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96
An Asymptotic Relation for Conformal Radii of Two Nonoverlapping Domains
A. V. Zherdev
Andrey V. Zherdev, https://orcid.org/0000-0003-2282-4169, Saratov State University, 83, Astrakhanskaya Str., Saratov, 410012, Russia; Petrozavodsk State University, 33, Lenin Str., Petrozavodsk, Republic of Karelia, 185910, Russia, jerdevandrey@gmail.com
We consider a family of continuously varying closed Jordan curves given by a polar equation, such that the interiors of the curves form an increasing or decreasing chain of domains. Such chains can be described by the Lowner-Kufarev differential equation. We deduce an integral representation of a driving function in the equation. Using this representation we obtain an asymptotic formula, which establishes a connection between conformal radii of bounded and unbounded components of the complement of the Jordan curve when the bounded component is close to the unit disk.
Key words: Lowner-Kufarev equation, conformal radius, asymptotic expansion, nonoverlapping domains. DOI: https://doi.org/10-18500/1816-9791-2018-18-3-274-283
INTRODUCTION
We denote by Br = {z e C : |z| < r} the open disk of the radius r > 0 and centered at the origin, B = B1. Let Q be a simply connected domain which is a proper subset of the complex plane and w0 e Q. According to the Riemann mapping theorem there are a unique number r > 0 and a unique function g conformally mapping Q onto the disk Br and such that g(w0) = 0, g(w0) = 1. This r is called a conformal radius of the domain Q with respect to the point w0. Let now Q be a domain in the extended complex plane with at least two boundary points and w0 = to e Q. There are a unique r > 0 and a unique function g, which is analytic in Q except to, where it has the expansion g(w) = w + c0 + c1w-1 + ..., and maps Q one-to-one onto {|z| > 1}. This r is called a conformal radius of the domain Q with respect to to. In both cases we denote the conformal radius of Q with respect to the point w0 by r(Q,w0). Note that if f maps conformally the unit disk B onto Q c C and f (0) = w0, f (0) > 0, then r(Q, w0) = f (0).
Let Q1, Q2 be disjoint simply connected domains in the extended complex plane, 0 e Q1, to e Q2. Then r(Q1,0)r(Q2, to) ^ 1, where the equality sign holds if and only if Q1 and Q2 are the bounded and unbounded components of the complement of a circle with the center at the origin. This results in a corollary of the theorem about nonoverlapping domains obtained by N. A. Lebedev using the area principle [1]. We will consider the case when Q1 and Q2 are the bounded and unbounded components of a closed Jordan curve, respectively. Let f : B ^ Q1 and F : {|z| > 1} ^ Q2 be conformal maps. The composition F-1 о f determines a homeomorphism of the unit circle which is called a conformal welding. We refer the reader to the works [2-7].
In the article, we use the Lowner-Kufarev parametric method to establish an asymptotic relation for conformal radii of two nonoverlapping domains. The Lowner equation
© Zherdev A. V., 2018
is a differential equation describing a continuously increasing sequence of simply connected domains of a special type, i.e. the so called slit domains [8]. Kufarev [9] and Pommerenke [10] generalized the Lowner equation to a wider class of domains.
Given a chain of simply connected domains 0(t), t e [0,T), such that 0 e 0(G) c C 0(t2), 0 ^ ti < t2 < T, the function f(z,t) = elz + conformally mapping D
onto 0(t) for each fixed t e [0,T), a.e. satisfies the (Lowner - Kufarev) equation [9,11]
df (z. о
dt
df (z,t) dz
p(z,t),
z e D, t e [0, T),
(1)
where, for all t e [0, T), p(z, t) is analytic in D with respect to z, p(0,t) = 1, Rep(z,t) > 0 and p(z, t) is measurable with respect to t for any z e D. A similar statement can be formulated for a decreasing chain of domains.
