On the equilibrium measure and the capacity
of certain condensers
Dimitrios Betsakos
November 29, 2002
FILE: cond.tex
1991 Mathematics Subject Classification. Primary: 31A15.
Key words and phrases: Condenser, capacity, equilibrium measure, Riesz
mass, harmonic measure.
Abstract
We prove some geometric estimates for the equilibrium measure
and the capacity of certain condensers. The proofs are based on the
interpretation of the equilibrium measure as the distributional Laplacian of the corresponding potential, on a formula of T.Bagby, and on
a method of Beurling and Nevanlinna that involves the transport of
the Riesz mass of a superharmonic function.
1
Introduction
A condenser is a triple (R, A, B), where R is a domain in the extended
complex plane C∞ , whose complement C∞ \ R is the union of the nonempty,
disjoint compact sets A and B. If R ⊂ C the capacity of (R, A, B) is defined
by
Z
(1.1)
cap (R, A, B) = cap R = inf
|∇u|2 dm,
u
R
where dm denotes the Lebesgue area measure and the infimum is taken over
all continuously differentiable functions u on R with boundary values 0 on A
and 1 on B.
1
It follows from classical results of Ahlfors and Beurling [?, p.65] that the
capacity of the condenser (R, A, B) is equal to the module of the family of
curves that lie in R and join A and B. In particular, cap R is conformally
invariant. The capacity of an arbitrary condenser (R ⊂ C∞ ) may be defined
by means of an auxiliary Möbius transformation.
Let S(R) denote the family of signed Borel measures of the form σ =
σA − σB , where σA is a unit measure on A and σB is a unit measure on B.
The energy (or transfinite modulus) of the condenser (R, A, B) with ∞ ∈
/ ∂R
is defined by (cf. [?, p.318])
(1.2)
where
(1.3)
E(R) = inf I(σ),
σ∈S(R)
Z
I(σ) =
Z
A∪B
A∪B
log
1
dσ(z)dσ(ζ).
|z − ζ|
If E(R) < ∞, then the infimum in (??) is attained uniquely for a signed
measure τ ∈ S(R) which is called the equilibrium measure of the condenser
[?, Lemma 4]. The logarithmic potential u of τ , defined by
Z
(1.4)
u(z) =
A∪B
log
1
dτ (ζ),
|z − ζ|
is called the equilibrium potential of the condenser (R, A, B).
By Theorems 1 and 2 of [?], if E(R) < ∞ and R is regular for the Dirichlet
problem, then u is continuous on C∞ , harmonic in R, and there exist finite
constants VB ≤ 0 ≤ VA such that
(1.5)
VB ≤ u(z) ≤ VA , z ∈ C∞ ,
(1.6)
u(z) = VA , z ∈ A,
(1.7)
u(z) = VB , z ∈ B,
(1.8)
E(R) = VA − VB .
The fundamental theorem of T.Bagby [?, Theorem 3] asserts that
(1.9)
E(R) =
2
2π
.
cap R
Geometric estimates for the capacity of condensers have been proved
by various authors including H.Grötzsch, O.Teichmüler, G.Pólya, G.Szegö,
V.Wolontis, and V.N.Dubinin. Our purpose is to use Bagby’s identity (??) to
prove inequalities which do not follow from the polarization and symmetrization results presented in [?, Note A], [?].
Our first result concerns the equilibrium measure of certain condensers.
Before stating it, we need to introduce some notation: If F ⊂ C∞ we denote
by Fb the set symmetric to F with respect to the real axis R, i.e. Fb = {z̄ :
z ∈ F }; if ∞ ∈ F , it is understood that ∞ ∈ Fb . For ζ ∈ C we write
ζF = {ζz : z ∈ F }. Also F+ = {z ∈ F : =z ≥ 0}, F− = {z ∈ F : =z ≤ 0}
and −F = {−z : z ∈ F }.
Theorem 1 Let A, B be two disjoint, compact sets in C∞ such that
d ⊂A ,
(a) A
+
−
d
(b) B− ⊂ B+ ,
(c) R := C∞ \ (A ∪ B) is a domain regular for the Dirichlet problem with
∞∈
/ ∂R.
Let τ = τA − τB be the equilibrium measure of the condenser (R, A, B).
Then
b
(1.10)
τA (E) ≥ τA (E),
f or E ⊂ A+ ,
and
(1.11)
τB (F ) ≥ τB (Fb ), f or F ⊂ B− ,
A+
B+
Fb
E
R
A−
b
E
F
¡
¡
¡ B−
Figure 1: An illustration for Theorem ??.
