Integral estimates of conformal derivatives and spectral properties of the Neumann-Laplacian

0.51 MB·

by  Vladimir Gol'dshtein

INTEGRAL ESTIMATES OF CONFORMAL DERIVATIVES AND
SPECTRAL PROPERTIES OF THE NEUMANN-LAPLACIAN
V.GOL’DSHTEIN,V.PCHELINTSEV,A.UKHLOV
7
1
0 Abstract. Inthispaperwestudyintegralestimatesofderivativesofconfor-
2 malmappingsϕ:D→ΩoftheunitdiscD⊂ContoboundeddomainsΩthat
satisfytheAhlforscondition.Theseintegralestimatesleadtoestimatesofcon-
n
stantsinSobolev-PoincaréinequalitiesandbytheRayleighquotientweobtain
a
spectralestimatesoftheNeumann-Laplaceoperatorinnon-Lipschitzdomains
J
(quasidiscs) in terms of the (quasi)conformal geometry of the domains. The
0 lower estimates of the first non-trivial eigenvalues of the Neumann-Laplace
1 operatorinsomefractaltypedomains(snowflakes)wereobtained.

]
P
A
h. 1. Introduction
t
a 1.1. Estimates of Conformal Derivatives. In the work [19] we obtained lower
m
estimates of the first non-trivial eigenvalues of the Neumann-Laplace operator in
[ thetermsofintegralsofcomplexderivatives(i.e. hyperbolicmetrics)ofconformal

mappings ϕ : D → Ω. Let us recall that the classical Koebe distortion theorem
1
v [7] gives the following estimates of the complex derivatives in the case of univalent
6 analytic functions (conformal homeomorphisms): ϕ : D → Ω normalized so that
1 ϕ(0)=0 and ϕ(cid:48)(0)=1:
6
2 1−|z| 1+|z|
0 ≤|ϕ(cid:48)(z)|≤ .
(1+|z|)3 (1−|z|)3
.
1
0 The example of the complex plane slit along the negative real axis shows that
7 these estimates don’t give even square integrability of the complex derivatives in
1 arbitrary simply connected planar domains. But if Ω ⊂ C is a simply connected
:
v planar domain of finite measure then by simple calculation
i ¨ ¨
X
|ϕ(cid:48)(z)|2 dxdy = J(z,ϕ) dxdy =|Ω|<∞.
r
a D D
(We identify the complex plane C and the real plane R2: C(cid:51)z =x+iy =(x,y)∈
R2.)
Hence in special classes of domains we have better integral estimates of the
complex derivatives than by the Koebe distortion theorem.
In the present work we study integral estimates of the complex derivatives in
domainsboundedbyJordancurvesthatsatisfytotheAhlforsthreepointscondition
0Key words and phrases: Sobolevspaces,conformalmappings,quasiconformalmappings,
ellipticequations.
02010 Mathematics Subject Classification: 35P15,46E35,30C65.
1
SPECTRAL PROPERTIES 2
[3]. For this study we introduce a notion of hyperbolic (integral) α-dilatation of Ω:
¨ ¨
Q(α,Ω):= |ϕ(cid:48)(z)|α dxdy = (cid:12)(cid:12)(cid:0)ϕ−1(cid:1)(cid:48)(w)(cid:12)(cid:12)2−αdudv.
(cid:12) (cid:12)
D Ω
The hyperbolic α-dilatation and its convergence hyperbolic interval
HI(Ω):={α∈R:Q(α,Ω)<∞}.
