Rearrangements of functions on unbounded domains. (English) Zbl 0818.49010

Douglas, R. J.

Proc. R. Soc. Edinb., Sect. A 124, No. 4, 621-644 (1994).

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 \(f\).

Reviewer: G.Stefani (Firenze)

MSC:

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

Keywords:

rearrangements; curtailments; unbounded domain; measure-preserving transformation; variational functional

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