# Ngô Quốc Anh

## May 2, 2010

### Symmetrization: The Decreasing Rearrangement

Given a measurable subset $E \subset \mathbb R^N$, we denote its $N$-dimensional Lebesgue measure by $|E|$.

Let $\Omega$ be a bounded measurable set. Let $u :\Omega \to \mathbb R$ be a measurable function. For $t \in \mathbb R$, the level set $\{u>t\}$ is defined as

$\displaystyle \{u>t\}=\{x\in \Omega: u(x)>t\}$.

The sets $\{u, $\{u \geqslant t\}$, $\{u=t\}$ and so on are defined by analogy. Then the distribution function of $u$ is given by

$\displaystyle \mu_u(t)=|\{u>t\}|$.

This function is a monotonically decreasing function of $t$ and for $t \geq {\rm esssup}(u)$ we have $\mu_u(t)=0$ while for $t\leqslant {\rm essinf}(u)$, we have $\mu_u(t)=|\Omega|$. Thus the range of $\mu_u$ is the interval $[0, |\Omega|]$.

Definition (Decreasing rearrangement). Let $\Omega \subset \mathbb R^N$ be bounded and let $u :\Omega \to \mathbb R$ be a measurable function. Then the (unidimensional) decreasing rearrangement of $u$, denoted by $u^\sharp$, is defined on $[0, |\Omega|]$ by

$\displaystyle {u^\sharp }(s) = \begin{cases} {\rm esssup} (u),& s = 0, \hfill \\ \mathop {\inf }\limits_t \left\{ {t:{\mu _u}(t) < s} \right\}, & s > 0. \hfill \\ \end{cases}$

Essentially, $u^\sharp$ is just the inverse function of the distribution function $\mu_u$ of $u$. The following properties of the decreasing rearrangement are immediate from its definition.

Proposition 1. Let $u : \Omega \to \mathbb R^N$ where $\Omega \subset \mathbb R^N$ is bounded. Then $u^\sharp$ is a nonincreasing and left-continuous function.

Proposition 2. The mapping $u \mapsto u^\sharp$ is non-decreasing, i.e. if $u\leqslant v$ in the sense that $u(x) \leqslant v(x)$ for all $x$, where $u$ and $v$ are real-valued functions on $\Omega$ then $u^\sharp \leqslant v^\sharp$.

We now see that $u^\sharp$ is indeed a rearrangement of $u$.

Proposition 3. The function $u : \Omega \to \mathbb R$ and $u^\sharp : [0,|\Omega|] \to \mathbb R$ are equimeasurable (i.e. they have the same distribution function), i.e. for all $t$

$\displaystyle |\{u >t\}|=|\{u^\sharp >t\}|$.