donotdependonchoiceofaconformalmappingϕ:D→Ωandcanbereformulated
in terms of the hyperbolic metrics [10]. Namely
¨ ¨
|ϕ(cid:48)(z)|α dxdy = (cid:18) λD(z) (cid:19)α dxdy
λ (ϕ(z))
Ω
D D
=¨ (cid:12)(cid:12)(cid:0)ϕ−1(cid:1)(cid:48)(w)(cid:12)(cid:12)2−αdudv =¨ (cid:32)λD(cid:0)ϕ−1(w)(cid:1)(cid:33)2−α dudv
(cid:12) (cid:12) λ (w)
Ω
Ω Ω
where λD and λΩ are hyperbolic metrics in D and Ω [6]. Let us recall that
λD(cid:0)ϕ−1(w)(cid:1)doesnotdependsonchoiceofaconformalhomeomorphismϕ:Ω→D
becauseanyotherconformalhomeomorphismψ−1 :Ω→Disacompositionofϕ−1
and a Möbius homeomorphism (that is an isometry of the hyperbolic metric).
Remark 1.1. For any bounded simply connected domain (−1.78,2] ⊂ HI(α,Ω)
[18, 20]. A more detailed discussion about the hyperbolic α-dilatation and its
convergence hyperbolic interval can be found in Appendix.
In [18] we proved that if a number α > 2 belongs to HI(Ω) then Ω has finite
geodesic diameter. By this reason we call domains that satisfy to a property 2 <
α∈HI(Ω) as conformal α-regular domains [10].
In this paper we obtain estimates of Q(α,Ω) for a large class of conformal α-
regulardomains(so-calledquasidiscs)withthehelpoftheinverseHölderinequality
for Jacobians of quasiconformal mappings. Recall that quasidiscs are images of
the unit discs under quasiconformal homeomorphisms of the complex plane. This
class includes all Lipschitz simply connected domains but also includes a class of
fractal domains (for example, so-called Rohde snowflakes). Hausdorff dimensions
of quasidiscs boundary can be any number of [1,2).
With the help of the estimates for Q(α,Ω) we obtain estimates of constants
for Sobolev-Poincaré inequalities and as a result we obtain lower estimates for
first nontrivial eigenvalues of the Laplace operator with the Neumann boundary
condition.
1.2. Spectral Estimates of the Neumann-Laplacian. The weak formulation
of the spectral problem for the Laplace operator with the Neumann boundary
condition is the following: a function u solves this spectral problem iff u∈W1(Ω)
2
and ¨ ¨
∇u(x,y)·∇v(x,y)dxdy =µ u(x,y)v(x,y)dxdy
Ω Ω
for all v ∈W1(Ω).
2
LetusgiveashorthistoricalremarkabouteigenvaluesestimatesfortheNeumann-
Laplace operator. In 1961 G.Polya [25] obtained upper estimates for eigenvalues in
SPECTRAL PROPERTIES 3
so-called plane-covering domains. Namely, for the first nontrivial eigenvalue µ (Ω)
1
it is:
µ (Ω)≤4π|Ω|−1.
1
The lower estimates for the µ (Ω) for convex domains were obtained in the
1
classical work [24]. It was proved that if Ω is convex with diameter d(Ω) (see, also
[11, 14, 28]), then
π2
µ (Ω)≥ .
1 d(Ω)2
In [19] was proved, that if Ω ⊂ C be a conformal α-regular domain, then the
spectrumofNeumann-LaplaceoperatorinΩisdiscrete,canbewrittenintheform
of a non-decreasing sequence
0=µ (Ω)<µ (Ω)≤µ (Ω)≤...≤µ (Ω)≤...,
0 1 2 n
and
(cid:18) (cid:19)2α−2
4 2α−2 α
(1.1) 1/µ (Ω)≤ √ Q(α,Ω)2/α.
