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ASYMPTOTIC RATIO OF HARMONIC MEASURES OF SLIT SIDES
D. V. Prokhorov, D. V. Ukrainskii
Saratov State University, 83, Astrakhanskaya st., 410012, Saratov, Russia, [email protected], [email protected]
The article is devoted to the geometry of solutions to the chordal Lowner equation which is based on the comparison of singular solutions and harmonic measures for the sides of a slit in the upper half-plane generated by a driving term. An asymptotic ratio for harmonic measures of slit sides is found for a slit which is tangential to a straight line under a given angle, and for a slit with high order tangency to a circular arc tangential to the real axis.
Key words: Lowner equation, singular solution, harmonic measure, half-plane capacity.
INTRODUCTION
The famous Lowner differential equation have been introduced in 1923 [1] and was aimed to give a parametric representation of slit domains. In this article we describe an asymptotic behavior of singular solutions and harmonic measures for the sides of a slit in domains generated by a driving term of the Lowner equation.
The chordal version of the Lowner equation deals with the upper half-plane H = {z : Imz > 0}, R = dH, and functions f (z,t) normalized near infinity by
f (z,t) = z + - + o(ZL
zz
which solve the chordal Lowner differential equation
df (z,t) 2
dt f (z,t) - A(t)'
f (z, 0) = z, t > 0, (1)
and map subdomains of H onto H. Here A(t) is a real-valued continuous driving term.
Let Yt := 7[0, t] = {7(x) : 0 < x < t} be a simple continuous curve in H U{0} with endpoints 7(0) = 0 and 7(t), 0 < t < T. Then there is a unique map f (z,t) : H \ Yt ^ H satisfying the chordal Lowner equation (1) with A(t) uniquely determined by 7[0, t]. The function f (z,t) can be extended continuously to RU7(t), and f (7(t),t) = A(t). The value t is called the half-plane capacity of the curve Yt, t = hcap(Yt), see, e.g. [2].
We say that 7t e Cn, n e N, on [0, S] if, for the arc-length parameter s of 7t, 7(t(s)) has a continuous derivative 7(n) in s on [0, S], t(S) = T. All the derivatives 7(k), 1 < k < n, at s = 0 are understood as one-side derivatives. A curve 7t e Cn, 7(0) = 0, is said to have at least n-order tangency with a ray Ie = {ei0s : s > 0}, 6 e R, at s = 0 if
7(t(s)) = eies + o(sn), s ^ +0.
Two curves y[0, s] e Cn and r[0, s] e Cn are said to have at least n-order tangency at s = 0 if derivatives y(k) and r(k) in s at s = 0 coincide, 0 < k < n.
The extended function / (z,t) maps Yt onto a segment I = I(t) = [/2(0, t), /1 (0, t)] while / (R) = R\I. The function f1 (0, t) is the maximal singular solution to the chordal Lowner equation (1), and f2(0, t) is the minimal singular solution to (1). Both of these solutions correspond to the singular point /(0,0) = 0 of equation (1).
The curve Yt has two sides Y1t and Y2t which define different prime ends at the same points, except for its tip. We say that Y1t is the left side of Yt if going along the boundary of the domain H \ Yt and moving along R from (-to) to 0, we first meet the side Y1t and then Y2t. In this case, Y2t is called the right side of Yt• The two parts [f2(0, t), A(t)] and [A(t),/1 (0, t)] of segment I(t) are the images of the two sides Y1t and Y2t of Yt under /(z,t), respectively.
The harmonic measures o>(/-1 (i,t); Yfct, H \ Yt) of Yfct at /-1 (i,t) with respect to H \ Yt are defined by the functions which are harmonic on H \ Yt and continuously extended on its closure except for the endpoints of Yt, |7fet = 1, wfc|MU(7t\7fet) =0, k = 1, 2, see, e.g., [3, § 3.6]. Denote
mfc(t) := w(/-1 (i, t); Yfct, H \ Yt), k =1, 2.
In Section 1, we prove the following theorem.
