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Maximisation and minimisation on classes of rearrangements*

Published online by Cambridge University Press:  14 November 2011

G. R. Burton  and
J. B. McLeod
G. R. Burton
Affiliation:
School of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, U.K.
J. B. McLeod
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh PA 15260, U.S.A.
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Synopsis

Maximisation and minimisation of the Dirichlet integral of a function vanishing on the boundary of a bounded domain are studied, subject to the constraint that the Laplacean be a rearrangement of a given function. When the Laplacean is two-signed, non-existence of minimisers is proved, and some information on the limits of minimising sequences obtained; this contrasts with the known existence of minimisers in the one-signed case. When the domain is a ball and the Laplacean is one-signed, maximisers and minimisers are shown to be radial and monotone. Existence of maximisers is proved subject additionally to a finite number of linear constraints, with particular reference to ideal fluid flows of prescribed angular momentum in a disc.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 3-4 , 1991 , pp. 287 - 300
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Brothers, J. E. and Ziemer, W. P.. Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384 (1988), 153–179.Google Scholar
2Burton, G. R.. Rearrangements of functions, maximization of convex functionals, and vortex rings. Math. Ann. 276 (1987), 225–253.CrossRefGoogle Scholar
3Burton, G. R.. Variational problems on classes of rearrangements and multiple configurations for steady vortices. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), 295–319.CrossRefGoogle Scholar
4Burton, G. R. and Ryan, E. P.. On reachable sets and extremal rearrangements of control functions. SIAM J. Control Optim. 26 (1988), 1481–1489.CrossRefGoogle Scholar
5McLeod, J. B.. Rearrangements (preprint).Google Scholar
6Ryff, J. V.. Orbits of L1-functions under doubly stochastic transformations. Trans. Amer. Math. Soc. 117 (1965), 92–100.Google Scholar
7Ryff, J. V.. Extreme points of some convex subsets of L 1(0, 1). Proc. Amer. Math. Soc. 18 (1967), 1026–1034.Google Scholar
8Ryff, J. V.. Majorized functions and measures. Indag. Math. 30 (1968), 431–437.CrossRefGoogle Scholar