1 απ2 α−2
Specifying by means of the inverse Hölder inequality and the Ahlfors condition
thisestimateforquasidiscsweobtaindiscreetnessofthespectrumoftheNeumann-
Laplaceoperatorinso-calledRohdesnowflakedomainsandgivesthelowerestimate
of first non-trivial eigenvalues:
Theorem A. Let S ⊂ C, 1/4 ≤ p < 1/2, be the Rohde snowflake. Then the
p
spectrum of Neumann-Laplace operator in S is discrete, can be written in the form
p
of a non-decreasing sequence
0=µ (S )<µ (S )≤µ (S )≤...≤µ (S )≤...,
0 p 1 p 2 p n p
and
1 C2e4(1+e2π(16/(1−2p))5) (cid:18)2α−2(cid:19)2αα−2
≤ α
µ (S ) 240π α−2
1 p
(cid:40)π2(2+π2)2e4(1+e2π(16/(1−2p))5)(cid:41)
×exp |S |,
241log3 p
for 2<α< 2K2 , where
K2−1
106 (cid:18)3π2 (cid:19)α α−2
C = , ν =104α e4(1+e2πC5) <1.
α [(α−1)(1−ν)]1/α 237 α−1
Thesuggestedapproachisbasedonestimatesofhyperbolicmetricinquasidiscs.
RecallthatadomainΩ⊂CiscalledaK-quasidiscifitistheimageoftheunitdisc
D under a K-quasiconformal homeomorphism of the complex plane C onto itself.
Note that quasidiscs represent large class domains including fractal type domains
like snowflakes.
Follow [4] a homeomorphism ϕ : Ω → Ω(cid:48) between planar domains is called K-
quasiconformal if it preserves orientation, belongs to the Sobolev class W1 (Ω)
2,loc
and its directional derivatives ∂ satisfy the distortion inequality
ξ
max|∂ ϕ|≤Kmin|∂ ϕ| a.e. in Ω.
ξ ξ
ξ ξ
SPECTRAL PROPERTIES 4
If Ω is a K-quasidisc, then a conformal mapping ϕ : D → Ω allows K2-
quasiconformal reflection. It is well known that Jacobians of quasiconformal map-
pings satisfy the weak inverse Hölder inequality [8]. On the base of the weak
inverse Hölder inequality and the estimates of the constants in doubling conditions
formeasuresgeneratedbyJacobiansofquasiconformalmappings(Proposition3.6)
we obtain
Theorem C. Let Ω ⊂ C be a K-quasidisc. Then the spectrum of Neumann-
Laplace operator in Ω is discrete, can be written in the form of a non-decreasing
sequence
0=µ (Ω)<µ (Ω)≤µ (Ω)≤...≤µ (Ω)≤...,
0 1 2 n
and
(1.2) 1 ≤ K2Cα2 (cid:18)2α−2(cid:19)2αα−2 exp(cid:26)K2π2(2+π2)2(cid:27)·(cid:12)(cid:12)Ω(cid:12)(cid:12),
µ (Ω) π α−2 2log3
1
for 2<α< 2K2 , where
K2−1
C = 106 , ν =104αα−2(cid:0)24π2K2(cid:1)α <1.
α [(α−1)(1−ν)]1/α α−1
The main technical problem of this estimate is evaluations of the quasiconfor-
mality coefficient K for quasidiscs. For this aim we use an equivalent description
of quasidiscs in the terms of the Ahlfors’s 3-point condition. This description of
quasidiscs allows to obtain the estimates for fractal type domains (Theorem A).
On the base Theorem A and Theorem C we can assert that eigenvalues of the
Laplace operator depend on quasi(conformal) geometry of planar domains.
2. Sobolev spaces and Poincaré-Sobolev inequalities
Let E ⊂ C be a measurable set. For any 1 ≤ p < ∞ we consider the Lebesgue
space of locally integrable functions with the finite norm
¨ 1/p
(cid:107)f |Lp(E)(cid:107):= |f(x,y)|p dxdy <∞.
