1. Introduction. In this paper we prove equality of the ... - Project Euclid

Tόhoku Math. J. 40 (1988), 1-25.

THE TRANSFINITE MODULI OF CONDENSERS IN SPACE GLEN D. ANDERSON AND MAVINA K.

VAMANAMURTHY

(Received May 13, 1985) Abstract. The authors prove the equality of the transfinite α-modulus and the discrete α-modulus of a condenser in p-space, 03. Two extremal properties for the transfinite modulus of the Teichmϋller ring are deduced from an expansion-contraction property of condensers. For the important examples of the spherical annulus and a ring of Grδtzsch type in p-space, p^S, the transfinite p-modulus is compared with the conformal modulus.

1. Introduction. In this paper we prove equality of the transfinite α-modulus and the discrete α-modulus (defined in §2) of a condenser in the finite p-space Rp, p ^ 3, 0 < a ^ p, and in the compact space Rp for the case a = p. We also prove an expansion-contraction property of condensers, from which we deduce two extremal properties of the Teichmϋller ring. In addition, we obtain bounds for the transfinite amodulus and study symmetric condensers. For the two important examples of the spherical annulus and a ring of Grotzsch type in p-space, p ;> 3, we show that the transfinite p-modulus is less than the conformal modulus but that their difference is bounded, whereas the two are equal in the plane [B2]. We also study the ratios of the transfinite p-moduli of these condensers to their conformal moduli. Some of these results are analogues of theorems proved by Bagby ([Bl], [B2]) for the case p = a = 2 and by the present authors [AV3] for the case a = 2, p ;> 3. p For p ^ 3, we let R denote the p-dimensional Euclidean space, and p P p R its one-point compactification JR U{°°}. By a condenser in R we shall mean an ordered pair (A, B) of nonempty, disjoint, compact subsets of p R such that oo g A. In particular, it is not required that the complement of AUB be connected. We shall say that (A, B) is a condenser in Rp if A and B are subsets of Rp. The sets A and B are called plates of the condenser. If A, B, and R = Rp \(A\JB) are each connected, then the condenser is called a ring. If (A, B) and (A\ B') are two condensers p in R such that Ac:A' and BaB' we shall say that (A, B)a(A', B'). AMS (MOS) subject classification (1980). Primary 31B15. Secondary 31B05, 28A33, 30C60.

2

G. D. ANDERSON AND M. K.

VAMANAMURTHY

The conformal capacity of a condenser (A, B) in Rp, with oo eJ5, is defined as \Vu\pdx ,

, B) = inf\ JRP

where the infimum is taken over all real-valued functions u that are continuous on Rp and are ACT (absolutely continuous in the sense of Tonelli [Sa]) in Rp with u = 0 on A and u = 1 on B (cf. [G3]). Since cap(^L, B) is a conformal invariant [C, p. 125] its definition can be extended to the case o o g i u S by means of an auxiliary Mobius transformation (cf. [Be, Definition 3.1.1, p. 22]) / of Rp such that oo ef(B). There is a unique extremal function u [G2, Theorem 1], which is a weak solution of the Euler equation ά\v(\Vu\p-2Vu) = 0 .

(2)

The (conformal) modulus of a condenser {A, B) is defined by mod(A, B) = ((Jp-i/capίA, B))l/{p~l) ,

(3)

where σp_x is the (p — l)-dimensional measure of the unit sphere in Rp. Next, we define the ikf. Riesz kernels ka on Rp by x \a~p

(4)

[ — \og\x\

for

0< a < p ,

for

a = p

(cf. [L, p. 431]). By abuse of notation we shall sometimes write k(r) instead of k{x) if r — \x\, for xeRp. Then (5)

Aka(x) =

ί(α - p)(a - 2)\x\a~p~2

for

0< a < p ,

((2 — p)\x\~2 for where Δ denotes the Laplacian. Thus in i2p\{0}, where 0 denotes the origin, ka is subharmonic for 0 < a < 2, harmonic for a = 2, and superp harmonic for 2 < a ^ p. For a compact set EaR we define the αRobin constant Va(E) of ί7 and the (potential-theoretic) a-capacity CJJS) of E by (6)

Va(E) = inf ί ( fcα(^ - y)dμ{x)dμ{y) μ

JEJE

and f^(^)-1 (exp(- Va{E))

if

if

0 0 we let Bp(x) denote the open ball with center at x and radius r and Sp~\x) its boundary sphere.

Then we set BP = Bpr(0), SΓ1 = SP~XO), and Bp = B?, S^1 p

=Sf~1.

