Estimates for Dirichlet Eigenfunctions

M. VAN DEN BERG E. BOLTHAUSEN

A

Estimates for the Dirichlet eigenfunctions near the boundary of an open, bounded set in euclidean space are obtained. It is assumed that the boundary satisfies a uniform capacitary density condition.

1. Introduction

Let D be an open, bounded set in euclidean spacem_(ml 2, 3, …) with boundary

cD. Let k∆_Dbe the Dirichlet laplacian for D. The spectrum ofk∆_Dis discrete and consists of eigenvalues λ"λ#… with a corresponding orthonormal set of eigenfunctionsoφ",φ#,…q. The behaviour of the eigenfunctions near the boundary cD of D has been investigated by several authors under a variety of assumptions on the geometry of D [1–8, 11–16].

In this paper we obtain pointwise bounds on the eigenfunctions under the assumption that cD satisfies a uniform capacitary density condition. Denote by Cap(A) the newtonian capacity of a compact set A9 m _(ml 3, 4, …) or the

logarithmic capacity of a compact set A9 #. For x ? m_{and r}
0 we define

B(x ; r)l oy ? m_:QykxQ  rq, _(1.1)

and for a non-empty set G9 m

diam(G)l supoQx"kx#Q:x"?G,x#?Gq. (1.2)

D 1.1. Let D 9 m_(ml 2, 3, …) be an open set with boundary cD.

Then cD satisfies an α-uniform capacitary density condition if for some α ? (0, 1] Cap((cD)EB(x; r))  α Cap(B(x; r)), x ? cD, 0 r diam(D). (1.3) Condition (1.3) guarantees that all points ofcD are regular, and in particular that lim_{x x}

!φj(x)l 0 for all x!?cD. Definition 1.1 has been introduced in [10] in a study

of the partition function of the Dirichlet laplacian on open sets with a non-smooth or fractal boundary.

Let d : D,- (0, _) denote the distance function

d(x)l minoQxkyQ: y ? cDq, (1.4)

and let R be the inradius of D, defined by Rl max

x?D

d(x). (1.5)

The main results of this paper are the following.

Received 26 January 1996 ; revised 2 October 1996. 1991 Mathematics Subject Classification 35J25, 60J65.

Part of this research was funded by the British Council and the Swiss National Science Foundation.

T 1.2. Let D be an open, bounded set in #. Suppose cD satisfies (1.3) for someα
0. Then for j l 1, 2, … and all x ? D such that d(x) λ−"/#

j Qφ_j(x)Q 

(

6λ_jlog(2\α#π) k1 log(d(x)λ"/# j )

*

"/# . (1.6)

T 1.3. Let D be an open, bounded set in m_(ml 3, 4, …). Suppose cD

satisfies(1.3) for someα
0. Let j l 1, 2, … and suppose x ? D satisfies d(x)λ"/# j 

0

α' 2"$

1

"+γ(m−")/(m−#) . (1.7) Then Qφ_j(x)Q  2λm/% j (d(x)λ"j/#)("/#)(("/γ)+(m−")/(m−#)) −" , (1.8) where γ l 3−m−"α log(2(2\α)"/(m−#)₎. (1.9)

The bounds in (1.6) and (1.8) are being complemented by the following well-known estimate [11, Lemma 3.1].

T 1.4. Let D be an open, bounded set in m_(ml 2, 3,…). Then for j l 1, 2, …

and x? D

Qφ_j(x)Q  λm/%

j . (1.10)

The bounds of Theorems 1.2 and 1.3 are in general not sharp. For example if D is open, bounded andcD is smooth then the eigenfunctions are comparable with d(x). If D is open, bounded and simply connected in# then it was shown by Ban4uelos [3, Corollary 2.3b] that φ" is comparable to the hyperbolic distance induced by the conformal map F from the unit disc onto D. By Koebe’s 1\4 theorem [17] one then obtains that

φ"(x)Cd(x)"/# _(1.11)

for some constant C depending on D. We will use the ideas of [3] to prove (in Section 5) the following refinement of (1.11).