E
We define the Sobolev space W1(Ω), 1 ≤ p < ∞, as a Banach space of locally
p
integrable weakly differentiable functions f : Ω → R equipped with the following
norm:
¨ ¨
(cid:18) (cid:19)1 (cid:18) (cid:19)1
p p
(cid:107)f |W1(Ω)(cid:107)= |f(x,y)|pdxdy + |∇f(x,y)|pdxdy .
p
Ω Ω
We also define the homogeneous seminormed Sobolev space L1(Ω), 1≤p<∞,
p
of locally integrable weakly differentiable functions f : Ω → R equipped with the
following seminorm:
¨
(cid:18) (cid:19)1
p
(cid:107)f |L1(Ω)(cid:107)= |∇f(x,y)|pdxdy .
p
Ω
Recall that the embedding operator i:L1(Ω)→L (Ω) is bounded.
p 1,loc
SPECTRAL PROPERTIES 5
Iff ∈W1(D),1≤p<∞,thenfor0≤κ=1/p−1/q <1/2thePoincaré-Sobolev
p
inequality
¨ 1 ¨ 1
q p
 |f(z)−fD|q dxdy ≤Bq,p(D) |∇f(z)|p dxdy
D D
holds (see, for example, [15, 19]) with the constant
2 (cid:18) 1−κ (cid:19)1−κ
B (D)≤ .
q,p πκ 1/2−κ
This estimate does not applicable in the critical case p = 1 and q = 2. Now
we obtain the upper estimate of the Poincaré constant in the Poincaré-Sobolev
inequality for the Sobolev space W1(D) in this critical case. We use the following
1
Gagliardo inequality for functions with compact support [13, 23]:
¨ 1 ¨
2
(cid:12) (cid:12)2 1
 (cid:12)f(z)(cid:12) dxdy ≤ √ |∇f(z)| dxdy,
2 π
Ω Ω
where Ω⊂C be a bounded Lipschitz domain.
Theorem2.1. Letf ∈W1(Ω). Thenforanyr >0andanyz ∈Ω:dist(z ,∂Ω)>
1 0 0
2r, the following inequality
 1
¨ 2 √ ¨
3 π3
(2.1)  |f(z)−f |2 dxdy ≤ |∇f(z)| dxdy
 D(z0,r)  4
D(z0,r) D(z0,r)
holds.
Proof. We prove this inequality in the case f = 0. In this case the inequity
D(z0,r)
(2.1) can be rewritten as
 1
¨ 2 √ ¨
3 π3
(2.2)  |f(z)|2 dxdy ≤ |∇f(z)| dxdy.
  4
D(z0,r) D(z0,r)
The inequality (2.2) is invariant under translations and similarities and it is suffi-
cient to prove one in for the disc D(0,1).
Denote by D(0,δ) an open disc of the radius δ > 1. Choose the cut function η
in the form η =η(|z|) such that
(cid:40)
1, if |z|<1,
η(z)=
0, if |z|>δ
and linear for z ∈R , where R :=D(0,δ)\D(0,1). Then
δ δ
(cid:40)
δ if z ∈R ,
|∇η(z)|= δ−1 δ
0 otherwise.
Let f(cid:101):Rδ →R be the extension function is defined by the rule
f(cid:101)(z)=f(w(z))η(z), w ∈Rδ,
SPECTRAL PROPERTIES 6
where w(z)=1/z is the inversion about the unit circle S(0,1).
Define the extension operator on Sobolev spaces
E :L1(D(0,1))→L1(D(0,δ))
1 1
by the formula
(cid:40)
f(z) if z ∈D(0,1),
(Ef)(z)=
f(cid:101)(z) if z ∈Rδ.
In order to estimate the norm (cid:107)E(cid:107) of the extension operator E :L1(D(0,1))→
1
L1(D(0,δ)) we have
1 ¨ ¨
(cid:107)E |L1(D(0,δ))(cid:107)= |∇f(z)| dxdy+ |∇f(cid:101)(z)| dxdy.
1
D(0,1) Rδ
Forestimateofthesecondintegralintherightsideofthisequalitybyelementary
calculations we have
(2.3) |∇f(cid:101)(z)|=|∇(f(w(z))η(z))|=|∇(f(w(z)))·η(z)+f(w(z))·∇η(z)|
δ
≤|∇f(w(z))|+ |f(w(z))|.