Theorem 1. Let Yt e C4, y(0) = 0, Im y(t) > 0 for t > 0, have at least 4-order tangency at the origin to the straight line under the angle | (1 — 0), —1 < 0 < 1, to the real axis R, and let /(z,t) map H \ Yt onto H and solve the chordal Lowner equation (1). Then
y m1 (t) 1+ 0 lim -= --,
t^+o m2 (t) 1 — 0
where
mfc (t) := w(/-1 (i,t); Yfct, H \ Yt), k = 1, 2, Y1t is the left side of Yt, and Y2t is the right side of Yt•
The most important argument in the proof is the comparison of asymptotic parametric representations of Yt in t and s at s = 0. This approach can be compared with the result by Earle and Epstein [4].
In Section 2, we solve a similar problem for a curve Yt which has at least 6-order tangency with a circular arc in H U {0} tangential to R at the origin. Since a scaling time change t ^ a21 in the Lowner equation (1) is accompanied by changing A(t) ^ -JA(a21) and /(z,t) ^ a/(az,a21), a e R, we can assume without loss of generality that the circular arc is of radius 1, and the argument of its points is increasing when going from 0. We prove the following theorem.
Theorem 2. Let Yt e C6, y(0) = 0, Im y(t) > 0 and Re y(t) > 0 for t > 0, have at least 6-order tangency at the origin to the circular arc of radius 1 centered at i, and let /(z,t) map H \ Yt onto H and solve the chordal Lowner equation (1). Then
Mi (t) o lim TT^V = 2n, t^+o M2 (t)
where
Mfc(t) := w(/-1 (i, t); Yfct, H \ Yt), k = 1, 2, Y1t is the left side of Yt, and Y2t is the right side of Yt•
1. PROOF OF THEOREM 1
Proof of Theorem 1. For 0 = 0, Theorem 1 has been proved [5]. The cases 0 > 0 and 0 < 0 are symmetric to each other, and we will stop only on 0 > 0.
The Lowner equation (1) can be integrated in quadratures in particular cases [6]. For example, if A(t) = cVt, c > 0, then a solution /c(z,t) to equation (1) maps H \ Yt onto H where Yt is parameterized as
Y[0,t] = {z = BVx : 0 < x < t},
with B = B(c) = |B(c)|ei0(c),
|B (c)| = 2
УС2Пб + л 2^2+Т6
Vc2 + 16 - cl
•(c) = 2(1 -7C27T6
Suppose that a C4-slit Yt satisfies the conditions of Theorem 1. Then there exists a driving function A(t) e Lip(2) such that a solution w = /(z,t) to equation (1) maps H \ Yt onto H. For the arc-length parameter s, y(t(s)) is represented as
Y(t(s)) = s + o(s4), s ^ +0.
(2)
Denote
J0(t) = : 0 < x < t}, 0 < 0 < t> 0.
There is c > 0 such that
в =
(3)
Vc2 + 16
for which 0 = 0(c) = 2(1 — 0). Then /c(z,r) maps H \ (|B(c)|VT) onto H. The length a(r) of /^(c)(|B(c)|VT) and the half-plane capacity t of /9(c)(|B(c)|^r) are related by
а(т) = |B(c)|VT, т > 0.
(4)
Let s denote the length of Yt(s) and let a denote the length of projection of Yt(s) onto i0(c) (T). There is a C4-dependence s = s(a),
Therefore,
s(0) = 0, s'(0) = 1, s(k) (0) = 0, k = 2, 3, 4.
s = a + o(a4), a ^ +0.
(5)
Asymptotic expansion (2) implies an asymptotic behavior of a distance between Yt and its projection on ,
dist(Yt(s),/*(c)(a(s))) = o(a4(s)), s ^ +0.