We consider a chain of bounded domains 0(t), 0 e 0(t), 0(0) = D with a boundary r(t) and a chain 0*(t), to e 0*(t), of unbounded domains with the same boundary r(t). The method of Lowner-Kufarev evolution can be used to establish a connection between conformal radii of these domains. In [6], it is shown that if 0(t) is decreasing, p(*, t) e C2(D) for t e [0,T), p(z, •) is continuous in [0, T) for z e D and p(z,t), p> (z,t) and p"(z, t) are bounded in D x [0, T), then ln(r(0*(0), to)) = t + o(t), t — +0.
We suppose now, that r(t) is given by the polar equation r = 7(), t). Let G(t), 0 e G(t) be a chain of domains bounded by a curve with the polar equation r = 7-1 (),t). Let f (z,t) = a(t)z + ... and g(z,t) = b(t)z + ... conformally map D onto 0(t) and G(t), respectively, where a(t) = r(0(t), 0), b(t) = r(G(t), 0) are positive, strictly monotone and continuous functions, a(0) = b(0) = 1. We can always choose the parameter t so that a(t) = el (a(t) = e-t in the case of a decreasing chain of domains). The following theorem gives the asymptotic expansion for b(t) in a neighbourhood of t = 0.
Theorem 1. Let 0(t), t e [0, T), be a chain of domains (increasing or decreasing), 0 e 0(t), r(0(t),0) = e± for each t e [0,T), 0(0) = D, and the boundary r(t) for each t e [0,T) given by the polar equation r = 7(),t), ) e [0,2n], where 7 e C3+a, a e (0,1). Let G(t) be a chain of domains bounded by the family of curves with the polar equation r = Y1(),t) = (7(),t))-1. Then
logr(G(t), 0) = Tt +
1
2n
2n
f ('У(^, 0))2 0
7(^, 0)
t2 + o(t2),
t —^ +0.
(2)
By 7, 7 we denote the first and second derivatives with respect to the parameter t, respectively. In general, we use the following convention. If f is a function of a real or complex variable and t is a parameter, then f denotes the derivative with respect to t, while f denotes the derivative with respect to another variable.
It is not difficult to see that r(0*(t), to) = r(G(t), 0) for t e [0,T). So, we have the following corollary of Theorem 1, which is the main result of the article.
Corollary 1. Let a chain of domains 0(t) and their boundaries r(t) be the same as in Theorem 1, and 0* (t) be a chain of the unbounded components of the complement
of Г (t). Then
log r(0* (t), ж) = Tt +
2n
ij (^0))2
0
7(p> 0) dp
t2 + o(t2),
t-> +0.
(3)
In Sect. 1 we deduce the integral representation for a driving function in the Lowner -Kufarev equation. We use it in Sect. 2, where Theorem 1 is proved.
1. LOWNER - KUFAREV EQUATION
The following theorem gives the integral representation for the driving function p(z, t) in the Lowner - Kufarev equation (1). Note that we do not suppose that f (0,t) = е±, as it is usually done.
Theorem 2. Let 0(t), t e [0, T) be a chain of domains (increasing or decreasing}, 0(0) = D, 0 e 0(t), with a boundary d0(t) given by the polar equation r = у(ф, t) = 1 + 5(ф, t), if e [0,2n], where 5 e C3+a, a e (0,1). Let f (z, t) = a(t)z + ... conformally map D onto 0(t), a(t) > 0. Then f is differentiable with respect to t for t e [0,T), z e D, and satisfies the equation (1) where
P(z,t)
2n
11
2n J \f'(e1^, t)| 0
e*v + z
5(ф(р,t),t) cos(e(ф(p,t),t))dp,
(4)
with ф(p,t) = arg f (e^,t) and в(ф, t) = - arctan().
Remark 1. Note that в(ф, t) is an angle between a normal to the boundary d0(t) at the point y(ф,^*^ and a radius vector of this point.
Remark 2. Here, the function p(z,t) is analytic in D with respect to z, Rep(z,t) > 0 if 0(t) is increasing and Rep(z,t) < 0 if 0(t) is decreasing, p(0,t) = ±1 if a(t) = e±*.