1

The (p — l)-dimensional measure of S ~ is denoted by σv_γ and has the value 2πp/2 σ

where Γ is Euler's Gamma function. Lebesgue measure m in Rp is extended to a Borel measure in Rp by setting m({°o}) = 0. When no range of integration is specified, an integral is to be taken over the entire support of the associated measure. Integration with respect to Lebesgue measure is indicated by dx, for example, instead of by dm(x). The authors wish to thank Professor F. W. Gehring for suggesting the problem studied in this paper. They also appreciate the careful reading of the manuscript and helpful comments by Professor M. Ohtsuka. 2. The transfinite α-modulus and the discrete α-modulus of a condenser. For an ordered quadruple of points x19 x2, a?8, x± in Rp with {xlf %i\ Π {%2, #3} = 0 a n d ft>r 0 < a ^ p, we define (8)

Ka(xlf

x2, x3, x,) = Kix,

- x4) + ka(x2

where ka is the kernel defined in (4).

- x3) - ka{xx - x2) - ka(x5

- x,) ,

4

G. D. ANDERSON AND M. K.

VAMANAMURTHY

To see the geometric meaning of (8) when a — p, let f:Rp-+Rp be a Mδbius transformation [Ah] with f{x2) = 0 and f(x4) = °o. Define the absolute

ratio

[Bl] of x19 x29 cc8, x± as

We note that if a?lf a?2, #3> #4 are all distinct and finite then \

x

x

x

x

1 -

\Xi

-

X*\\%z

-

X±\

Also the absolute ratio is invariant under Mδbius transformations of Rp [Ah, p. 19]. If a = p, then (8) reduces to (9)

x2, χS9 x,) = loglίCx, x i 9 xZ9 x,\ ,

Kp(xlf

which is a conformal invariant. Hence we use (9) to define Kp(xlf x2, xZf xd for any quadruple in Rp satisfying the condition {x19 x±} Π {x2, %Ά} = 0 • For 0 < a < p, Ka is invariant under isometries of Rp, but not under general Mobius transformations of Rp. Given a condenser (A, B) in Rp, we denote by Sf{A9 B) the family of all signed Borel measures σ = σA — σB, where σA and σB are unit positive measures on A and B, respectively. For any σ e ^ ( A , B) define (10)

Ia{σ) = ( ( ( ( K a ( a 1 9 b2, b19 JBJBJAJA

where the integral is taken with respect to the product measure σxσx (-σ)x(-σ). To see that this integral exists, we note first that (10) may be simplified to (11)

Ia{σ) = (\\

\JAJA

+\\

JBJB

+\\

JAJB

+\\

)ka(x -

JBJA/

y)dσ(x)dσ(y)

Let diam(AU^) = r0 and dist(A, B) = s0. Then 0 < s0 ^ r0 < oo, Hence 0 < ka(r0) ^ ka(x ~y)^oo if χ,yeAl)B and 0 < ka(x - y) ^ fcα(s0) < oo for (a?, | / ) e i x 5 . It follows that Ia(σ) ^ 2ka(r0) - 2ka(s0) > - oo, so that (10) exists, either as a finite number or as +oo. A proof of the following lemma may be found in [L, Theorem 1.15] for 0 0. The above proof can then be modified in an obvious way to show that the relation mdp(A, B) = WP(A, B) holds also. • REMARK. The proof of the theorem shows that the limiting signed measure τ is indeed the unique minimizing equilibrium measure for mdα(A, B) for each α, 0 < a ^ p. COROLLARY 2. If ((An, Bn)} decreases monotonically to the condenser (A, B) in Rp, then limn Wa(An, Bn) = Wa(A, B), 0 < a ^ p. For a = p we can replace Rp by Rp. PROOF.

This follows from Lemma 3 and Theorem 1.

•

If the plates A and B expand and approach each other, the discrete modulus has the following interesting monotone behavior. THEOREM

2. Let {A, B) and (A', Br) be condensers in Rp, and let

TRANSFINITE MODULI OF CONDENSERS

9 f

/: A[jB~>A'\JB' be an injection such that /(i)cA', f(B)(zB , satisfying \Ax)-Λv)\^\x-v\

for all

(x, y) e(AxA)U(BxB)

for all

(x,y)eAxB

,

and \f(x) -Λv)\

£\x-y\ f

Then Wa(A, B) ^ Wa(A', B ) for each a, 0 < a ^ p. p p we can replace R by R .

. For the case a = p

Let a*eA, 6 ^ 5 , 1 0 we let (Ao, Bo) denote the Teichmϋller ring bounded by the segment Ao = [ — e19 0] and the ray Bo = [te19 oo] along the α^-axis. 4. Let (A, B) be any condenser in Rp for which —e1 and 0 lie in one component of A while a and oo He in one component of B, where \a — t. Then COROLLARY

10

G. D. ANDERSON AND M. K. VAMANAMURTHY md p (A, B) ^ mάp(A0,

Bo) .

