Symmetrization: Schwarz symmetrization

May 14, 2010

Symmetrization: Schwarz symmetrization

Filed under: Giải tích 8 (MA5206) — Tags: symmetrization — Ngô Quốc Anh @ 15:34

Given a measurable subset $E \subset \mathbb R^N$ , we denote its $N$ -dimensional Lebesgue measure by $|E|$ . We will denote by $E^\star$ the open ball centered at the origin and having the same measure as $E$ , i.e. $|E^\star|=|E|$ . The norm of vector $x \in \mathbb R^n$ will be denoted by $|x|$ . Finally, we will denote by $\omega_N$ the volume of the unit ball in $\mathbb R^N$ . It is worth recalling that

$\displaystyle \omega_N=\frac{\pi^\frac{N}{2}}{\Gamma \left(\frac{N}{2}+1\right)}$

where $\Gamma$ us the usual gamma function.

Definition (Schwarz symmetrization). Let $\Omega \subset \mathbb R^N$ be a bounded domain. Let $u : \Omega \to \mathbb R$ be a measurable function. Then, its Schwarz symmetrization (or the spherically symmetric and decreasing rearrangement) is the function $u^\star : \Omega^\star \to \mathbb R$ defined by

$u^\star(x)=u^\sharp (\omega_N|x|^N), \quad x \in \Omega^\star$ .

Observe that if $R$ is the radius of $\Omega^\star$ , then

$\displaystyle\begin{gathered} \int_{{\Omega ^ \star }} {{u^ \star }(x)dx} = \int_{{\Omega ^ \star }} {{u^\sharp }({\omega _N}{{\left| x \right|}^N})dx} \hfill \\ \qquad= \int_0^R {{u^\sharp }({\omega _N}{{\left| x \right|}^N})N{\omega _N}{\tau ^{N - 1}}d\tau } \hfill \\ \qquad= \int_0^{|{\Omega ^ \star }|} {{u^\sharp }(s)ds} \hfill \\ \qquad= \int_0^{|\Omega |} {{u^\sharp }(s)ds} . \hfill \\ \end{gathered}$

We obviously have the following properties of Schwarz symmetrization

$u^\star$ is radially symmetric and decreasing.
$u, u^\sharp, u^\star$ are all equi-measurable.
If $F: \mathbb R \to\mathbb R$ is a Borel measurable function such that either $F \geqslant 0$ or $F(u) \in L^1(\Omega)$ then
$\displaystyle\int_{{\Omega ^ \star }} {F({u^ \star }(x))dx} = \int_\Omega {F(u(x))dx}$ .

In particular, $u$ and $u^\star$ have the same $L^p$ -norm and

$\displaystyle\int_{{\Omega ^ \star }} {{u^ \star }(x)dx} = \int_\Omega {u(x)dx}$ .

when $u$ is integrable over $\Omega$ .
If $\psi : \mathbb R \to \mathbb R$ is a non-decreasing function, then
$\displaystyle {(\psi (u))^ \star } = \psi ({u^ \star })$ .
The mapping $u : \mapsto u^\star$ is a non-expansive mapping from $L^p(\Omega)$ into $L^p(\Omega^\star$ for all $1 \leqslant p \leqslant \infty$ .
If $E \subset \Omega$ is a measurable subset, then
$\displaystyle\int_E {u(x)dx} \leqslant \int_0^{|E|} {{u^\sharp }(s)ds} = \int_{{E^ \star }} {{u^ \star }(x)dx}$ .

Equality occurs iff

$\displaystyle (u\big|_E)^\star=u^\star\big|_{E^\star}$ .
The Hardy-Littlewood inequality holds, i.e.
$\displaystyle\int_\Omega {f(x)g(x)dx} \leqslant \int_0^{|\Omega |} {{f^\sharp }(s){g^\sharp }(s)ds}=\int_{\Omega^\star} {f^\star(x)g^\star(x)dx}$

for any $f\in L^p(\Omega)$ and $g \in L^q(\Omega)$ where $\frac{1}{p}+\frac{1}{q}=1$ and $1 \leqslant p,q\leqslant \infty$ .

Similarly, we have the following variant.

Definition (Schwarz symmetrization). Let $\Omega \subset \mathbb R^N$ be a bounded domain. Let $u : \Omega \to \mathbb R$ be a measurable function. Then, its Schwarz symmetrization (or the spherically symmetric and increasing rearrangement) is the function $u_\star : \Omega^\star \to \mathbb R$ defined by

$u_\star(x)=u_\sharp (\omega_N|x|^N), \quad x \in \Omega^\star$ .

Now if we replace a bounded domain $\Omega$ by the whole space $\mathbb R^N$ , we have the following variant

Definition (Schwarz symmetrization for the whole space). Let $f : \mathbb R^N \to \mathbb R$ be a measurable function which is vanishing at infinity in the sense that for any $t>0$

$\displaystyle | \{|f|>t\}| < \infty$ .

Then, its Schwarz symmetrization (or the spherically symmetric and increasing rearrangement) is the function $f^\star : \mathbb R^N \to \mathbb R$ defined by

$\displaystyle {f^ \star }(x) = \int_0^\infty {{\chi _{{{\left\{ {|f| > t} \right\}}^ \star }}}(x)dt}$ .

Interestingly, $f^\star$ preserves all properties decreasing rearrangement or increasing rearrangement. The most important property of this rearrangement is the Riesz inequality. The simplest form is the following

$\displaystyle\int_{{\mathbb{R}^N}} {\int_{{\mathbb{R}^N}} {f(x)g(x - y)h(y)dxdy} } \leqslant \int_{{\mathbb{R}^N}} {\int_{{\mathbb{R}^N}} {{f^ \star }(x){g^ \star }(x - y){h^ \star }(y)dxdy} }$

for any non-negative Borel measurable functions vanishing at infinity.

3 Comments »

Hi, Ngo
I port to your attention this nice notes on rearrangement inequalities by Almut Burchard from Toronto.

http://www.math.toronto.edu/almut/rearrange.pdf

Comment by Fab — February 13, 2012 @ 19:18

Reply
Thanks Fab. A short but interesting note.

Comment by Ngô Quốc Anh — February 13, 2012 @ 22:34

Reply
Hi Ngo,
it will be nice to continue this post on Schwarz symmetrization with the classical Polya Szego inequality!

Comment by fab — March 7, 2012 @ 23:36

Reply

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Ngô Quốc Anh

May 14, 2010