Remark 3. Differentiating (1) with respect to z and putting z = 0, we obtain an equation for the conformal radius r(0(t), 0) = a(t)
2n
d i^i
—ioga(t)=p(0,t) = 2^y \f,(e,v t)\5(ф(p,t),t)cos(eсф(р,t),t))dp.
0
First, we prove the following lemma.
Lemma 1. Let 0l c 0 be domains bounded by simple closed curves Г, Г1 given by the polar equations r = y(ф), r = y1 (ф), ф e [0,2n], y, Y1 e C3+a, a e (0,1). Let f and f1 conformally map D onto 0 and 01, respectively, f (0) = f1 (0) = 0, f'(0) > 0, fl(0) > 0. Let 5(ф) = y(ф) - Yi(ф) satisfy \5(ф)\ < e, \5'(ф)\ < e, \5"(ф)\ < e. Then
fi(z) = f
2n
i-L
1
2n J \f'(e^)\
0
+O(e2), z e D,
e^ + z
5(ф(р)) cos в(ф(р)) ——: dp
e*v — z
+
e —— +0,
(5)
with ф(p) = arg f (e^) and в(ф) = — arctan().
We need the following theorem obtained by Siryk [12] (see also [13, p. 371]). It provides the asymptotic representation for functions conformally mapping D onto domains close to D.
Theorem 3. [12] Let Q be a domain that contains 0 and is bounded by a curve given by the polar equation r = 1 — 5(ф), 0 ф ф ф 2n, where ф is twice differentiable and satisfies the conditions
|5(ф)| <6, |5'(ф)| <6, |5''(ф)| <6.
Then a function f : D ^ Q, f (0) = 0, f (0) > 0, mapping D conformally onto Q has the asymptotic representation
f (z) = z
1 — —
2n
2n
/ 5(ф) фф~z #
0
+ Оф2),
6 ^ +0.
(6)
Proof of Lemma 1. We denote the inverse function by g = f-1. Since 7 e C3+a, f, f', f'', f(3) can be continuously extended to D [14, p. 49] and f' does not vanish there [14, p. 48]. Hence g, g', g", g(3) can be continuously extended to D.
The function g maps the curve Г1 onto a simple closed curve in D, which has the following equation
и(р) = g(Yi(ф(рЖФЫ) = g(f (вг1р) — 5(ф(р))вгф{^)), 0 ф р ф 2п. (7)
We have
и (
(р) — e* = g(f (e*) — 5 (ф (р)КФ(^) — g(f (e*))
= — g (f (e’v ))5 (ф (р))е ,:ф(ф) + О(б2) = — f^—у 5(ф( р)КФ^> + О(е2) = 5(ф(р))е‘(^-в«'М)) + О(е2), е ^ +0,
|f (e
where в(ф(р)) = arg f (e *”)e V
(8)
f (<tv)
— arctan
i (ф(у)) y (ф(я>У)
. Differentiating (7), we obtain
и (р) = g (f (e*) — 5(ф(р)КФ(ф> )[f (e* )ieiv — ф' (рКФ<*> (5' (ф(р)) + iS (ф (р)))], и''(р) = g' '(f (eg — 5(ф( рЖ^ )[f (e*v )iei ф — ф' (рУФ(ф> (5' (ф(р)) + i5 (ф (р)))]2+ +g (f (e*) — 5 (ф (р)7ф^)[—f (e^ )e2i^ — f (e* )e*—
—ф"(рКФЫ (5' (ф(р)) + ^'(/ФЫФ — (ф' ^fe^ (5''(ф(р)) + 2i5'(ф (р)) — 5(ф(р)))].