PROOF. By the axiom of choice we may define /: AOΌBO->A\JB in the following way. For x e Ao let f(x) einSf*7\ and for x eBo let f(x) e BΓΪS*^ ' Since the minimum and maximum distances between points on two concentric spheres occur when they lie on the same diameter, we then have 1

-y\

if

(x,y)e(AoxAo)[J(BoxBo)

and \Aχ)-Ay)\£\χ-v\ if (χ,y)eAoxBo. The result then follows from Theorem 2.

•

COROLLARY 5. For each φ, 0 ^ φ ^ π, let (Aφi B) be the condenser in Rp for which Aφ = {te: 0 ^ t ^ 1} for e e S^"1 such that φ is the angle between e and the positive x^axis and for which B is the ray [beλ1 °°], b > 1, along the positive xλ-axis. Then

mάp(A0, B) ^ mdp(A9, B) ^ mdp(Ar, B) . PROOF. For the left inequality we let f: Aφ{jB-+AQ{jB be the mapping such that f\A is rotation of Aψ onto AQ and such that f\B is the identity. Then / satisfies the hypotheses of Theorem 2. The right inequality is proved similarly. • REMARK. Inequalities analogous to those in Corollary 5 were obtained by F. W. Gehring for the conformal modulus [V, Lemma 2.58].

3. Symmetric condensers. Following Bagby [Bl, p. 44] we shall say that a condenser (A, B) is symmetric if AaRp+ = {x: xp > 0} and B — A is the reflection of A in the hyperplane xp = 0. If (Af B) is a symmetric condenser for which mdα(A, B) = Ia(τ) < °°, we shall prove that the extremal measure τ has the same symmetry. That is, if Έ is a subset of Rp and if E is the reflection of E in xp — 0, then τ{E) = —τ(E). From this property of symmetric condensers we shall be able to derive a representation for the transfinite p-modulus of certain condensers in terms of the hyperbolic distance function. 5. Let (A, B) be a symmetric condenser in Rp, and let mdα(A, B) = Ia(τ) < °°. Then τ is symmetric with respect to the hyperplane xp = 0. LEMMA

PROOF. Define a new signed measure τ' e S?(A, B) by τ\E) = —τ(E) for each Borel set EaRp. Then

TRANSPINITE MODULI OP CONDENSERS

( \ ka(x - y)dτ'{x)dτ\y)

JAJA

= \ [ k α ( x - y)dτ(x)dτ(y)

11

,

JAJA

U A0» - y)dτ'{x)dτ\y) = \ \ kα(x - y)dτ(x)dτ(y) ,

JAJA

JAJA

and \ \jcα{x - y)dτ\x)dτ\y) = \\ kα(x - y)dτ{x)dτ{y) .

JAJA

JAJA

Hence mdα(A, B) = Iα(τ) = Iα(τ'), and by Lemma 2 we have τ = τr.

Π

p

6. For α symmetric condenser {A, A) in R with mάα(A, A) < oo 9 we have mdα(A, Ά) = min{Ia(σ): σ 6 &ftA, Ά)}, where £K(A, A) is the subset of S?(A, A) consisting of symmetric signed measures. COROLLARY

3. Let (A, A) be a symmetric condenser in Rp with mda(A, A) = Ia(τ) < °°, where τ = μ — v. Then THEOREM

m d a ( A , A) = 2 j j A ( k a ( x - y ) - ka(x - y))dμ{x)dμ{y) PROOF.

.

Since τ is symmetric, we have

LLfc«(» - y)dτ(x)dτ(y) = \ \ ka(x - y)dτ(x)dτ{y) JAJA

JAJA

and \ \^ka(x - y)dτ{x)dτ{y) = -\ \ ka(x - y)dτ{x)dτ{y) .

JAJA

•

JAJA

Next, we define the hyperbolic distance function [T, p. 94] in Rp+ by [xf y] = \x — y\l\x — y\ (also called point-pair invariant [Ah, p. 6]). By conformal in variance this definition can be extended to Bp. COROLLARY

7. For a symmetric condenser (A, A), Ip(σ) = 2Ϊ \ k,([x, y])dσ{x)dσ{y) JAJA

for each σ e S^0(A9 A). p COROLLARY 8. If (A, A) is a symmetric condenser in R and mdα(A, A) = Ia(τ) < °°, then

f

\ (K(x - y) - K(x - y))dτ{y) , f \ K([x, y])dτ(y) , a =p. JA

0< a α y ),

and at edA, 1 ^ i ^ n, is any n-tuple of points. PROOF. First let 2 ai9 αy) + 2 Σ (|a? - αj""* - |a - 3,1-*)

(1)

for xeA, where Σ means that points with subscript 1 do not appear in the sum. Hence / is lower semicontinuous on A, and Af(x) = 2(α - p)(a - 2) Σ (|a - α j α " p " 2 - \x - S,|α"p~2) ^ 0 B) = 1 ™ mod(A, B)

(

PROOF.