T 1.5. Let D be an open, simply connected set in # with Šolume QDQ. Then for jl 1, 2, …

Qφ_j(x)Q  2*/#π"/%jQDQ"/%R−#d(x)"/#. _(1.12)

The following example (see [13, 4.6.7]) shows that Theorem 1.5 is sharp.

E 1.6. Let U 9 m_{be the conical region in polar coordinates defined by}

Ul o(r, ω): 0 r 1,ω ? Ωq, (1.13)

whereΩ is an open subset of the unit sphere Sm−". Then

}"(r,ω)0rβ(Ω)_{, (1.14)}

whereβ(Ω) is the positive solution of

β(βjmk2) l λ"(Ω), (1.15)

and whereλ"(Ω) is the first eigenvalue of the Laplace–Beltrami operator on Ω with Dirichlet conditions ofcΩ. In particular if m l 2 and

then

λ"(Ω!)l"% (1.17)

and by (1.15)

β(Ω!)l"#, (1.18)

which shows that the exponent in (1.12) cannot be improved.

The main idea of the proofs of Theorems 1.2 and 1.3 goes back to Brossard and Carmona [10] who obtained estimates for the Dirichlet heat kernel p_D(x, x ; t) for x near D. We improve their lemma [10, Lemma 3.5] and its proof (see also [9]). In the proof of Theorem 1.3 we also require a refinement of Wiener’s test [18].

Let (B(s), s 0; _x, x? m_{) be a brownian motion associated to}k∆jc\ct. Let T D

denote the first exit time of the brownian motion from D :

T_Dl infos  0: B(s) ? cDq. (1.19)

For a compact set K we also define the first entry time

τ_Kl infos
0: B(s) ? Kq. (1.20)

Then

_x[T_D
t] l

&

D

p_D(x, y ; t) dy. (1.21)

By the eigenfunction expansion of the heat kernel and by the semigroup property we have e−tλjφ# j(x)  _ j=" e−tλjφ# j(x)l pD(x, x ; t)l

&

D p#_D(x, y ; t\2) dy. (1.22) Since the Dirichlet heat kernel is monotone in D

p_D(x, y ; t\2)  p_m(x, y ; t\2) (2πt)−m/#. (1.23) Hence e−tλjφ# j(x) (2πt)−m/#

&

D p_D(x, y ; t\2) dy l (2πt)−m/#  x[TD
t\2]. (1.24) The choice tl 2λ−" j (1.25) yields for ml 2, 3, … Qφ_j(x)Q  e(4π)−m/% λm/% j (x[TD
λ−j"])"/#  λjm/%(x[TD
λ−j"])"/#. (1.26)

This proves Theorem 1.4 since_x[T_D
λ−"

j ] 1.

In Sections 2 and 3 we obtain the upper bounds for _x[T_D
λ−"

j ] in the cases

ml 2 and m l 3, 4, … respectively. In the proof of Theorem 1.3 we use the following modification of Wiener’s test. See also [20, Theorem 4.7, p. 73] for related refinements of Wiener’s test.

T 1.7. Let D be an open, bounded set in m_(ml 3, 4, …). Suppose cD

satisfies(1.3) for someα
0. If x ? D satisfies d(x)  (α'\2"$) diam(D), then for any a?

9

2"$

α', diam(D)\d(x)

:

(1.27)

one has

_x[τ_(cD)EB(x;_ad(x)) _]  1k2a−γ, (1.28)

The proof of Theorem 1.7 will be deferred to Section 4.

