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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 $$f$$.

##### MSC:
 49J40 Variational inequalities 26A99 Functions of one variable 28D05 Measure-preserving transformations 76B47 Vortex flows for incompressible inviscid fluids
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##### References:
 [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
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