vol. II (New York: Interscience, 1962).Google Scholar
6Crooke, P. S. and Sperb, R. P.. Isoperimetric inequalities in a class of nonlinear eigenvalue problems. SIAM J. Math. Anal. 9 (1978), 671–681.Google Scholar
7Damlamian, A.. Application de la dualite non-convexe à un problème nonlineare à frontière libre. C.R. Acad. Sci. Paris 286 (1978), 153–155.Google Scholar
8Fraenkel, L. E.. On steady vortex rings of small cross-section in an ideal fluid. Proc. Roy. Soc. London Ser. A 316 (1970), 29–62.Google Scholar
9Fraenkel, L. E. and Berger, M. S.. A global theory of steady vortex rings in an ideal fluid. Acta Math. 132 (1974), 13–51.Google Scholar
10Fraenkel, L. E.. On Steiner symmetrization and a lower bound for electrostatic capacity in the plane, submitted for publication.Google Scholar
11Gallouet, T.. Quelques remarques sur une equation apparaissant en physique des plasmas. C.R. Acad. Sci. Paris 286 (1978), 739–741.Google Scholar
12Hardy, G. H., Littlewood, J. E. and Polya, G.. Inequalities (Cambridge Univ. Press, 1934).Google Scholar
13Hille, E.. Analytic function theory, Vol. II (New York: Chelsea, 1962).Google Scholar
14Isaacson, E. and Keller, H. B.. Analysis of numerical methods (New York: Wiley, 1966).Google Scholar
15Keady, G. and Norbury, J.. A semilinear elliptic eigenvalue problem, I. Proc. Roy. Soc. Edinburgh Sect. A (1980), 87, 65–82.Google Scholar
16Keller, H. B.. Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of bifurcation theory (ed. P. H., Rabinowitz) (New York: Academic, 1976).Google Scholar
17Kinderlehrer, D., Nirenberg, L. and Spruck, J.. The shape and smoothness of stable plasma configurations, II, submitted for publication.Google Scholar
18Kinderlehrer, D. and Spruck, J.. The shape and smoothness of stable plasma configurations, I. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 131–150.Google Scholar
19Krasnoselskii, M. A.. Positive solutions of operator equations (Groningen: Noordhoff, 1964).Google Scholar
20Norbury, J.. Steady vortex pairs in an ideal fluid. Comm. Pure Appl. Math. 28 (1975), 679–700.Google Scholar
21Osserman, R.. A note on Hayman's theorem on the bass note of a drum. Comment. Math. Helv. 52 (1977), 545–555.CrossRefGoogle Scholar
22Osserman, R.. The isoperimetric inequality. Bull. Amer. Math. Soc. 84 (1978), 1182–1238.Google Scholar
23Payne, L. E. and Philippin, G. A.. On some maximum principles involving harmonic functions and their derivatives. SIAM J. Math. Anal. 10 (1979), 96–104.Google Scholar
24Payne, L. E. and Rayner, M. E.. An isoperimetric inequality for the first eigenfunction in the fixed membrane problem. Z. Angew. Math. Phys. 23 (1972), 13–15.Google Scholar
25Payne, L. E. and Stakgold, I.. On the mean value of the fundamental mode in the fixed membrane problem. Applicable Anal. 3 (1973), 295–306.Google Scholar
26Polya, G. and Szego, G.. Isoperimetric inequalities in mathematical physics (Princeton Univ. Press, 1951).Google Scholar
27Puel, J.. Un problème de valeurs propres nonlinéares et de frontière libre. C.R. Acad. Sci. Paris 284 (1977), 861–863.Google Scholar
28Rabinowitz, P. H.. Some global results for nonlinear eigenvalue problems. J. Functional Analysis 7 (1971), 487–513.Google Scholar
29Rado, T.. Sur la representation conforme des domaines variables. Acta Litt. Sect. Sci. Math., R. Univ. Hungar. Francisco-Josephina 1 (1923), 180–186.Google Scholar
30Schaefer, P. W. and Sperb, R. P.. Maximum principles and bounds in some inhomogeneous elliptic boundary-value problems. SIAM J. Math. Anal. 8 (1977), 871–878.Google Scholar
31Schaeffer, D. G.. Nonuniqueness in the equilibrium shape of a confined plasma. Comm. Partial Differential Equations 2 (1977), 587–599.Google Scholar
32Teman, R.. Remarks on a free boundary value problem arising in plasma physics. Comm. Partial Differential Equations 2 (1977), 563–583.CrossRefGoogle Scholar