2. Proof of Theorem 1.2 Let ml 2 and define the Green function

g(x, y)lk(2π)−" log QxkyQ. _(2.1)

The equilibrium measure on a compact set K9 # is the unique probability measure µ_Kconcentrated on K for which

u_K(x)l

&

K

g(x, y)µ_K(dy) (2.2)

is constant on the regular points of K. The function u_Kis the equilibrium potential of Kand its value R(K ) on the regular points of K is the Robin constant. The logarithmic capacity is defined by

Cap(K )l e−R(K)_{. (2.3)}

Then

Cap(K )l expk

(

inf

µ?P(K)

&

K

&

K

g(x, y)µ(dx) µ(dy)

*

, (2.4)

where P(K ) is the set of all probability measures supported by K. Moreover

Cap(B(x ; r))l r"/(#π) _(2.5)

[18, Chapter 3, Proposition 4.11]. Define

Bm(x; r) l oy ? m_:QykxQ _rq. _(2.6) Let a
4. Then _x[T_D
t]  _x[τ_(cD)EB(x;#_d(x))
t] l _x[τ_(cD)EB(x;#_d(x))
t, τ₍cD)EB(x;#_d(x))
T_Bm(x;_ad(x))] j_x[τ_(cD)EB(x;#_d(x))
t, τ₍cD)EB(x;#_d(x)) T_Bm(x;_ad(x))]  1k_x[τ_(cD)EB(x;#_d(x)) T_Bm(x;_ad(x))]j_x[T_Bm(x;_ad(x))
t]. (2.7) Let H be an open half space in# containing Bm(x; ad(x)) such that cH is tangent to cBm(x; ad(x)). Then

_x[T_Bm(x;_ad(x))
t]  _x[T_H
t] l (πt)−"/#

&

[!,ad(x))

e−q#/(%t)_dq ad(x) (πt)−"/#. _(2.8)

For compact sets K"7K# we have by the variational formula (2.4)

Cap(K")Cap(K#). (2.9)

Let x!?cD be such that d(x)lQxkx!Q. Then B(x;2d(x)):B(x!;d(x)), and by (1.3), (2.5) and (2.9)

Cap((cD)EB(x; 2d(x)))  Cap((cD)EB(x!;d(x)))

 α Cap(B(x!;d(x)))lα(d(x))"/(#π)_{. (2.10)}

By (2.3) and (2.10)

By (2.3) and (2.9) we have for K"7K# R_(K")R(K#). (2.12) Hence R((cD)EB(x; 2d(x)))  R(B(x; 2d(x))) k(2π)−" log(2d(x)). _(2.13) Moreover, by (2.1) and (2.2) supou_(cD)EB(x;#_d(x))( y) : y? cB(x; ad(x))q k(2π)−" log((ak2) d(x))

&

(cD)EB(x;#_d(x)) µ₍cD)EB(x;#_d(x))(dy) lk(2π)−" log((ak2) d(x)). _(2.14)

Following the proof of [11, Lemma 3.5] we define for r
0

m(r)lk(2π)−" log((ak2) r), _(2.15)

and h :# ,-  by

h( y)l (R((cD)EB(x; 2d(x)))km(d(x)))−"(u

(cD)EB(x;#_d(x))( y)km(d(x))). (2.16) Now h is superharmonic, harmonic outside (cD)EB(x; 2d(x)), equal to one on (cD)EB(x; 2d(x)), and by (2.14) negative on cB(x; ad(x)). Hence

_x[τ_(cD)EB(x;#_d(x)) T_Bm(x;_ad(x))] h(x) l u_(cD)EB(x;#_d(x))(x)km(d(x)) R((cD)EB(x; 2d(x)))km(d(x)). (2.17) But u_(cD)EB(x;#_d(x))(x)k 1 2πlog(2d(x)), (2.18) so that by (2.11), (2.15), (2.17) and (2.18) _x[τ_(cD)EB(x;#_d(x)) T_Bm_(x;_ad(x))] log(ak2)klog 2 log(ak2)klog α#π. (2.19) Hence by (2.7), (2.8) and (2.19) _x[T_D
λ−" j ] log(2\α#π) log((ak2)\α#π)jπ−"/#ad(x) λ"j/#. (2.20)

We make the following choice for a :

ak2 l 2 λ"/# j d(x)

0

1jlog

0

1 λ"/# j d(x)

11

−" . (2.21) Let zl λ−"/#

j d(x)−". Then d(x)  λ−j"/#implies z 1, and z  1jlog(z) implies a  4.