Lind, Marshall and Rohde [2] studied the closeness of half-plane capacities for two curves which are close together. According to Lemma 4.10 [2], we have that
t(s) — t(a(s)) = o(s2), s ^ +0, where ct(t) is given by (4). Hence, due to (4) and (5),
t(s(a)) = t(a) + o(s2(a)) = t(a) + o(a2) = |B(c)|-2s2 + o(s2), s ^ +0. Take into account (2) and rewrite the last relation in the form
Y (t) = |B (c)|Vt + a(t)Vt, t lim+o a(t) = 0.
t—^+o
Choose an arbitrary sequence {xn} of positive numbers xn, xn ^ to as n ^ to, and denote
= gn(w,t) := ^Xnf
Т
w t
— , n = 1, 2,
(6)
(7)
The function z = /-1 (w,t) maps H onto H \ y[0,t], /-1 (A(t),t) = y(t). So the functions gn(w,t) map H onto H \ y(n) [0,t] where
Y(n)(t) = VXnY f—) = ei0(c)|B(c)|V + a ) Vt,
\ xn / \ xn /
gn (^vxn ^xn) ,t) = y(n)(t), 0 <t < T.
We see that
7(n) (t) - ei0(c) |B (c)|V = a(f) V ^ 0, n ^ to,
and the convergence is uniform with respect to t e [0,T].
The Rado theorem [7], see also [8, p. 60], states that a sequence {hn} of conformal mappings hn from the unit disk D onto simply connected domains Dn bounded by Jordan curves dDn, 0 e Dn, hn(0) = 0, h'n(0) > 0, converges uniformly on the closure of D to h : D ^ D, dD is bounded by a Jordan curve, if and only if Dn converges to the kernel D and, for every e > 0, there exists N > 0 such that, for all n > N, there is a one-to-one correspondence zn : dDn ^ dD, |zn(Z) — Z| < e, Z e dDn. Markushevich [9] generalized the Rado theorem to domains with arbitrary boundaries.
Apply the Rado - Markushevich theorem to gn o p with a conformal mapping p from D onto H and obtain that the sequence {gn(w,t)} converges to fc-1 (w,t) as n ^ to uniformly on compact subsets of H U R.
Denote by r[0,t(t)] the left side of the segment I0(c)(|B(c)|VT) and denote by r2[0,r(t)] the right side of this segment. Similarly, denote 7in [0, t] the left side of 7(n) [0, t], and denote 72n [0, t] the right side of 7(n) [0, t]. The functions g-1 (z,t) map 71n [0, t] and 72n [0, t] onto segments I1n = I1n(t) c R and I2n = I2n(t) c R, respectively. It is known, see, e.g. [5], that slit sides r1[0, t(t)] and r2[0,r(t)] are mapped by fc(z,t) onto
Ii = Ii (t) =
c - л/с^ + Гб
2
and /2 = I2(t) =
2
respectively. The uniform convergence of gn to fc-1 implies that I1„(t) tend to I1 (t), and I2„ (t) tend to I2(t) as n ^ to.
Denote by 71n[0, t] and 72n[0, t] the left and the right sides of 7[0, X-], respectively. The function f (z, X-) maps slit sides 71 n [0,t] and 72n[0,t] onto segments I1 n = I1 n(tfc R and I2n = I2n(t) c R, respectively. Compare Ikn (t) and I'kn (t) by (7) and conclude that Ikn (t) = s/xT^n (t), and so
meas Ikn(t) = VXn meas I'n(t), k = 1, 2, n > 1, 0 <t < T.
The harmonic measure is invariant under conformal transformations. This gives that
m1 (Xn) = "(f Xn ),71 [0, Xn ], H \ 7 (Xn)) = "M1 n (t), H) m2(Xn) "(f-1(i, Xn), 72[0, Xn], H \ 7(Xn)) I2n(t) H)'
For k = 1, 2, n > 1, the harmonic measure "(i; I'n(t), H) of I'n(t) at i with respect to H equals the angle divided over n under which the segment I'n(t) is seen from the point i. Similarly, the harmonic measure "(i; Ikn(t), H) of Ikn(t) at i with respect to H equals the angle divided over n under which the segment Ikn(t) is seen from the point i, see, e.g. [8, p. 334]. This shows that the last term in the chain of equalities has a limit as n ^ to, and
lim "MMH = lim w^M'n(*).H)) = lim measJ' f) = lim measInM.
