Symmetrization methods

The classical isoperimetric inequality problem asks: Given all shapes of a given area, which of them has the minimal perimeter. The conjectured answer was the disk and Steiner in 1838 showed this to be true using the Steiner symmetrization method. From this many other isoperimetric problems sprung. We will study the following three areas: Rayleigh's conjecture (1877) that the first eigenvalue of the Dirichlet problem is minimized for the ball was proved independently by G. Faber and E. Krahn. Polya and G. Szego (1951) proved that for a fixed volume, the ball has the minimum electrostatic capacity. Some of the symmetrization methods used are described below.

Symmetrization

Let $\Omega \subset \mathbb {R} ^{n}$ be measurable, then we denote by $\Omega ^{*}$ the symmetrized version of $\Omega$ i.e. a ball $\Omega ^{*}:=B_{r}(0)\subset \mathbb {R} ^{n}$ such that $vol(\Omega ^{*})=vol(\Omega )$. We denote by $f^{*}$ the symmetric decreasing rearrangement of nonnegative measurable function f and define it as $f^{*}(x):=\int _{0}^{\infty }1_{\{f(x)>t\}^{*}}(t)dt$. The following methods have been proved to transform $\Omega$ to $\Omega ^{*}$ i.e. given a sequence of symmetrization transformations $\{T_{k}\}$ we have $\lim \limits _{k\to \infty }d_{Ha}(\Omega ^{*},T_{k}(K))=0$, where $d_{Ha}$ is the Hausdorff distance (for discussion see Burchard (2009) )

Steiner Symmetrization

Steiner symmetrization was introduced by Steiner (1838) to solve the isoperimetric theorem stated above. Let $H^{n-1}\subset \mathbb {R} ^{n}$ be a hyperplane through the origin. Rotate space so that $H^{n-1}$ is the $x_{n}=0$ hyperplane. For each $x\in H$ let the perpendicular line through $x\in H$ be $L_{x}=\{x+ye_{n}:y\in \mathbb {R} \}$. Then by replacing each $\Omega \cap L_{x}$ by a line centered at H and with length $|\Omega \cap L_{x}|$ we obtain the Steiner symmetrized version.

$St(\Omega ):=\{x+ye_{n}:x+ze_{n}\in \Omega ~{\text{for some z and}}~|y|\leq {\frac {1}{2}}|\Omega \cap L_{x}|\}.$

We denote by $St(f)$ the Steiner symmetrization wrt to $x_{n}=0$ hyperplane of nonnegative measurable function $f:\mathbb {R} ^{d}\to \mathbb {R}$ and for fixed $x_{1},...,x_{n-1}$ define it as

$St:f(x_{1},...,x_{n-1},\cdot )\mapsto (f(x_{1},...,x_{n-1},\cdot ))^{*}.$

Circular Symmetrization

Circular symmetrization of set Omega

A popular method for symmetrization in the plane is Polya's circular symmetrization. After we will describe its generalization to higher dimensions. Let $\Omega \subset \mathbb {C}$ be a domain then we define its circular symmetrization $Circ(\Omega )$ wrt to the positive real axis as follows: Let

$\Omega _{t}:=\{\theta \in [0,2\pi ]:te^{i\theta }\in \Omega \}$ 

i.e. contain the arcs of radius t contained in $\Omega$. So we define

• If $\Omega _{t}$ is the full circle, then $Circ(\Omega )\cap \{|z|=t\}:=\{|z|=t\}$.
• If the length is $m(\Omega _{t})=\alpha$, then $Circ(\Omega )\cap \{|z|=t\}:=\{te^{i\theta }:|\theta |<{\frac {\alpha }{2}}\}$.
• $0,\infty \in Circ(\Omega )$ iff $0,\infty \in \Omega$.

In higher dimensions $\Omega \subset \mathbb {R} ^{n}$, we define its spherical symmetrization $Sp^{n}(\Omega )$ wrt to positive axis of $x_{1}$ as follows: Let $\Omega _{r}:=\{x\in \mathbb {S} ^{n-1}:rx\in \Omega \}$ i.e. contain the caps of radius r contained in $\Omega$. Also, for the first coordinate let $angle(x_{1}):=\theta$ if $x_{1}=rcos\theta$. So as above

• If $\Omega _{r}$ is the full cap, then $Sp^{n}(\Omega )\cap \{|z|=r\}:=\{|z|=r\}$.
• If the surface area is $m_{s}(\Omega _{t})=\alpha$, then $Sp^{n}(\Omega )\cap \{|z|=r\}:=\{x:|x|=r$ and $0\leq angle(x_{1})\leq \theta _{\alpha }\}=:C(\theta _{\alpha })$ where $\theta _{\alpha }$ is picked so that its surface area is $m_{s}(C(\theta _{\alpha })=\alpha$. In words, $C(\theta _{\alpha })$ is a cap symmetric around the positive axis $x_{1}$ with the same area as the intersection $\Omega \cap \{|z|=r\}$.
• $0,\infty \in Sp^{n}(\Omega )$ iff $0,\infty \in \Omega$.

Polarization

Polarization of set Omega

Let $\Omega \subset \mathbb {R} ^{n}$ be a domain and $H^{n-1}\subset \mathbb {R} ^{n}$ be a hyperplane through the origin. Denote the reflection across that plane to the positive halfspace $\mathbb {H} ^{+}$ as $\sigma _{H}$ or just $\sigma$ when it is clear from the context. Also, we denote the reflected $\Omega$ across hyperplane H as $\sigma \Omega$. Then, we denote the polarized $\Omega$ as $\Omega ^{\sigma }$ and define it as follows

• If $x\in \Omega \cap \mathbb {H} ^{+}$, then $x\in \Omega ^{\sigma }$.
• If $x\in \Omega \cap \sigma (\Omega )\cap \mathbb {H} ^{-}$, then $x\in \Omega ^{\sigma }$.
• If $x\in (\Omega \setminus \sigma (\Omega ))\cap \mathbb {H} ^{-}$, then $\sigma x\in \Omega ^{\sigma }$.

In words, we simply reflect $(\Omega \setminus \sigma (\Omega ))\cap \mathbb {H} ^{-}$ to the halfspace $\mathbb {H} ^{+}$.

References

• Morgan, Frank (2009). "Symmetrization". Retrieved November 2015. {{cite web}}: Check date values in: |accessdate= (help)
• Burchard, Almut (2009). "A Short Course on Rearrangement Inequalities" (PDF). Retrieved November 2015. {{cite web}}: Check date values in: |accessdate= (help)

• Kojar, Tomas (2015). "Brownian Motion and Symmetrization". Retrieved November 2015. {{cite web}}: Check date values in: |accessdate= (help)