δ−1
Hence
¨ ¨ ¨
δ
|∇f(cid:101)(w(z))| dxdz ≤ |∇f(w(z))| dxdy+ |f(w(z))| dxdy.
δ−1
Rδ Rδ Rδ
First consider the integral
¨ ¨
|∇f(w(z))| dxdy = |∇f|(w(z))|w(cid:48)(z)|dxdy =
Rδ ¨ Rδ
= |∇f|(w(z))|w(cid:48)(z)||J(z,w)||J(z,w)|−1dxdy
Rδ ¨
|w(cid:48)(z)|
≤ sup |∇f|(w(z))|J(z,w)| dxdy
|J(z,w)|
z∈Rδ
Rδ ¨
|w(cid:48)(z)|
= sup |∇f|(w) dudv,
|J(z,w)|
z∈Rδ
Dδ
where w : R → D , w(z) = 1/z and D = {w ∈ C : 1/δ < |z| < 1}. We calculate
δ δ δ
the norm of the derivative of mapping w by the formula
|w(cid:48)(z)|=|w |+|w |
z z
and the Jacobian of mapping w by the formula
J(z,w)=|w |2−|w |2.
z z
Here
(cid:18) (cid:19) (cid:18) (cid:19)
1 ∂w ∂w 1 ∂w ∂w
w = −i and w = +i .
z 2 ∂x ∂y z 2 ∂x ∂y
By elementary calculations
1
w =0 and w =− .
z z z2
SPECTRAL PROPERTIES 7
Hence
1 1
|w(cid:48)(z)|= and |J(z,w)|= .
|z|2 |z|4
Finally we get
¨ ¨ ¨
|∇f(w(z))| dxdy ≤ sup |z|2 |∇f|(w) dudv ≤δ2 |∇f|(w) dudv.
z∈Rδ
Rδ Dδ D(0,1)
In order to estimate the integral
¨
|f(w(z))| dxdy
A
we will use the change of variable formula.
We obtain
¨ ¨
|f(w(z))| dxdy = |f(w(z))||J(z,w)||J(z,w)|−1 dxdy
Rδ Rδ ¨
1
≤ sup |f(w(z))||J(z,w)| dxdy
|J(z,w)|
z∈Rδ
Rδ ¨ ¨
1
= sup |f(w)| dudv ≤δ4 |f(w)| dudv.
|J(z,w)|
z∈Rδ
Dδ D(0,1)
Using the following Poincaré–Sobolev inequality [1]
¨ ¨
d
|f(z)| dxdy ≤ |∇f(z)| dxdy
2
Ω Ω
where Ω be a convex domain with diameter d and f ∈W1(Ω), finally we obtain
1
¨ ¨
|f(w(z))| dxdy ≤δ4 |∇f(w)| dudv.
Rδ D(0,1)
Thus
¨ ¨ ¨
δ5
|∇f(cid:101)(z)| dxdy ≤δ2 |∇f(w)| dudv+ |∇f(w)| dudv,
δ−1
Rδ D(0,1) D(0,1)
and consequently
¨
(cid:18) δ5 (cid:19)
(cid:107)E(f)|L1(D(0,δ))(cid:107)≤ 1+δ2+ |∇f(z)| dxdy
1 δ−1
D(0,1)
(cid:18) δ5 (cid:19)
= 1+δ2+ (cid:107)f |L1(D(0,1))(cid:107).
δ−1 1
SPECTRAL PROPERTIES 8
Now, by the Gagliardo inequality [13, 23] we obtain
 1
¨ 2
 |f(z)|2 dxdy
 
D(0,1)
 1
¨ 2 ¨
1
≤ |Ef(z)|2 dxdy ≤ √ |∇Ef(z)| dxdy
  2 π
D(0,δ) D(0,δ)
¨ ¨
δ5+δ3−δ2+δ−1
≤ √ |∇f(z)| dxdy ≤C(δ) |∇f(z)| dxdy.