n ^^o "(i,I2n(t), H) n ^^o tan(n"(i,I2n(t), H)) n ^^o measI2n(t) n^^ measI2n(t)
This limit exists for every sequence {xn} tending to infinity. So there exists a limit for the ratio of m1 (t) and m2 (t) as t ^ +0, and
T m1 (t) m1 (") meas I1 (t) Vc2 + 16 + c 1 + £
lim -= lim --n— = lim -= . - = --,
t^+o m2 (t) n^^ m2 (n) t^+o meas I2 (t) Vc2 + 16 — c 1 — £
where £ is given by (3). This leads to the conclusion desired in Theorem 1 and completes the proof. 2. PROOF OF THEOREM 2
Proof of Theorem 2. The Lowner equation (1) admits an explicit integration [10] in the case when 70[0,t] is a circular arc centered at i, 70(0) = 0, with an implicitly given driving function A(t). To be concrete, we will consider 70 [0,t] such that the argument of (70 [0,t] — i) increases in t. Let a
solution /0(z, t) to equation (1) map H \ y0 [0,t] onto H. Its inverse /0 1 (w,t) is represented [10] by the Christoffel - Schwarz integral
1
(1 — Aq w)dw
r-1
(w,t)
1 w — ßi , ß2 + ßi 1 = — log-— +
(1 — ßi w)2(1 — ß2 w) 2п Öw — ß2 ß2 — ßl w — ßi'
where ß1 = ß1(t) and ß2 = ß2 (t) are expanded in powers of ffi,
ßi (t) = Ai ^t2 + a 2t + A3 ^t4 + ..., Ai = —
j/9
and
02 (t) = Bi + B2 ^t2 + Bi =
The driving function Ao (t) is evaluated by
Aq (t) = 20i (t) + 02 (t) = Ci tf + C2 ^t2 + ..., C1 = Bi.
Suppose that a C6-slit Yt satisfies the conditions of Theorem 2. Then there exists a driving function A(t) such that a solution w = f (z,t) to equation (1) maps H\ Yt onto H. For the arc-length parameter s, represent a transformation of y(t(s)),
s + ), s ^ +0.
The function /0(z,t) maps H \ yo [0,t] onto H. Hence,
2/c"i (w,t )
(8)
go (w,t) =
2 + i/o-i (w,t )
maps H onto the exterior of the disk of radius 1 centered at (—i) and slit along the segment [0, a] c R. The length a(T) of [0, a] and the half-plane capacity t of y0 [0, t] are related by
а(т ) =
2/c) (Ao(t),t)
2 + /-i (Ac (t ),t )
= bi + o( 3 t 2 ),
t +0.
(9)
Let s denote the length of y[0, s], and let a denote the length of projection of y[0, s] onto [0, a]. There is a C6-dependence s = s(a),
Therefore,
s(0) = 0, s'(0) = 1, s(k)(0) = 0, k = 2,...,6.
s = a + o(a6), a ^ +0.
(10)
Asymptotic expansion (8) implies an asymptotic behavior of a distance between y and its projection on [0,a],
dist(Y[0, s], [0, a(s)]) = o(a6(s)), s ^ +0. According to Lemma 4.10 [2], we have that
t(s) — t(a(s)) = o(s3), s ^ +0,
where а(т) is given by (9). Hence, due to (9) and (10),
t(s(a)) = t (a) + o(s3 (a)) = t (a) + o(a3 ) = B-3 s3 + o(s3 ), s ^ +0.
Take into account (8) and rewrite the last relation in the form
Y(t) = Bi ffi + a(t) л/t, lim a(t) = 0. (11)
t^+0
i
w
0
Choose an arbitrary sequence {xn} of positive numbers xn, xn ^ to as n ^ to, and denote
z = gn (w,t):= Vxnf-1( VU—Y n = 1, 2,... . (12)
xn xn
The function z = f-1 (w,t) maps H onto H \ 7[0,t], f-1(A(t),t) = 7(t). So the functions
Gn (w,t):= 2gn (w/t) , n = 1, 2,...,
nV 7 2 + ign(w,t) ' ' '
map H onto the exterior of the disk of radius 1 centered at (—i) minus 7(n) (t) where
7(n)(t) = Vx"7 = B Vt + a Vt,
xn xn
Gn(vx" AQ") ,t) = ^(n)(t), 0 <t < T.