Since, g'', g(3) can be continuously extended to D and |5(ф)| < е, |5'(ф)| < е, |5''(ф)| <е we obtain the following estimates
и'(р) = [g'(f (ei^)) + O(e)][f'(ei^)ie^ + О(е)] = ie^ + О(е), е ^ +0,
(9)
и
(р) = [g''(f (ei^)) + О(е)][/(ei^ )iei^ + О(е)]2 + [g' (f (e*)) + ОИИ—/'^ )e2i^—
f" (e**)
-f' (e** )e** + 0(e)]
Г 1
(f '(e**))3
+ 0(e)
[-(f' У* ))2 e2** + 0(e)] +
+
f'(e**)
+ 0(e)
[-f"(e**)e2** - f (e**)e** + O(e)]
Л£*) e2i* fCei*) e2i* e,
f' (e**) f' (e**)
Dividing (9) by и(у) gives
e“^ - e** + O(e) = -e** + O(e), e — +0.
(10)
therefore
d ie** + O(e)
dV 10^"(") = = i + 0(e), e — +0,
d
—— arg и(у) = 1 + O(e), e —— +0,
ду
(ii)
(12)
Hence, the curve g(Г1) can be given by a polar equation for e small enough. Denote м(у) = arg и(у) - у, 0 ^ у ^ 2n. Using (8) we obtain
/'■Ы = arg
g(f (e**) - 5y((p))e‘v
e**
arg 1 +
g(f (e**) - 5(ф(v))e‘*(*>) - e**'
= arg 1 -
1
|f'(e**)|
5(ф(y))e-*e<^<*)) + O(e2 ) ) ,
ei*
e —— +0,
Therefore, it is not difficult to see that м(у) = O(e2). From (12) we conclude that м'(у) = O(e). Let r = 1 - Д(у) be the polar equation of g(Г1). Then Д(у) = 1 - |и(у1)|, where у1 is a unique solution of arg и(у1) = у. Hence ^(у1 )+у1 = у and у1 = у+O(e2). Applying (8) gives
Д(у) = 1 - |и(у1)| = 1 - |и(у + O(e2))| = 1 - |и(у) + O(£2)| =
= 1 - |e** + (и(у) - e**) + O(e2)| = 1 - |e** - 1 ..«(ф^У*-5<^<*))) + O(e2)| =
|f' (e**)|
= 1 - |1 - МД)e-*m’<*))) + O(e2)| =
5 (ф (у)) If' (e
If '(e
cos в(ф(у)) + O(e2), 0 ^ у ^ 2n, e — +0.
(13)
From (13) it easily follows that |Д(у)| = O(e). Now we want to deduce similar estimates for |Д'(у)|, |Д''(у)|. It follows from (8)-(10) that |и(у)|' = O(e), |и(у)|'' = O(e). Differentiating the equation Д(у + м(у)) = 1 - |и(у)| we obtain
Д' (у + М(у))(1 + м'(у)) = -|и(у)|',
Д'' (у + м(у))(1 + м' (у))2 + Д' (у + Му)У (у) = -|и(у)|Я
Hence |Д'(у)| = O(e), |Д''(у)| = O(e).
Thus, the curve g(Г1) has the polar equation r = 1 - Д(у) where |Д(у)| = O(e), |Д'(у)| = O(e), |Д''(у)| = O(e), e — +0. Therefore, we can apply Theorem 3 for
h = g о f, h(0) = 0, Ы(0) > 0, conformally mapping D onto the domain bounded by g(r). Hence, (6) gives
2n
h(z) =z (1 - cos e Ml» d| + O(£2),
0
GD, £ +0.
Since fi = f о h, we obtain (5). □
Proof of Theorem 2. Let D(t) be an increasing chain. Fix t e [0,T), h such that t + h e [0, T). First, let h be negative, so D(t + h) e D(t). Denote A(^, h) = = у(^,t) - y(^,t + h). Since y e C3+a, we conclude that |A(^,h)| = O(h), |A'(^,h)| = O(h), |A"(^,h)| = O(h). So we can apply Lemma 1. By (5), we obtain
2n
f (z, t + h) = f I z I 1 - У* s(|, z, t)A(^(|, t), h)d| I , t I + o(h)
2n
f I z I 1 + h у s(|, z, t)(j(^(|, t), t)d| I , t I + o(h),
where
Therefore,
s(l,z,t) =
11
2n |f'(ei^,t)|
2n
e*v +
cos(e (^(|,t),t)^^
e«V — z
f (z, t + h) - f (z, t) = f (z + zh / s(|, z, t)(j(^(|, t), t)d|, t) - f (z, t) + o(h) =
2n
= f (z,t)zh / s(|, z, t)(j(^(|, t), t)d| + o(h), h ^ 0.