3
Theorem ?? has an electrostatic interpretation: Assume for simplicity
d = A , and that B lies in the upper half-plane:
that A is symmetric: A
+
−
B− = ∅. Then, since the negative charge −τB attracts the positive charge
τA , the set E ⊂ A+ has more charge than Eb has. The components of A and
B are assumed to be connected by very thin wires so that the charge can
move from one component to the other towards equilibrium.
Theorem 2 Let 0 ≤ ρ ≤ b1 ≤ b01 < b02 ≤ b2 < a1 < a2 < ∞ and consider
the sets A = [−a2 , −a1 ] ∪ [a1 , a2 ] and S = {z : |z| ≤ ρ}. For θ ∈ [0, π/2],
let Iθ = eiθ [b1 , b2 ] ∪ ei(θ+π) [b01 , b02 ] and Rθ = C∞ \ (A ∪ Iθ ∪ S). The function
f (θ) := cap (Rθ , A, S ∪ Iθ ) is strictly decreasing on [0, π/2].
Iθ
#Ã
r
−a2
A
r
−a1
S
"!
r
a1
A
r
a2
Iθ
Figure 2: The condenser (Rθ , A, S ∪ Iθ ) of Theorem ??.
For the proof of Theorem 2 we will need an inequality for harmonic measure. Let ω(z, K, D) denote the harmonic measure of a Borel set K ⊂ C
with respect to a domain D ⊂ C∞ at the point z ∈ D \ K; i.e. ω(z, K, D) is
the Perron solution in D \ K of the Dirichlet problem for the Laplacian with
boundary values 1 on K and 0 on ∂D \ K.
Theorem 3 Let A, S, Iθ , Rθ be as in Theorem ??. For z ∈ Rθ , define w(z) =
ω(z, A, Rθ ). Then
(1.12)
w(iy) ≤ w(−iy), y > 0;
(1.13)
w(x) ≤ w(−x), x > 0, x ∈ Rθ ;
(1.14)
w(ζ) + w(ζ̄) ≤ w(−ζ) + w(−ζ̄), ζ ∈ Rθ , <ζ ≥ 0.
For the proof of Theorem 1 in Section 2 we will use the fact that the
equilibrium measure of a condenser is given by the distributional Laplacian
of the equilibrium potential. Theorem ?? will be proved in Section 3. It
4
is a simple consequence of the Markov property of harmonic measure. The
main tool in the proof of Theorem 2 (in Section 4) is Bagby’s identity (??).
We will also use Theorem 3 and a method of Beurling and Nevanlinna that
involves the transport of the equilibrium measure.
We conclude this section with some comments on higher dimensional
extensions of our results. The necessary theory for the equilibrium potential
of space condensers has been developed in [?] and [?, pp.194-195]. Theorem 1
holds in Rn , n ≥ 3 without any essential change: one only needs to replace the
assumptions (a) and (b) with assumptions involving reflection with respect to
an (n − 1)-dimensional plane. With some obvious modifications Theorem ??
also holds in higher dimensions. Apropos of Theorem 2, we note that there
are (at least) two different notions of capacity in Rn , n ≥ 3, the Newtonian
capacity and the conformal capacity. A weaker version of Theorem 2 for
conformal capacity has been proved in [?]. The proof of Theorem 2 can
be modified to give monotonicity results for the Newtonian capacity or the
α-moduli [?] of certain space condensers.
2
Proof of Theorem 1
We will prove (??). The proof of (??) is similar. Let u be the equilibrium
potential of the condenser (R, A, B). Since u is harmonic in R and satisfies
(??), (??), it is superharmonic in D := R ∪ A. Let µ be the Riesz mass of
u (that is, the measure appearing in the Riesz decomposition theorem for
superharmonic functions, see [?, ch.1, §5]).
Lemma 1 µ = τA .
For now we assume that Lemma ?? is true and we defer its proof for the
end of this section. From the proof of the Riesz theorem in [?] it follows that
µ = −∆u/(2π), where ∆u is the distributional Laplacian of u. For r > 0, we
consider the functions ∆r u defined in a neighborhood V of A by
(2.1)
Then [?, p.102]
(2.2)
∆r u(z) =
·
¸
4
1 Z 2π
it
u(z)
−
u(z
+
re
)
dt
.
r2
2π 0
∆r u → 2πµ,
5
in the sense that [?, ch.1, §1]
Z
Z
lim
r→0 V
f ∆r u dm = 2π
V
f dµ,
for all real, continuous functions f with compact support in V .
By Lemma ?? and (??), it suffices to prove that
Z
Z
(2.3)
E
∆r u dm ≥
b
E
∆r u dm,
for all sufficiently small r and all Borel sets E ⊂ A+ .
Taking into account (??), (??), and (??), we see that it suffices to show
that
(2.4)
u(ζ) ≤ u(ζ̄), for ζ ∈ R+ .