2 π(δ−1)
D(0,1) D(0,1)
Taking the optimal δ =5/4 we have
 1
¨ 2 √ ¨
3 π3
 |f(z)|2 dxdy ≤ |∇f(z)| dxdy.
  4
D(0,1) D(0,1
(cid:3)
3. The Doubling Condition and the Hölder Inequality
Recallnecessaryfactsaboutconformalcapacity. Awell-orderedtriple(F ,F ;Ω)
0 1
of nonempty sets, where Ω is an open set in C, and F , F are closed subsets of Ω,
0 1
is called a condenser on the complex plane C.
The value ˆ
cap(E)=cap(F ,F ;Ω)=inf |∇v|2dxdy,
0 1
Ω
wheretheinfimumistakenoverallLipschitznonnegativefunctionsv :Ω→R,such
that v = 0 on F , and v = 1 on F , is called conformal capacity of the condenser
0 1
E = (F ,F ;Ω). If the set of admissible functions is empty, then cap(F ,F ;Ω) =
0 1 0 1
∞. For finite values of capacity 0 < cap(F ,F ;Ω) < +∞ there exists a unique
0 1
continuous weakly differentiable function u (an extremal function) such that:
¨0
cap(F ,F ;Ω)= |∇u |2dxdy.
0 1 0
Ω
Quasiconformal mappings can be characterized in capacitary terms (see, for ex-
ample[17]). Namely,ahomeomorphismϕ:Ω→Ω(cid:48),Ω,Ω(cid:48) ⊂CisK-quasiconformal,
if and only if
K−1cap(F ,F ;Ω)≤cap(ϕ(F ),ϕ(F );Ω(cid:48))≤K cap(F ,F ;Ω)
0 1 0 1 0 1
for any condenser E =(F ,F ;Ω).
0 1
We will need the following estimate of the conformal capacity (see, for example
[17, 29]).
Denote D(z ,r) an open disc with center z of radius r and D(z ,r) its closure.
0 0 0
(cid:16) (cid:17)
Lemma3.1. [17]LetR>r >0. Thencap D(z ,r),C\D(z ,R);C =2π(logR/r)−1.
0 0
Consider capacity estimates for Teichmüller type condensers in C.
SPECTRAL PROPERTIES 9
Lemma 3.2. Fix 0<r <R. Let F and F be continuums in C such that
0 1
F ∩S(0,ρ)(cid:54)=∅ and F ∩S(0,ρ)(cid:54)=∅
0 1
for all r <ρ<R, where S(0,ρ) is the circle of radius ρ. Then
2 R
cap(F ,F ;D(0,R)\D(0,r))≥ log .
0 1 π r
Denote by R (t)=C\{[−1,0]∪[t,∞]}, t>1, the Teichmüller ring in C.
T
Lemma 3.3. Let R (t) be the Teichmüller ring in C. Then
T
2π
capR (t)= ,
T logΦ(t)
where Φ satisfies the conditions
t+1≤Φ(t)<32t, t>1.
Using this capacity estimates we obtain estimates for a constant in the inverse
Hölder inequality. We start from the weak Hölder inequality.
Lemma 3.4. Let ϕ : Ω → Ω(cid:48) be a K-quasiconformal mapping of planar domains
Ω,Ω(cid:48) ⊂ C. Then for every disc D(z ,r) such that D(z ,2r) ⊂ Ω the weak inverse
0 0
Hölder inequality for the first generalized derivatives of ϕ
 1
¨ 2 ¨
1 24π2K
(3.1)  |ϕ(cid:48)(z)|2 dxdy ≤ |ϕ(cid:48)(z)| dxdy
|D(z ,r)|  |D(z ,2r)|
0 0
D(z0,r) D(z0,2r)
holds.