We see that
7(-)(t) - Bi VVt = a(X-) VVt — 0, n —> то,
and the convergence is uniform with respect to t e [0,T].
Apply the Rado - Markushevich theorem to gnop with a conformal mapping p from D onto H and obtain that the sequence {Gn(w,t)} converges to G0(w,t) which implies that {gn(w,t)} converges to f0-1 (w,t) as n ^ to uniformly on compact subsets of H U R.
Denote by r1 [0,t(t)] the left side of the circular arc 70[0,t] and denote by T2[0,t(t)] the right side of this circular arc. Similarly, denote 71n [0,t] the left side of
in) [0t]:= 2Y[0, t]
7[0, t] :=
2 - ¿7[0,t]
and denote 72-[0,t] the right side of 7(-) [0, t]. The functions g-1 (z,t) map Yi-[0,t] and 72- [0, t] onto segments /1n = /1n(t) с R and /2- = /2-(t) с R, respectively. It is shown [10] that slit sides Г1 [0, т(t)] and Г2 [0, т(t)] are mapped by /0 (z,t) onto
/1 = /1 (t) = в (t),Ao (t)] and /2 = /2 (t) = [Ao (t),^2 (t)],
respectively. The uniform convergence of g- to /-1 implies that /1-(t) tend to /1 (t), and /2- (t) tend to /2(t) as n — то.
Denote by y1-[0,t] and y2-[0,t] the left and the right sides of 7[0, X-], respectively. The function /(z, XL) maps slit sides y1 - [0,t] and y2-[0,t] onto segments /1 - = /1 -(tf с R and /2- = /2-(t) с R, respectively. Compare /k-(t) and /k-(t) by (12) and conclude that /k-(t) = Vx-/k-(t), and so
meas /fc-(t) = Vх-meas /'-(t), k = 1, 2, n > 1, 0 <t < T.
The harmonic measure is invariant under conformal transformations. This gives that
M2 (Xn) = ^2(/-1 (i Xn ),71 [0, Xn ], H \ 7 (Xn)) = (i,/1 - (t), H) M2(Xn) / (i,Xn),72[0,Xn],H\7(Xn)) "(^-(t^H)'
For k = 1, 2, n > 1, the harmonic measure w(i; /'-(t), H) of /'-(t) at i with respect to H equals the angle divided over n under which the segment /'-(t) is seen from the point i. Similarly, the harmonic measure w(i; /k-(t), H) of /k-(t) at i with respect to H equals the angle divided over n under which the segment / -(t) is seen from the point i. This shows that the last term in the chain of equalities has a limit as n — то and
lim ^^ = lim ^^f'/'"Д = "m ^МШ = Ип1 .
w(i,/2-(t),H) tan(nw(i,/2-(t),H)) meas/2-(t) meas/2-(t)
This limit exists for every sequence {xn} tending to infinity. So there exists a limit for the ratio of M2 (t) and M2 (t) as t ^ +0, and
lim MM = lim Mii|) = lim meas21M = lim - ßw)2
t^+Q M2 (t) n^œ M2 ( П ) t^+0 meas J2(t) t^+0 ß2 (t) - A0 (t)
_ lim (Bi fft + (C - Ai ) ^t2 + ... )2 = B2 = = 2n (B2 - C2)/t2 + ... B2 - C2 -2A i n
This leads to the conclusion desired in Theorem 2 and completes the proof.
D. V. Prokhorov has been supported by the RF Ministry of Education and Science (project no. 1.1520.2014K). D. V. Ukrainskii has been supported by the Russian/Turkish grant RFBR/TUBITAk no. 14-01-91370.
References
1. Löwner K. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann., 1923, vol. 89, no. 1-2, pp. 103-121.
2. Lind J., Marshall D. E., Rohde S. Collisions and spirals of Loewner traces. Duke Math. J., 2010, vol. 154, no. 3, pp. 527-573.
3. Hayman W. K., Kennedy P. B. Subharmonic Functions, vol. 1, London, New York, Academic Press, 1976.
4. Earle C. J., Epstein A. L. Quasiconformal variation of slit domains. Proc. Amer. Math. Soc., 2001, vol. 129, no. 11, pp. 3363-3372.