This inequality follows at once from the maximum principle applied to the
d.
function u(ζ) − u(ζ̄) on the domain R+ \ A
−
Therefore it remains only to prove Lemma ??.
Proof of Lemma 1
By the Riesz decomposition theorem [?, Theorem 1.220 ],
Z
u(z) =
(2.5)
A
log
1
dµ(ζ) + h1 (z), z ∈ Ω,
|z − ζ|
where Ω is a neighborhood of A in D and h1 is a function harmonic in Ω.
On the other hand, by definition,
Z
(2.6)
u(z) =
A
log
Z
1
1
dτA (ζ) − log
dτB (ζ), z ∈ D.
|z − ζ|
|z − ζ|
B
R
Since B ∩ Ω = ∅, the function h2 (z) = B log |z − ζ| dτB (ζ) is harmonic in Ω.
Setting ν = µ − τA , we infer from (??) and (??) that the potential
Z
v(z) =
A
log
1
dν(ζ)
|z − ζ|
is equal to the harmonic function h2 − h1 in Ω, and hence [?, Th. 1.13] ν = 0.
Thus the lemma is proved.
6
3
Proof of Theorem 3
First we prove (??). Let I1 = eiθ [b1 , b2 ], I2 = ei(θ+π) [b01 , b02 ], I3 = (−I1 ) \ I2 ,
and G = Rθ \ I3 . By the strong Markov property of harmonic measure (see
e.g. [?]) we have
Z
(3.1)
w(iy) = ω(iy, A, G) +
I3
ω(iy, dξ, G) ω(ξ, A, Rθ )
and similarly
Z
(3.2)
w(−iy) = ω(−iy, A, G) +
I3
ω(−iy, dξ, G) ω(ξ, A, Rθ ).
Because of the symmetries G = −G and A = −A,
(3.3)
ω(iy, A, G) = ω(−iy, A, G).
Also, again by the Markov property, for every interval Ξ ⊂ I3 ,
Z
(3.4)
ω(iy, Ξ, G) =
R\A
ω(iy, dt, G+ ) ω(t, Ξ, G)
and
Z
(3.5) ω(−iy, Ξ, G) = ω(−iy, Ξ, G− ) +
R\A
ω(−iy, dt, G− ) ω(t, Ξ, G).
By symmetry, for every interval T ⊂ R \ A,
(3.6) ω(iy, T, G+ ) + ω(iy, −T, G+ ) = ω(−iy, T, G− ) + ω(−iy, −T, G− ).
From (??), (??), and (??) we obtain
(3.7)
ω(iy, Ξ, G) ≤ ω(−iy, Ξ, G).
Finally, (??) follows from (??), (??), (??), and (??).
The proof of (??) is similar to the proof of (??). To prove (??) we apply
the maximum principle to the function w(ζ) + w(ζ̄) − w(−ζ) − w(−ζ̄). Then
(??) follows from (??).
7
4
Proof of Theorem 2
Let 0 ≤ φ < ψ ≤ π/2. By (??), it suffices to show that
(4.1)
E(Rφ ) > E(Rψ ).
Let τ ψ = τ1ψ − τ2ψ be the equilibrium measure of (Rψ , A, S ∪ Iψ ), and let u be
the corresponding equilibrium potential. As in the proof of Theorem ??, we
have τ ψ = −∆u/(2π) in the sense of distributions. Because of the properties
(??)-(??) of u, we have
(4.2)
ω(z, A, Rθ ) =
u(z) − V2
,
V1 − V2
for all z ∈ Rθ and some constants V1 ≤ 0 ≤ V2 .
Theorem ?? and (??) imply
(4.3)
u(ζ) + u(ζ̄) ≤ u(−ζ) + u(−ζ̄), ζ ∈ Rθ , <ζ ≥ 0.
Using (??) and the method of proof of Theorem ?? we obtain
τ1ψ (−E) ≤ τ1ψ (E), E ⊂ A ∩ {z : <z ≥ 0}.
(4.4)
We define the signed measure τ̃ = τ̃1 − τ̃2 on A ∪ S ∪ Iφ as follows:
τ̃1 (K) = τ1ψ (K), for K ⊂ A,
τ̃2 (L) = τ2ψ (L), for L ⊂ S,
τ̃2 (L) = τ2ψ (ei(ψ−φ) L), for L ⊂ Iφ .
(4.5)
(4.6)
(4.7)
A similar transport of the Riesz mass occurs in the proof of the BeurlingNevanlinna projection theorem for harmonic measure; see [?, ch.IV, §5].
We will prove that
(4.8)
I(τ ψ ) ≥ I(τ̃ ).
Note that I(τ̃ ) ≥ E(Rφ ) by the definition of E(Rφ ). Hence (??) implies (??),
and therefore it remains only to prove (??).
