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Rearrangements of functions on unbounded domains

Published online by Cambridge University Press:  14 November 2011

R. J. Douglas
R. J. Douglas
Affiliation:
School of Mathematics, University of Bath, Claverton Down, Bath BA 27AY, U.K.
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Abstract

A characterisation is provided for the weak closure of the set of rearrangements of a function on an unbounded domain. The extreme points of this convex, weakly compact set are classified. This result is used to study the maximising sequences of a variational problem for steady vortices.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 4 , 1994 , pp. 621 - 644
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

1Adams, R. A.. Sobolev Spaces (New York: Academic Press, 1975).Google Scholar
2Amick, C. J. and Fraenkel, L. E.. The uniqueness of Hill's spherical vortex. Arch. Rational Mech. Anal. 92(1986), 92–119.CrossRefGoogle Scholar
3Benjamin, T. B.. The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics. Applications of Methods of Functional Analysis to Problems in Mechanics, Lecture Notes in Mathematics 503, 8–29 (Berlin: Springer, 1976).CrossRefGoogle Scholar
4Burton, G. R.. Vortex rings in a cylinder and rearrangements. J. Differential Equations 70 (1987), 333–348.CrossRefGoogle Scholar
5Burton, G. R. and Ryan, E. P.. On reachable sets and extremal rearrangements of control functions. SIAM J. Control Optim. 6 (1988), 1481–1489.CrossRefGoogle Scholar
6Cesari, L.. Optimization Theory and Applications (New York: Springer, 1983).CrossRefGoogle Scholar
7Crowe, J. A., Zweibel, J. A. and Rosenbloom, P. C.. Rearrangements of functions. J. Funct. Anal. 66, (1986), 432–438.CrossRefGoogle Scholar
8Douglas, R. J.. Rearrangements and nonlinear analysis of vortices (Ph.D Thesis, Bath University, 1992).Google Scholar
9Dunford, N. and Schwartz, J. T.. Linear Operators, Part 1 (New York: Interscience, 1967).Google Scholar
10Ekeland, I. and Temam, R.. Convex Analysis and Variational Problems (Amsterdam: North-Holland, 1976).Google Scholar
11Eydeland, A., Spruck, J. and Turkington, B.. Multiconstrained variational problems of nonlinear eigenvalue type: new formulations and algorithms. Math. Comp. 55 (1990), 509–535.CrossRefGoogle Scholar
12Fraenkel, L. E. and Berger, M. S.. A global theory of steady vortex rings in an ideal fluid. Acta Math. 132(1974), 14–51.CrossRefGoogle Scholar
13Halmos, P. R.. Measure Theory (New York: Springer, 1974).Google Scholar
14Hardy, G. H., Littlewood, J. E. and Polya, G.. Inequalities (Cambridge: Cambridge University Press, 1934).Google Scholar
15Lions, P.-L.. Symétrie et compacité dans les espaces de Sobolev. J. Funct. Anal. 49 (1982), 315–344.CrossRefGoogle Scholar
16Royden, H. L.. Real Analysis, 2nd edn (London: Collier–Macmillan, 1963).Google Scholar
17Royden, H. L.. Real Analysis, 3rd edn (London: Collier–Macmillan, 1988).Google Scholar
18Ryff, J. V.. Measure preserving transformations and rearrangements. J. Math. Anal. Appl. 31 (1970), 449–458.CrossRefGoogle Scholar
19Ryff, J. V.. Majorized functions and measures. Indag. Math. 30 (1968), 431–437.CrossRefGoogle Scholar