By (2.20) and (2.21) _x[T_D
λ−" j ]

0

log 2 α#π

10

log 1 α#πjlog(2z)klog(1jlog z)

1

−" j2π−"/#z−"j2π−"/#(1jlog z)−". _(2.22) L 2.1. For z  1

Proof. Inequality (2.23) is equivalent to

4z (1jlog z)#. (2.24)

But (2.24) holds for zl 1. Moreover for z  1

4 2(1jlog z)\z. (2.25)

Integration of (2.25) over [1, z] yields (2.24) and hence (2.23). For z 1, 1\z  (log 1\z)−". Hence by Lemma 2.1 and (2.22)

_x[T_D
λ−" j ]

0

2 log

0

2 α#π

1

j 4 π"/#

1

(log z)−"  6

0

log

0

2 α#π

11

(log z)−". (2.26)

Theorem 1.2 follows from (1.26), (2.26) and by the definition of z. 3. Proof of Theorem 1.3

We define for ml 3, 4, … the Green function on m_by

g(x, y)l 1

c(m)QxkyQ#−m, (3.1)

where

c(m)l 4πm/#(Γ((mk2)\2))−". _(3.2)

For a compact set K9 m_{the equilibrium measure} µ

Kis the unique non-negative

measure on K satisfying

_x[τ_K _] l

&

g(x, y)µ_K(dy). (3.3)

The newtonian capacity of K is defined by

Cap(K )l µ_K(K ). (3.4)

The capacity of a ball is

Cap(B(0 ; r))l c(m) rm−#. _(3.5)

Again, there is a variational description Cap(K )l

(

inf

µ?P(K)

&&

g(x, y)µ(dx) µ(dy)

*

−", (3.6) where P(K ) is the set of all probability measures supported by K. If K",K# are compact sets with K"7K# then Cap(K")Cap(K#). For these facts, see for example [18, Chapter 3].

To prove Theorem 1.3 we adapt [10, Lemma 3.5]. Let b
a
1. Then by the strong Markov property

_x[T_D
t]  1k_x[τ_(cD)EB(x;_ad(x)) t]

 1k_x[τ_(cD)EB(x;_ad(x)) T_Bm(x;_bd(x))]j_x[T_Bm(x;_bd(x))
t]  1k_x[τ_(cD)EB(x;_ad(x)) _]

j_x[_B(T

To estimate the third term in the right-hand side of (3.7) we let y be such that QykxQ l bd(x). Then _y[τ_(cD)EB(x;_ad(x))] _y[τ_B(x;_ad(x)) _] l

0

a b

1

m−# . (3.8)

The fourth term in (3.7) is again estimated by (2.8). Hence _x[T_D
t]  1k_x[τ_(cD)EB(x;_ad(x)) _]j

0

a b

1

m−#

jbd(x) (πt)−"/#. _(3.9)

Choose a and b as follows :

al Ad(x)−β_{", (3.10)}

bl B(d(x))β#−", _(3.11)

whereβ", β#, A and B are the solutions of

β"γl(1kβ"kβ#)(mk2)lβ#, (3.12) A−γl

0

A B

1

m−# l Bλ"/# j . (3.13) From (3.12) we obtain β#lβ"γl

0

1γj mk1 mk2

1

−" , (3.14)

and from (3.13) we obtain Bλ"/#

j l A−γl λ("/#)(("/γ)+(m−")/(m−#))

−_"

j . (3.15)

If we can show that (1.7) implies (1.27) and the requirement b a, then by (3.9)–(3.11) and Theorem 1.7 _x[T_D
λ−" j ] 2A−γd(x)β"γj

0

A B

1

m−# d(x)("−β_"−β_#)(m−#) jBd(x)β_#λ"/# j . (3.16)

Substitution of the values ofβ", β#, A and B respectively in (3.16) gives _x[T_D
λ−"

j ] 4A−γd(x)β"γl 4(d(x) λ"j/#)(("/γ)+(m−")/(m−#))

−"

. (3.17)

Estimate (1.8) follows from (1.26) and (3.17).