Let now h be positive. Then D(t) e D(t + h), and A(^,h) = y(^, t + h) - y(^, t) satisfies all the conditions of Lemma 1. So we have
2n
f (z, t) = f ( z ( 1 - s(|, z, t + h)A(^(|, t + h), h)d| I , t + hi + o(h) =
0
2n
f ( z ( 1 - h у s(|, z, t + h) j(^(|, t + h), t)d| I , t + h I + o(h), So we obtain
f (z, t + h) - f (z, t) = f (z, t + h)-
2n
—f (z - zh J s(|, z, t + h)(j(^(|, t + h), t)d|, t + h) + o(h) = 0
2п
= f'(z,t + h)zh I s(v,z,t + h)5(V(v,t + h),t)dp + o(h), h ^ 0,
Thus, we have shown that f is differentiable with respect to t and satisfies (1). One can similarly repeat the proof for a decreasing chain of domains. □
2. PROOF OF THEOREM 1
Let, for each t e [0,T), f(-,t) and g(-,t) conformally map D onto Q(t) and G(t), respectively, f (0, t) = g(0, t) = 0, f (0, t) > 0, g'(0, t) > 0. Denote 5(V,t) = 7(V,t) _ 1, hi(V,t) = 71 (V,t) — 1. By Theorem 2 and Remark 3, conformal radii satisfy the equations
dt loS r(n(t)>0) = P(0,t)> d bg r(G(t),0) = q(°,t),
where p(z,t), q(z, t) are given by
2n
P(z, t)
1
1
-5(ф(р, t), t) cos(e(V(v, t), t))
e*^ + eiv —
d(p,
2W If'(e^,t)|
0
2n
1 f 1 . e*^ +
q(z,t) = ^ 1 ,, ^ 15i(7i(^,t),t)cos(e 1 C0i(zVMY——-d^,
2W Iff' (e’v,t)|
e*v — z
(14)
(15)
(16)
0
with 7(^7) = argf(e’:<z,t), в(V,t) = — arctan(77т), Vi(v,t) = argg(e’*,t),
Yi
7i (^,i)
^ Y (^,t)
ei(V,t) = — arctan(7l(^’f)). First, we want to prove the following equalities
2n
1 r . e’V + z
P(z 0) = — q(z, 0) = — J <5(^, 0) e.iy _ z dZ z
0
2n
9q(z,0) 9p(z,0) 1 /(5( 0))2 0)e'v + zd
—------------dm = nj(5(v0)) _0)m—dv
0
Elementary calculations lead us to the formulas
5i (V, 0) = _5(V, 0),
5i (V, 0) = -5' (V, 0),
e D,
€ D.
5i(V, 0) = -5(V, 0) + 2(<5(V, 0))2.
(17)
(18)
(19)
(20) (21)
Since f'(e*^, 0) = 1, V(v, 0) = v, в(V, 0) = 0, representation (15) gives
2n
e,m + z
z e D.
4 ■ ' e*^ — z
0
Similarly we obtain
1 r . e*v + z
P(z, 0) = 2n/m 0) ^ ^
2n
1 r . e*v + z
q (z- 0) = 2n/5i (v. 0) m
0
D.
Thus, (19) gives (17).
Denote by P(t) the expression under the integral in (15). Elementary calculations yield the following result
d
P(0) = - dt (|f' (evp ,t)|)
((p, 0) + ('(p, 0)p(p, 0) + d(p, 0).