(48)

?

lim mά(A> B) - 0 ™ mod(A, B) ~

From Corollary 16 and [Anl, Theorem 1] we have md( A, B ) < j ^ ^ - ^ mod(A, B) ,

and the upper bound follows. The lower bound follows from Corollary 16 and the p-dimensional version of [Gl, Lemma 8] (cf. [C, Lemma 9, p. 235]). The first limit in (47) is an immediate consequence of (46), while the second follows from (48) and [Anl, Theorem 3]. • For two special condensers we have found the transfinite and conformal moduli to be asymptotically equal as r tends to 0. Our final result shows that the transfinite modulus of the spherical condenser behaves much differently as r tends to 1. COROLLARY 20. For p ^ 3 let (A, B) denote the p-dimensional spherical condenser studied in Theorem 6 and let (A', Bf) be the Grotzschtype condenser introduced in Corollary 16. Then

TRANSFINITE MODULI OF CONDENSERS I™ md(A, B)(mod(A',

fAQ\

r

_ „(

B )f

(mod(A, B)f

23

1 Y _ o λoπκcx

WΎ/

for p = 3 and md(A

(50) /or p ^ 4. PROOF.

2

lim > *>» = P " r-i (mod(A, B)pf 2(p-3) If p = 3, it follows from (37) and ΓHδpitaΓs rule twice

that lim md(A, B\ r-i (1 - r) 2 log(l - r)

=

_ JL 2 '

But since mod(A, B\ — log(l/r) ^ 1 - r as r tends to 1 and since limlog(l - r)(mod(A', B')3)2 = - 2 i ί ( ^ L [Anl, Theorem 3] we obtain (49). Finally, let p ^ 4. For re(1/2, 1) and 9>e(0, TΓ) by Cauchy's mean value theorem there exists rx 6 (r, 1) such that 1 j (1 — r) 2

1 + r 2 — 2r cos y 2r(l — cos φ)

1 + rι 2rx(l + r 2 — 2r1 cos φ) Ί[(1 ~ n)2 + 2r x (l - cos φ)] -COSΦ) 2r2(l — cos

V

\2//

Thus 1 - r2

log -log

2r(l - cos

Since p ^ 4, 1 sinp~4(^>/2)cί9 < 00. So we can apply Lebesgue's dominated Jo

convergence theorem to obtain 2 lii mmm d ^( ' ^)> i= £ n - i ?( S i n l - i? , lmi m S

_nn

r

( i — r )2 ) 2 (i r

σ ^ i JJo o σ

sin 2σ?>_1 Jo

X

1 — rr22 r-i 1

log log 2 r ( l — cos

lim φ lim

ί±ϊ 2

r-i r ( l + r — 2 r cos

- 3)

24

G. D. ANDERSON AND M. K. VAMANAMURTHY REFERENCES

[Ah]

[B2]

L. V. AHLFORS, Mόbius Transformations in Several Dimensions, University of Minnesota Lecture Notes, Minneapolis, 1981. G. D. ANDERSON, Extremal rings in w-space for fixed and varying n, Ann. Acad. Sci. Fenn., Ser. AI, 575 (1974), 1-21. G. D. ANDERSON, Limit theorems and estimates for extremal rings of high dimension, Lecture Notes in Mathematics, No. 743, Springer-Verlag, Berlin-Heidelberg-New York, 1979, 10-34. G. D. ANDERSON AND J. S. FRAME, Numerical estimates for a Grotzsch ring constant, Constructive Approximation (to appear). G. D. ANDERSON AND M. K. VAMANAMURTHY, Estimates for the asymptotic order of a Grotzsch ring constant, Tόhoku Math. J. 34 (1982), 133-139. G. D. ANDERSON AND M. K. VAMANAMURTHY, Inequalities for elliptic integrals, Publ. Inst. Math. (Beograd) (N.S.) 37 (51) (1985), 61-63. G. D. ANDERSON AND M. K. VAMANAMURTHY, The Newtonian capacity of a space condenser, Indiana Univ. Math. J. 34 (1985), 753-776. T. BAGBY, Conformal invariants for condensers, Ph.D. Dissertation, Harvard University, 1966. T. BAGBY, The modulus of a plane condenser, J. Math. Mech. 17 (1967), 315-329.

[BBH]

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TRANSFINITE MODULI OF CONDENSERS [We]

J. WERMER, Potential Theory, Lecture Notes in Mathematics, No. 408, SpringerVerlag, Berlin-Heidelberg-New York, 1974.

MICHIGAN STATE UNIVERSITY EAST LANSING, MI 48824

U.S.A.

25

AND

UNIVERSITY OF AUCKLAND AUCKLAND

NEW ZEALAND