Note that b a is (by (3.10), (3.11)) equivalent to showing that B

A d(x)"−

β_"−β_{#. (3.18)}

It follows from (3.14) that

0 β"jβ#l 1jγ 1jγmk1

mk2

1. (3.19)

Since (1.7) implies d(x) λ−"/#

j it is (by (3.19)) sufficient to check that (3.18) holds for

d(x)l λ−"/# j , that is, B A λ−("/#)("− β_"−β_#) j . (3.20)

We see by (3.15) and (3.19) that (3.20) holds with the equality sign.

It remains to check the validity of (1.27). Since D is bounded, D is contained in a hypercube of sidelength diam(D). By monotonicity of the Dirichlet eigenvalues [19, Chapter XIII.15, Proposition 4(a)]

λ_{j λ"l(diam(D))}mπ# _#
(diam(D))1 _#. (3.21) Hence the first inequality in (1.27) is satisfied if

1 λ"/# j d(x)  a. (3.22) By (3.10), (3.14) and (3.15) al (λ"/# j d(x))−("/γ)(("/γ)+(m−")/(m−#)) −_" . (3.23) Hence (3.22) is satisfied if (d(x)λ"/# j )"−("/γ)(("/γ)+(m−")/(m−#)) −"  1. (3.24)

This is indeed the case because (1.7) implies d(x)λ"/#

j  1 and 1k1 γ

0

1 γj mk1 mk2

1

−"
0. (3.25)

The second inequality in (1.27) follows directly from (3.23) and (1.7). 4. Proof of Theorem 1.7

For s
r
0 we define the annulus

B(x ; r, s)l B(x; s)BBm(x; r), (4.1)

and the set

cD_i(x)l (cD)EB(x; bi_{, b}i+"), _(4.2)

where b
1 will be specified later. Let A_i(x) be the event A_i(x)l oτ_cD

i(x) _q. (4.3)

Define

Nl maxok ? : bk+"  a(d(x)) d(x)q. _(4.4)

Then for any n N

oτ₍cD)EB(x;_a(d(x))d(x)) _q :  N i=n A_i(x). (4.5) If bi+" _{d(x) then A} i(x)l 6. We will choose nl minok ? : bk 2d(x)q. _(4.6)

We choose a ‘ spacing ’ s? +_{, s} 2 (to be specified later) and use

N i=n A_i(x): [(N−n)/s] j=! A_n+js(x), (4.7) to obtain _x[τ_(cD)EB(x;_a(d(x))d(x)) _]  1k_x

9

 [(N−n)/s] j=! Ac n+js(x)

:

. (4.8)

For technical reasons we replace A_j(x) by A`<sub>j</sub>(x) which are defined by A`_j(x)l oτ_cD j(x) τB(x;bj+s,_)q. (4.9) Note that _x[A`_j(x)BA_j(x)]l 0, (4.10) and therefore _x[τ_(cD)EB(x;_a(d(x))d(x)) _]  1k_x

9

 [(N−n)/s] j=! A`c n+js(x)

:

. (4.11)

Next we derive a lower bound for_y(A`_j(x)) for QxkyQ  bj_.

L 4.1. Let

bl 2

0

2 α

1

"

/(m−#)

. (4.12)

Then for j n satisfying b(bjjd(x))  2 diam(D) and any y satisfying QykxQ  bj

_y[A`_j(x)] 2:3−mα. _(4.13)

Proof. _{Let x!?cD be such that d(x)lQxkx!Q. One easily checks that if} bj 2d(x)

cD_j(x): (cD)EB(x!;r,br\2), (4.14)

where

rl bjjd(x). _(4.15)

From this we obtain, by the monotonicity and subadditivity of the newtonian capacity,

Cap(cD_j(x)) Cap((cD)EB(x!;r,br\2))

 Cap((cD)EB(x!;br\2))kCap(B(x!;r))