Proof. Derivatives of K-quasiconformal mappings satisfy the following inequality
([8], pp 274)
 1
¨ 2
1
 |ϕ(cid:48)(z)|2 dxdy
|D(z ,r)| 
0
D(z0,r)
 1
¨ ¨ 2
≤ 4K  1 (cid:12)(cid:12)(cid:12)ϕ(z)− 1 ϕ(z)(cid:12)(cid:12)(cid:12)2 dxdy .
r |D(z ,2r)| (cid:12) |D(z ,2r)| (cid:12) 
0 0
D(z0,2r) D(z0,2r)
Applying to the right side of this inequality the Poincaré-Sobolev inequality (2.1),
we obtain the required inequality (3.1). (cid:3)
By Lemma 3.4 and Theorem 4.2 from [8] we have the following weak Hölder
inequality:
Theorem 3.5. Let ϕ:Ω→Ω(cid:48) be a K-quasiconformal mapping of planar domains
Ω,Ω(cid:48) ⊂ C. Then for every disc D(z ,r) such that D(z ,2r) ⊂ Ω and for some
0 0
SPECTRAL PROPERTIES 10
σ >2 the inequality
 1
¨ σ
1
(3.2)  |ϕ(cid:48)(z)|σ dxdy
|D(z ,r)| 
0
D(z0,r)
 1
¨ 2
1
≤C  |ϕ(cid:48)(z)|2 dxdy
σ|D(z ,2r)| 
0
D(z0,2r)
holds, where
C = 106 , ν =104σσ−2(cid:0)24π2K(cid:1)σ <1.
σ [(σ−1)(1−ν)]1/σ σ−1
Forestimatesoftheleftsideoftheinequality(3.2)weusethedoublingproperty
of measures generated by Jacobians of quasiconformal mappings (see, for example,
[21]). Recall that a Borel measure µ on a set Ω is doubling if there exist a constant
C ≥1 so that the inequality
µ
µ(D(z ,2r))≤C ·µ(D(z ,r))<∞
0 µ 0
hold for all discs D(z ,r) in Ω.
0
In the following lemma we give an estimate of the constant C in the measure
µ
doubling condition.
Proposition 3.6. Let ϕ : C → C be a K-quasiconformal mapping. Then for any
z ∈C and any r >0 we have
0
¨ ¨
(cid:26)Kπ2(2+π2)2(cid:27)
(3.3) |J(z,ϕ)| dxdy ≤exp |J(z,ϕ)| dxdy.
2log3
D(z0,2r) D(z0,r)
Proof. Because the inequality (3.3) is invariant under translations and similarities
we can suppose that z =0 and radius r =1. It is sufficient to show that
0
(cid:26)Kπ2(2+π2)2(cid:27)
|ϕ(D(0,2))|≤exp |ϕ(D(0,1))|.
2log3
Since ϕ is a quasiconformal mapping, ϕ(0) is an interior point of the open con-
nected set U =ϕ(D(0,1)). Denote by λ =dist(ϕ(0),∂U) and let
0
λ1 = max|ϕ(0)−y|, where, U(cid:101) =ϕ(D(0,2)).
y∈∂U(cid:101)
Then 2λ1 ≥ diam(U(cid:101)). Next let F0 be a line segment of length λ0 joining ϕ(0) to
∂U and F1 be a continuum in C=C∪{∞} joining a point in ∂D(ϕ(0),λ1)∩U(cid:101) to
∞ in C\D(ϕ(0),λ ).
1
Denote by w the point of intersection of F and ∂U. Then z = ϕ−1(w ) =
0 0 0 0
(1,θ ), where (1,θ ) are the polar coordinates of the point z .
0 0 0
Consider the ψ :R2 →R2 defined by
ψ(r,θ)=(r,θ−(r−1)θ ), 0≤θ ≤π.
0 0
The transformation ψ is a bi-Lipschtz mapping with the Lipschitz coefficient L ≤
(cid:112)
(2+π4)/2.