By the definition of I(τ ψ ) we have
Z
Z
ψ
I(τ ) =
A∪Iψ ∪S
A∪Iψ ∪S
log
8
1
dτ ψ (z) dτ ψ (ζ) =
|z − ζ|
Z
Z
1
dτ ψ (z) dτ ψ (ζ)
|z
−
ζ|
A∪S A∪S
Z
Z
1
+
log
dτ ψ (z) dτ2ψ (ζ)
|z − ζ| 2
Iψ ∪S Iψ ∪S
Z Z
1
−2
log
dτ1ψ (z) dτ2ψ (ζ) =
|z − ζ|
A Iψ
=
log
=: J1ψ + J2ψ − 2J3ψ .
Similarly, for I(τ̃ ) we have
Z
Z
1
dτ̃ (z) dτ̃ (ζ)
|z − ζ|
A∪S A∪S
Z
Z
1
log
+
dτ̃2 (z) dτ̃2 (ζ)
|z − ζ|
Iφ ∪S Iφ ∪S
Z Z
1
log
−2
dτ̃1 (z) dτ̃2 (ζ) =
|z − ζ|
A Iφ
=: J˜1 + J˜2 − 2J˜3 .
I(τ̃ ) =
log
By (??) and (??) we have J1ψ = J˜1 and J2ψ = J˜2 . Therefore, to prove
(??), it suffices to show that
(4.9)
J3ψ ≤ J˜3 .
We use (??) to write
(4.10)
J3ψ
Z
Z
#
"
1
1
dτ1ψ (x) dτ2ψ (ζ)
=
+ log
log
| − x − ζ|
|x − ζ|
[a1 ,a2 ] Iψ
Z
Z
1
log
+
dσ(x) dτ2ψ (ζ),
|x − ζ|
[a1 ,a2 ] Iψ
where σ is a positive measure on [a1 , a2 ].
Also, by (??) and a change of variables
(4.11)
J˜3 =
Z
Z
"
#
1
1
log
+ log
dτ1ψ (x) dτ2ψ (ζ)
i(φ−ψ)
i(φ−ψ)
| − x − ζe
|
|x − ζe
|
[a1 ,a2 ] Iψ
Z
Z
1
dσ(x) dτ2ψ (ζ),
+
log
i(φ−ψ)
|x − ζe
|
[a1 ,a2 ] Iψ
Now (??) follows from (??), (??), and the following lemma whose elementary
proof is omitted.
9
Lemma 2 Let 0 < t < x. For θ ∈ [0, π/2], define
1
1
+ log
,
iθ
| − x − te |
|x − teiθ |
1
g2 (θ) = log
.
|x − teiθ |
g1 (θ) = log
The functions g1 , g2 are both strictly decreasing.
Aknowledgements: This research was supported by grants from the
Finnish Ministry of Education (C.I.MO.) and the Finnish Academy of Sciences. The author thanks the Department of Mathematics of the University
of Helsinki and especially Professor Matti Vuorinen for their hospitality.
References
[Ahl]
L.V.Ahlfors, Conformal Invariants. McGraw-Hill, New York, 1973.
[AV1]
G.D.Anderson, M.K.Vamanamurthy, The Newtonian capacity of
a space condenser, Indiana Univ. Math. J. 34 (1985), 753-776.
[AV2]
G.D.Anderson, M.K.Vamanamurthy, The transfinite moduli of
condensers in space, Tôhoku Math. J. 40 (1988), 1-25.
[Bag]
T.Bagby, The modulus of a plane condenser, J. Math. Mech. 17
(1967), 315-329.
[Be1]
D.Betsakos, Polarization, conformal invariants, and Brownian motion. Ann. Acad. Sci. Fenn. Ser. A I Math. 23 (1998), 59-82.
[Be2]
D.Betsakos, On conformal capacity and Teichmüller’s modulus
problem in space. To appear in J. Analyse Math.
[Dub]
V.N.Dubinin, Symmetrization in the geometric theory of functions
of a complex variable, Russian Math. Surveys 49 (1994), 1-79.
[Lan]
N.S.Landkof, Foundations of Modern Potential Theory. SpringerVerlag, Berlin, 1972.
10
[Nev]
R.Nevanlinna, Analytic Functions. Springer-Verlag, New York,
1970.
[PóSz]
G.Pólya, G.Szegö, Isoperimetric Inequalities in Mathematical
Physics. Princeton Univ. Press, Princeton, 1951.
[PoSt]
S.C.Port, C.J.Stone, Brownian Motion and Classical Potential
Theory. Academic Press, New York, 1978.
Department of Mathematics, FIN-00014 University of Helsinki, Finland.
[email protected]
11