 α Cap(B(x!;br\2))kCap(B(x!;r)), (4.16)

since br\2  diam(D) by assumption. By the choice of b and by (3.5)

Cap(cD_j(x)) αc(m) (br\2)m−#kc(m) rm−# l c(m) rm−#  c(m) bj(m−#)_{. (4.17)}

For z? cD_j(x) andQykxQ  bj_{we have since b} 2

QykzQ  QzkxQjQxkyQ  bj+"jbj 3bj+"\2. _(4.18) Hence by (4.17) and (4.18) _y[A_j(x)]l

&

cDj(x) c(m)−" QykzQm−# µcDj(x)(dz)  c(m)−"(3\2)#−m_b(#−m)(j+")_Cap(cD j(x))  c(m)−"(3\2)#−m_b(#−m)(j+")_{c(m) b}j(m−#) l (3b\2)#−m_{. (4.19)} Furthermore _y[A`<sub>j</sub>(x)] <sub>y</sub>[A<sub>j</sub>(x)]k<sub>w</sub>[τ<sub>B(x</sub>;<sub>b</sub>j+"<sub>)</sub> _], (4.20) whereQwkxQ l bj+s<sub>. Hence</sub> <sub>y</sub>[A`_j(x)] (3b\2)#−mkb(m−#)("−s)_{. (4.21)}

From now on we choose sl 3. Then by (4.21) and (4.12) _y[A`_j(x)] 3#−mα 2kα#2#("−m) 2:3−mα. (4.22) Let _j σ(B(t): t  τ_B(x;_bj_,_)). (4.23)

By definition of A`<sub>j</sub>(x), we have that A`_j(x) is _j+

$-measurable. Let x be such that

Nkn  3. (4.24)

Then for any k? o1, 2, …, [(Nkn)\3]q we have _x

9

k j=! A`c n+$j(x)

:

l x

9

x[A`cn+$k(x)Q n+$k] ; k−" j=! A`c n+$j(x)

:

l _x

9

_B(τ_B(x;_bn+$k,_))(A` c n+$k) ; k−" j=! A`c n+$j(x)

:

 (1k2:3−mα)  x

9

 k−" j=! A`c n+$j(x)

:

. (4.25)

From this we finally obtain _x

9

[(N−n)/ $] j=! A`c n+$j(x)

:

 (1k2:3−mα)[(N−n)/$]  exp ko[(Nkn)\3] 2:3−mαq. _(4.26) Since Nkn  3, [(Nkn)\3]  (Nkn)\6, (4.27) and _x

9

[(N−n)/ $] j=! A`c n+$j(x)

:

 exp ko(Nkn) 3−m−"αq. (4.28)

By the choice of N and n we have

bN+#
a(d(x)) d(x)  bN+", _(4.29) bn−" _2d(x) bn_{, (4.30)} and hence bN−na(d(x)) 2b$ . (4.31) By (4.28) and (4.31) we obtain _x

9

[(N−n)/ $] j=! A`c n+$j(x)

:



0

a(d(x)) 2b$

1

−α$−m−"/logb

l a(d(x))−γ_eα$−m+α$−m−"(log#)/logb

 a(d(x))−γe%α$−m−" 2a(d(x))−γ_{, (4.32)}

by definition ofγ and the fact that b  2, m  3 and α 1. Estimate (1.28) follows from (4.11) and (4.32) provided x? D is such that (i) b(bjjd(x))  2 diam(D) for

jl n, …, N, (ii) (4.24) holds. But (i) is satisfied if

bN+"j2bN−n+"d(x)  2 diam(D), _(4.33)

since b 2. But bN+"  a(d(x)) d(x) and 2bN−n+"  a(d(x)). So (4.33) and hence (i)

are clearly satisfied if

For (4.24) to hold we have to have bN+#−(n−") b'. This is the case by (4.29) and

(4.30) if

a(d(x)) 2b'. (4.35)

But b 4\α since 0 α 1 and ml 3, 4, …. Hence (4.35) and (4.24) hold if a(d(x))2"$

α'. (4.36)

This completes the proof of Theorem 1.7.

