zbMATH — the first resource for mathematics

Rearrangements of functions on unbounded domains. (English) Zbl 0818.49010
The paper gives a characterization for the set of the rearrangements \(R(f)\) of a function \(f\) defined on the halfline to be weakly closed. It is proved that the set of the extreme points of the weak closure of \(R(f)\) is the set of rearrangements of curtailments. The statements are extended to a function defined on an unbounded domain \(\Omega\) of \(\mathbb{R}^ n\) by establishing the existence of a measure-preserving transformation between \(\Omega\) and the halfline and by considering the problem on the halfline.
The results are used to show that a variational functional attains a maximum value relative to the weak closure of \(R(f)\), where \(f\) is a given function. Moreover, the author shows that all maximizers belong to the set of rearrangements of curtailments of \(f\).

49J40 Variational inequalities
26A99 Functions of one variable
28D05 Measure-preserving transformations
76B47 Vortex flows for incompressible inviscid fluids
Full Text: DOI
[1] DOI: 10.1016/0022-1236(86)90067-4 · Zbl 0612.46027 · doi:10.1016/0022-1236(86)90067-4
[2] Ekeland, Convex Analysis and Variational Problems (1976) · Zbl 0322.90046
[3] DOI: 10.1137/0326086 · Zbl 0664.49015 · doi:10.1137/0326086
[4] DOI: 10.1016/0022-0396(87)90155-0 · Zbl 0648.35029 · doi:10.1016/0022-0396(87)90155-0
[5] DOI: 10.1007/BFb0088744 · doi:10.1007/BFb0088744
[6] DOI: 10.1007/BF00251252 · Zbl 0609.76018 · doi:10.1007/BF00251252
[7] Adams, Sobolev Spaces (1975)
[8] Ryff, Indag. Math. 30 pp 431– (1968) · doi:10.1016/S1385-7258(68)50051-9
[9] DOI: 10.1016/0022-247X(70)90038-7 · Zbl 0214.13701 · doi:10.1016/0022-247X(70)90038-7
[10] Royden, Real Analysis (1988)
[11] Royden, Real Analysis (1963)
[12] DOI: 10.1016/0022-1236(82)90072-6 · Zbl 0501.46032 · doi:10.1016/0022-1236(82)90072-6
[13] Hardy, Inequalities (1934)
[14] Halmos, Measure Theory (1974)
[15] Dunford, Linear Operators, Part 1 (1967)
[16] DOI: 10.1007/BF02392107 · Zbl 0282.76014 · doi:10.1007/BF02392107
[17] DOI: 10.1090/S0025-5718-1990-1035931-7 · doi:10.1090/S0025-5718-1990-1035931-7
[18] Cesari, Optimization Theory and Applications (1983) · doi:10.1007/978-1-4613-8165-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.