Girish Mahajan (Editor)

Symmetric decreasing rearrangement

Updated on
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit
Covid-19

In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function.

Contents

Definition for sets

Given a measurable set, A , in Rn one can obtain the symmetric rearrangement of A , called A , by

A = { x R n : ω n | x | n < | A | } ,

where ω n is the volume of the unit ball and where | A | is the volume of A . Notice that this is just the ball centered at the origin whose volume is the same as that of the set A .

Definition for functions

The rearrangement of a non-negative, measurable real-valued function f whose level sets f 1 ( y ) ( y R 0 ) have finite measure is

f ( x ) = 0 I { y : f ( y ) > t } ( x ) d t ,

where I A denotes the indicator function of the set A. In words, the value of f ( x ) gives the height t for which the radius of the symmetric rearrangement of { y : f ( y ) > t } is equal to x. We have the following motivation for this definition. Because the identity

g ( x ) = 0 I { y : g ( y ) > t } ( x ) d t ,

holds for any non-negative function g , the above definition is the unique definition that forces the identity I A = I A to hold.

Properties

The function f is a symmetric and decreasing function whose level sets have the same measure as the level sets of f , i.e.

| { x : f ( x ) > t } | = | { x : f ( x ) > t } | .

If f is a function in L p , then

f L p = f L p .

The Hardy–Littlewood inequality holds, i.e.

f g f g .

Further, the Szegő inequality holds. This says that if 1 p < and if f W 1 , p then

f p f p .

The symmetric decreasing rearrangement is order preserving and decreases L p distance, i.e.

f g f g

and

f g L p f g L p .

Applications

The Pólya–Szegő inequality yields, in the limit case, with p = 1 , the isoperimetric inequality. Also, one can use some relations with harmonic functions to prove the Rayleigh–Faber–Krahn inequality.

References

Symmetric decreasing rearrangement Wikipedia


Similar Topics
Children of the Damned
Daniel Brel
Nonini
Topics
 
B
i
Link
H2
L