5. Proof of Theorem 1.5 Let G_D(:, :) be the Green function for k∆_D. Then

G_D(x, y)l

&

_ !

p_D(x, y ; t) dt, (5.1)

and any Dirichlet eigenfunction ofk∆_Dsatisfies }_j(x)l λ_j

&

D

G_D(x, y)}_j( y) dy. (5.2)

By the Cauchy–Schwarz inequality Q}_j(x)Q  λ_j

(&

D G#_D(x, y) dy

*

"/# (5.3) since R}_jR#l1. L 5.1. For j l 1, 2, … λ_j 8πjR−#. _(5.4)

Proof. By definition of R, D contains an open ball with radius R. Hence D contains an open square with sidelength RN2. Since the Dirichlet eigenvalues are monotone in D,λ_jis bounded from above by the jth eigenvalue of this square. The eigenvalues for this square are given by

λ_k,ll π#(k#jl#)\(2R#), k ? +_{, l}? +_{. (5.5)}

By definition

jl Fo(k, l): k#jl#  2λ_jR#\π#q. (5.6)

Suppose j 4. Then k#jl#  8 since j l 1 corresponds to (k, l) l (1, 1) and j l 2, 3 corresponds to (k, l )l (2, 1) and (k, l) l (1, 2). Hence

λ_j4_R#π#, j 4. (5.7)

But the right-hand side of (5.6) is equal to the number of lattice points in the first quadrant of the disc with radius R(2λ_j\π#)"/#. Hence by (5.6) and (5.7) we have

for j 4

jπ

4((2λjR#\π#)"/#k2"/#)#  λj

R#

8π . (5.8)

Let F be the conformal map from the unit disc onto D with F(0)l x. Then by the results of [3,§1]

G_D(x, y)l 1

2πlog coth(ρD(x, y)), (5.9)

where )_D(x, y)l inf γ

&

" ! Qγh(t)Q QFh(0)Qdt, (5.10)

and where the infimum is taken over all rectifiable curves γ in D with γ(0) l x, γ(1) l y, and where Fh(0) is evaluated at γ(t). By Koebe’s 1\4 theorem

d(γ(t))  QFh(0)Q  4d(γ(t)). (5.11)

Without loss of generality we may assume thatγ has a parametrisation with constant speed c. Then for any suchγ we have

d(γ(t))  d(x)jtc. (5.12) By (5.10)–(5.12) )_D(x, y)1 4

&

" ! c d(x)jtcdtl 1 4log

0

1j c d(x)

1

. (5.13)

Since c QxkyQ we have by (5.9) and (5.13) G_D(x, y) 1

2πlog

(d(x)jQxkyQ)"/#jd(x)"/#

(d(x)jQxkyQ)"/#kd(x)"/#. (5.14)

We note that the right-hand side of (5.14) is positive and strictly decreasing inQxkyQ for x fixed. Hence the square of the right-hand side of (5.14) is strictly decreasing in QxkyQ for x fixed. Let R! be defined by

πR#!lQDQ. (5.15) By spherical-symmetric rearrangement

&

D G#_D(x, y) dy 1 2π

&

R_! ! rdr

0

log(d(x)jr)"/#jd(x)"/# (d(x)jr)"/#kd(x)"/#

1

# l2d(x)#_π

&

_ d(x)/R_! dr r$(log((1jr)"/#jr"/#))# 8d(x) R!π , (5.16) since log((1jr)"/#jr"/#)  2r"/#. The theorem follows from (5.3), (5.4) and (5.16).

C 5.2. Let D be open, simply connected in # with finite Šolume QDQ. Then _x[T_D] 2$/#π−$/%QDQ$/%d(x)"/#. _(5.17)

Proof. By the Cauchy–Schwarz inequality

_x[T_D]l

&

D

G_D(x, y) dy QDQ"/#

(&

D

G#_D(x, y) dy

*

"/# (5.18)

Acknowledgements. The first author wishes to thank R. Ban4 uelos for helpful discussions.

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