5. Prokhorov D., Zakharov A. Harmonic measures of sides of a slit perpendicular to the domain boundary. J. Math. Anal. Appt., 2012, vol. 394, no. 2, pp. 738-743.
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9. Markushevich A. I. Sur la representations conforme des domaines a frontieres variables. Rec. Math. [Mat. Sbornik] N.S., 1936, vol. 1(43), no. 6, pp. 863-886.
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УДК 517.54
Асимптотическое отношение гармонических мер сторон разреза
Д. В. Прохоров Д. В. Украинский2
1 Доктор физико-математических наук, заведующий кафедрой математического анализа, Саратовский государственный университет им. Н. Г. Чернышевского, [email protected]
2Студент механико-математического факультета, Саратовский государственный университет им. Н. Г. Чернышевского, [email protected]
Статья посвящена геометрии решений хордового уравнения Левнера, основанной на сравнении сингулярных решений и гармонических мер берегов разреза в верхней полуплоскости, порожденного управляющей функцией. Найдено асимптотическое отношение гармонических мер берегов разреза, касательного к прямой под заданным углом, и разреза, имеющего высокий порядок касания к дуге окружности, касающейся действительной оси.
Ключевые слова: уравнение Левнера, сингулярное решение, гармоническая мера, емкость в полуплоскости.
Д. В. Прохоров получил поддержку Министерства образования и науки РФ (проект № 1.1520.2014К). Д. В. Украинский поддержан российско-турецким грантом РФФИ/ТЮБИТАК № 14-01-91370.
Библиографический список
1. Löwner K. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I // Math. Ann. 1923. Vol. 89, № 1-2. P. 103-121.
2. Lind J., Marshall D. E., Röhde S. Collisions and spirals of Loewner traces // Duke Math. J. 2010. Vol. 154, № 3. P. 527-573.
3. Hayman W. K., Kennedy P. B. Subharmonic Functions. Vol. 1. London, New York : Academic Press, 1976.
4. Earle C. JEpstein A. L. Quasiconformal variation of slit domains // Proc. Amer. Math. Soc. 2001. Vol. 129, № 11. P. 3363-3372.
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УДК 517.584
О НЕКОТОРЫХ ИНТЕГРАЛЬНЫХ СВОЙСТВАХ МОДИФИЦИРОВАННЫХ ФУНКЦИЙ БЕССЕЛЯ
Ю. М. Раппопорт
Кандидат физико-математических наук, доцент, старший научный сотрудник, Институт автоматизации проектирования Российской Академии наук, Москва, [email protected]
Представлены новые интегральные тождества для модифицированных функций Бесселя произвольного комплексного порядка. Изучены свойства интегральных преобразований Лебедева-Скальской.
Ключевые слова: модифицированные функции Бесселя комплексного порядка, интегральные преобразования Конторо-вича-Лебедева, интегральные преобразования Лебедева-Скальской.
1. НЕКОТОРЫЕ СВОЙСТВА ФУНКЦИЙ Ие Ка+гв(х) И 1т Ка+гв(х)
Можем записать вещественную и мнимую часть модифицированных функций Бесселя комплекс-
ного порядка в виде
Re Ka+ie (x) =
Ka+ie (x) + Ka-ie (x)
2
Im Ka+ie (x) =
Kq+ie (x) - Kq-ie (x) 2i
где К (х) — модифицированная функция Бесселя второго рода (также называемая функцией Мак-дональда).
Функции (х), Ие (х) и 1т (х) имеют интегральные представления [1,2]
.-х cosht CQS(et) dt,
Re Kq+ie (x) = / e-x cosh 4 cosh(at) cos(£t) dt,
Kie(x) = / e '0
—x cosh t
(1)
Г<Х>
1т Ка+гв(х)= / е-х С°вЬ4 втЬ(^) 8т(^) (2)
Jo
Из (1), (2) следует, что возможно переписать Ие(х) в виде косинус-преобразования Фурье
ReKq+ie(x) = (I)1/2 Fe[e-xcosh4 cosh(at); t ^ в]
(3)
и
ОС
