In mathematical analysis, the Pólya–Szegő inequality (or Szegő inequality) states that the Sobolev energy of a function in a Sobolev space does not increase under symmetric decreasing rearrangement. The inequality is named after the mathematicians George Pólya and Gábor Szegő.
u:\Rn\to\R+,
u*:\Rn\to\R+,
t\in\R,
u*{}-1((t,+infty))
0\in\Rn
u-1((t,+infty)).
Equivalently,
u*
u
The Pólya–Szegő inequality states that if moreover
u\inW1,p(\Rn),
u*\inW1,p(\Rn)
\int | |
\Rn |
|\nablau*|p\leq
\int | |
\Rn |
|\nablau|p.
The Pólya–Szegő inequality is used to prove the Rayleigh–Faber–Krahn inequality, which states that among all the domains of a given fixed volume, the ball has the smallest first eigenvalue for the Laplacian with Dirichlet boundary conditions. The proof goes by restating the problem as a minimization of the Rayleigh quotient.
The isoperimetric inequality can be deduced from the Pólya–Szegő inequality with
p=1
The optimal constant in the Sobolev inequality can be obtained by combining the Pólya–Szegő inequality with some integral inequalities.[1]
Since the Sobolev energy is invariant under translations, any translation of a radial function achieves equality in the Pólya–Szegő inequality. There are however other functions that can achieve equality, obtained for example by taking a radial nonincreasing function that achieves its maximum on a ball of positive radius and adding to this function another function which is radial with respect to a different point and whose support is contained in the maximum set of the first function. In order to avoid this obstruction, an additional condition is thus needed.
It has been proved that if the function
u
\{x\inRn:u(x)>0and\nablau(x)=0\}
u
a\inRn
The Pólya–Szegő inequality is still valid for symmetrizations on the sphere or the hyperbolic space.[3]
The inequality also holds for partial symmetrizations defined by foliating the space into planes (Steiner symmetrization)[4] [5] and into spheres (cap symmetrization).[6] [7]
There are also Pólya−Szegő inequalities for rearrangements with respect to non-Euclidean norms and using the dual norm of the gradient.[8] [9] [10]
The original proof by Pólya and Szegő for
p=2
u
\Rn.
\varepsilon>0
\begin{align} C\varepsilon&=\{(x,t)\in\Rn x \R:0<t<\varepsilonu(x)\}
* | |
\\ C | |
\varepsilon |
&=\{(x,t)\in\Rn x \R:0<t<\varepsilonu*(x)\} \end{align}
These sets are the sets of points who lie between the domain of the functions
\varepsilonu
\varepsilonu*
* | |
C | |
\varepsilon |
C\varepsilon
-1 | |
\int | |
u*{ |
((0,+infty))}1+\sqrt{1+\varepsilon2|\nablau*|2}\le
\int | |
u-1((0,+infty)) |
1+\sqrt{1+\varepsilon2|\nablau|2}.
Since the sets
u-1((0,+infty))
u{}*{}-1((0,+infty))
1 | |
\varepsilon |
-1 | |
\int | |
u*{ |
((0,+infty))}\sqrt{1+\varepsilon2|\nablau*|2}-1\le
1 | |
\varepsilon |
\int | |
u-1((0,+infty)) |
\sqrt{1+\varepsilon2|\nablau|2}-1.
The conclusion then follows from the fact that
\lim\varepsilon
1 | |
\varepsilon |
\int | |
u-1((0,+infty)) |
\sqrt{1+\varepsilon2|\nablau|2}-1=
1 | |
2 |
\int | |
\Rn |
|\nablau|2.
The Pólya–Szegő inequality can be proved by combining the coarea formula, Hölder’s inequality and the classical isoperimetric inequality.[12]
If the function
u
\int | |
\Rn |
|\nablau|p=
+infty | |
\int | |
0 |
\int | |
u-1({t |
)}|\nablau|pdl{H}ndt,
where
l{H}n-1
(n-1)
\Rn
t\in(0,+infty)
l{H}n-1\left(u-1(\{t\})\right)\le
\left(\int | |
u-1(\{t\ |
)}|\nablau|p\right
| ||||
) |
\left(\int | |
u-1(\{t\ |
)}
1 | |
|\nablau| |
\right
| ||||||
) |
.
Therefore, we have
\int | |
u-1(\{t\ |
)}|\nablau|p-1\ge
l{H | |
n |
\left(u-1(\{t\})\right
p}{\left(\int | |
) | |
u-1(\{t\ |
)}
1 | |
|\nablau| |
\right)p
Since the set
u*{}-1((t,+infty))
u-1((t,+infty))
l{H}n-1\left(u*{}-1(\{t\})\right)\lel{H}n\left(u-1(\{t\})\right).
Moreover, recalling that the sublevel sets of the functions
u
u*
-1 | |
\int | |
u*{ |
(\{t\})}
1 | |
|\nablau*| |
=
\int | |
u-1(\{t\ |
)}
1 | |
|\nablau| |
,
and therefore,
\int | |
\Rn |
|\nablau|p\ge
+infty | |
\int | |
0 |
l{H | |
n-1 |
\left(u*{}-1
-1 | |
(\{t\})\right) | |
u*{ |
(\{t\})}
1 | |
|\nablau*| |
\right)p-1
Since the function
u*
l{H | |
n-1 |
\left(u*{}-1(\{t\})\right)p}
-1 | |
{\left(\int | |
u*{ |
(\{t\})}
1 | |
|\nablau*| |
\right)p-1
and the conclusion follows by applying the coarea formula again.
When
p=2
\int | |
\Rn |
|\nablau|2=\limt
1 | |
t |
\left
(\int | |
\Rn |
|u|2-
\int | |
\Rn |
\int | |
\Rn |
Kt(x-y)u(x)u(y)dxdy\right),
where for
t\in(0,+infty)
Kt:\Rn\to\R
z\in\Rn
Kt(z)=
1 | |||||||||
|
| ||||
e |
.
Since for every
t\in(0,+infty)
Kt
\int | |
\Rn |
\int | |
\Rn |
Kt(x-y)u(x)u(y)dxdy\le
\int | |
\Rn |
\int | |
\Rn |
Kt(x-y)u*(x)u*(y)dxdy
Hence, we deduce that
\begin{align} \int | |
\Rn |
|\nablau|2&=\limt
1 | |
t |
\left
(\int | |
\Rn |
|u|2-
\int | |
\Rn |
\int | |
\Rn |
Kt(x-y)u(x)u(y)dxdy\right)\\[6pt] &\ge\limt
1 | |
t |
\left
(\int | |
\Rn |
|u|2-
\int | |
\Rn |
\int | |
\Rn |
Kt(x-y)u*(x)u*(y)dxdy\right)\\[6pt] &=
\int | |
\Rn |
|\nablau*|2. \end{align}