(22)
t=0
Similarly, we denote by Q(t) the expression under the integral in (16) and obtain
Q(0) = - | (\g' (e*p ,t)\)
(1(P? 0) + (P, 0)p 1(P, 0) + (P, 0)-
(23)
t=0
It easily follows from (1) that p(p, 0) = Imp(eip, 0), p 1 (p, 0) = Im q(eip, 0). Therefore, (17) gives
p 1 (p, 0) = — p(p, 0), 0 ф p ф 2n. (24)
Differentiating (1) with respect to z and putting t = 0 we conclude that
= p(z, 0) + zp'(z, 0). Hence
t=0
• . d
f'(eip, 0) = -f'(eip,t)
= p(eip, 0)+ eipp'(eip, 0).
t=0
Since
we see that
f '(eip, t) = f'(eip, 0) + f'(eip, 0)t + o(t), t ^ +0,
|f'(eip,t)| = |eip + f'(eip, 0)t| + o(t) =
= 1 + | f' (eip, 0)| cos(arg(f'(eip, 0)) — p)t + o(t), t ^ +0.
Thus,
d
rn |f' (e!p ,t)|
= |f'(e'p, 0)| cos(arg(f'(e'p, 0)) — P), 0 ф P ф 2n
t=0
From (25) and (17) we deduce
t=0
d
— d | f (eiP ^ |
0 ф p ф 2n.
(25)
(26)
t=0
Formulas (19)-(24) and (26) show that Q(0) — P(0) = —2<5(p, 0)+2(((p, 0))2, which leads to (18). One can deduce (2) from (14) and (17), (18). Indeed, let, for example, U(t) be an increasing chain of domains. Using (14) and (17) we obtain
dt *og r(G(tr0)
q (0,0) = —p(0,0) = — 1,
t=0
Similarly, using (18) we obtain
d2
dt2
- logr(G(t), 0)
t=0
|q (0-t)
t=0
2n
П (((P.0))2 — <5(P. 0) dp.
Acknowledgements: This work was supported by the Russian Science Foundation (project no. 17-11-01229).
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Cite this article as:
Zherdev A. V. An Asymptotic Relation for Conformal Radii of Two Nonoverlapping Domains. Izv. Saratov Univ. (N. S.), Ser. Math. Meoh. Inform., 2018, vol. 18, iss. 3, pp. 274-283. DOI: https://doi.org/10.18500/1816-9791-2018-18-3-274-283
УДК 517.54
АСИМПТОТИЧЕСКОЕ СООТНОШЕНИЕ ДЛЯ КОНФОРМНЫХ РАДИУСОВ ДВУХ НЕНАЛЕГАЮЩИХ ОБЛАСТЕЙ
А. В. Жердев
Жердев Андрей Владимирович, аспирант кафедры математического анализа, Саратовский национальный исследовательский государственный университет имени Н. Г. Чернышевского, Россия, 410012, Саратов, Астраханская, 83; Петрозаводский государственный университет, Россия, 185910, Республика Карелия, Петрозаводск, просп. Ленина, 33, jerdevandrey@gmail.com
В статье рассматривается семейство замкнутых жордановых кривых, заданных в полярной системе координат и непрерывно зависящих от параметра, и такое, что области, ограниченные этими кривыми, образуют возрастающее или убывающее семейство. Такое семейство областей описывается дифференциальным уравнением Левнера-Куфарева. Для рассмотренного случая получено интегральное представление для управляющей функции в этом уравнении. Используя это представление, получено асимптотическое соотношение, связывающее конформные радиусы ограниченной и неограниченной компоненты дополнения к жордановой кривой, когда ограниченная компонента близка к единичному кругу.
Ключевые слова: уравнение Левнера-Куфарева, конформный радиус, асимптотическое разложение, неналегающие области.
Благодарности. Работа выполнена при финансовой поддержке Российского научного фонда (проект № 17-11-01229).
Образец для цитирования:
Zherdev A. V. An Asymptotic Relation for Conformal Radii of Two Nonoverlapping Domains [Жердев А. В. Асимптотическое соотношение для конформных радиусов двух неналегающих областей] // Изв. Сарат. ун-та. Нов. сер. Сер. Математика. Механика. Информатика. 2018. Т. 18, вып. 3. С. 274-283. DOI: https://doi.org/10.18500/1816-9791-2018-18-3-274-283
