THE CAPACITY–VOLUME INEQUALITY OF
POINCARÉ–FABER–SZEGÖ
JEFFREY L. JAUREGUI
Abstract. The well-known isoperimetric inequality relates the area to the volume for
regions in Rn . The lesser-known concept of capacity gives an alternative method of measuring the “size” of a region. We recall the idea of capacity and give a complete proof
of the sharp inequality relating capacity to volume that is due to Poincaré, Faber, and
Szegö, following the reference [1].
1. Introduction
The famous isoperimetric inequality for a bounded open set Ω ⊂ Rn with smooth boundary states that hypersurface area A of ∂Ω is related to the n-volume V of Ω by
1
1
n−1
A
V n
≥
,
(1)
ωn−1
βn
where βn and ωn−1 are the volume and area of a unit ball in Rn and its boundary, respectively. Moreover, equality holds in (1) precisely for a round ball. Note that if Ω is uniformly
stretched by a factor of R, both sides of (1) scale by R.
Abstractly, we may think of A and V as different ways of quantifying the notion of the
“size” of the set Ω. Another similar quantity associated to any such Ω is the capacity1:
Z
1
C=
|∇f |2 dV,
(n − 2)ωn−1 Rn \Ω
where the infimum is taken over all locally-Lipschitz functions that vanish on ∂Ω and
approach the value 1 at infinity. It is not difficult to show that the infimum is achieved by
the unique harmonic function ϕ on Rn \ Ω that vanishes on ∂Ω and approaches 1 at infinity.
The following properties of capacity are not difficult to prove:
(1) The capacity of a round ball of radius R is Rn−2 .
(2) If Ω1 ⊂ Ω2 , then
R C(Ω1 ) ≤ C(Ω2 ).
1
(3) C = (n−2)ωn−1 ∂Ω ∂ν ϕdA, where ∂ν is the unit outward normal directional derivative of ϕ, and dA is the area form on ∂Ω.
Based on properties (1) and (2), one could think of the capacity as providing yet another
quantification of the “size” of an open set.
Date: August 7, 2012.
1Historically, this has been called the electrostatic capacity. In the literature, it occasionally is referred
to as the harmonic capacity.
1
2
JEFFREY L. JAUREGUI
It is natural to ask whether there is any relationship between the capacity C and volume
V of Ω (or the capacity and area). The answer to the former case is yes, given by the
inequality of Poincaré–Faber–Szegö:
Theorem 1 (Poincaré–Faber–Szegö). Let Ω be a bounded open set in Rn with smooth
boundary. Then
n−2
n
V
C≥
.
(2)
βn
Moreover, equality holds if and only if Ω is a round ball.
As for any possible relationship between the capacity and area, we leave it as an exercise
to check that the scale-invariant ratio
1
1
C n−2 /A n−1
can be made both arbitrarily large and arbitrarily small for appropriate Ω.
2. Proof of the inequality
The following is based entirely on the dimension three case of [1].
Proof. Let ϕ be the unique function on Rn \ Ω that vanishes on ∂Ω, is harmonic on Rn \ Ω,
and approaches 1 at infinity. Then
Z
(n − 2)ωn−1 C =
|∇ϕ|2 dV.
Rn \Ω
For t ∈ [0, 1), let Σt be the level set ϕ−1 (t) (smooth for almost every t). By the co-area
formula2,
Z
Z 1Z
1
2
|∇ϕ| dV =
|∇ϕ|2
dAt dt,
(3)
|∇ϕ|
Rn \Ω
0
Σt
where dAt is the area form on Σt . By the Schwarz inequality,
Z
Z
1
2
|Σt | ≤
|∇ϕ|dAt
dAt ,
(4)
Σt
Σt |∇ϕ|
where |Σt | is the area of Σt . Combining (3) and (4) produces
Z
Z 1
|Σt |2
R
dt.
|∇ϕ|2 dV ≥
1
Rn \Ω
0
Σt |∇ϕ| dAt
Let V (t) be the volume of the region bounded by Σt ; again by the co-area formula,
Z tZ
1
V (t) = vol(Ω) +
dAt dt,
0
Σt |∇ϕ|
and therefore
V 0 (t) =
Z
Σt
1
dAt .
|∇ϕ|
2For an orthogonal foliation of an open set with speed η and parametrized by t, the volume form dV
decomposes as η dt dAt , where dAt is the hypersurface area form on the leaves of the foliation.
THE CAPACITY–VOLUME INEQUALITY OF POINCARÉ–FABER–SZEGÖ
3
Combining the above gives
Z
1
(n − 2)ωn−1 C ≥
0
Z
1
≥
0
|Σt |2
dt
V 0 (t)
2(n−1)
n
V (t)
2
ωn−1
βn
V 0 (t)
dt,
where we have used the isoperimetric inequality on the second line. Let R(t) be the radius of the sphere that has volume equal to V (t), i.e., V (t) = βn R(t)n . Then V 0 (t) =
nβn R(t)n−1 R0 (t), so
Z 1
ωn−1 R(t)n−1
(n − 2)ωn−1 C ≥
dt,
(5)
R0 (t)
0
having used the fact nβn = ωn−1 .
Now, let Ω̃ be the open ball about the origin with the same volume as Ω. Let Σ̃t be
the sphere about the origin of radius R(t), with area form dÃt . Let ϕ̃ be the function
that
R
equals t on Σ̃t . We continue inequality (5), using the fact that ωn−1 R(t)n−1 = Σ̃t dÃt and
the observation that |∇ϕ̃| = R01(t) on Σ̃t :
Z 1Z
|∇ϕ̃|dÃt dt
(by (5))
(n − 2)ωn−1 C ≥
0
Σ̃t
Z
=
|∇ϕ̃|2 dV
(co-area formula)
Rn \Ω∗
≥ (n − 2)ωn−1 C̃,
where on the last line we used the definition of the capacity of Ω̃ (denoted C̃) and the fact
that ϕ̃ is locally-Lipschitz with appropriate boundary conditions. It is easy to check that
equality in (2) holds for round balls. Since Ω has the same volume as Ω̃, the proof of the
inequality is complete.
The case of equality follows readily, since in particular, equality holds in our use of the
isomperimetric inequality on Σt .
References
[1] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics
Studies, no. 27, Princeton University Press, Princeton, N. J., 1951.
Department of Mathematics, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia,
PA 19104,
E-mail address: [email protected]