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67Probl. Anal. Issues Anal. Vol. 7 (25), Special Issue, 2018, pp. 31-39 31
DOI: 10.15393/j3.art.2018.5530
The paper is presented at the conference "Complex analysis and its applications" (COMAN 2018), Gelendzhik - Krasnodar, Russia, June 2-9, 2018.
UDC 517.54
A. Kh. Fatykhov, P. L. Shabalin
SOLVABILITY HOMOGENEOUS RIEMANN-HILBERT BOUNDARY VALUE PROBLEM WITH SEVERAL POINTS OF TURBULENCE
Abstract. We consider the so called Hilbert boundary value problem with infinite index in the unit disk. Its coefficient is assumed to be Holder-continuous everywhere on the unit circle excluding a finite set of points. At these points its argument has power discontinuities of orders less than one. We obtain formulas for the general solution and describe completely the solvability picture in a special functional class. Our technique is based on the theory of entire functions and the geometric theory of functions.
Key words: Riemann-Hilbert problem, maximum principle, infinite index, entire functions
2010 Mathematical Subject Classification: 30E25
1. Statement of the problem. Let D be the unit disk in the plane of complex variable z, L = dD. We consider the Riemann-Hilbert boundary value problem for analytic in D functions with the boundary condition
a(t) Re $(t) - b(t) Im $(t) = 0, t G L.
The given coefficients a, b satisfy the Holder condition everywhere on the unit circle except vicinities of a finite set of singular points tj = el9j, j = l,n, where their arguments (arg[a(t) — ib(t)]) have diskontinuities of the second kind. Thus, the problem under consideration belongs to the class of Hilbert boundary value problems with undefined index. A great body of publications on this subject beginning from the fundamental N. V. Govorov's investigations [3] (see also [14], [12], [5]) are dealing with the case, where the problem is stated on the real axis and the turbulence
© Petrozavodsk State University, 2018
(i. e., the essential discontinuity of argument of coefficients) is located in infinity.
A boundary value problem with infinite index and more complicated asymptotics of argument of the coefficient is investigated by V. N. Monahov, E. V. Semenko [6]. In the same monograph the problem with infinite index is firstly considered on Riemann surfaces. M. I. Zhuravleva [15], [16] studied the problems with infinite sets of zeros, poles and jumps of the coefficient on a half of the real axis.
The Hilbert problem for the half-plane with two-sided turbulence of power order less than one is considered by I. E. Sandrygaylo [11]. He obtained formulas for its general solution and described the main cases of solvability by means of the N. I. Muskhelishvili method and results of [12]. R. B. Salimov and P. L. Shabalin [9] applied a more successful formula for the general solution, deduced in [8] by means of the F. D. Gahov regularizing multiplier. As a result, they obtained the full picture of solvability of the problem. Alehno A. G. [1] used the same technique for solving the problem with the logarithmic turbulence. Sevruk A. B. [13] solved the Riemann-Hilbert problem for piecewise analytic functions with many-sided turbulence at the infinity point.
The Hilbert boundary value problem with several points of turbulence was formulated first by R. B. Salimov, A. Kh. Fatykhov and P. L. Shabalin in [10]. The authors described the general solution, existence and uniqueness of solutions, and the set of solutions in the case of non-uniqueness. This article continues this investigation. We define more accurately the general solution and clarify the conditions of solvability. Our results are of interest for the research of boundary-value problems for generalized Cauchy-Riemann system of equations with singular manifolds. For instance, A. B. Rasulov [7] found out that solvability of certain boundary-value problem for generalized analytic functions is immediately connected with solvability of corresponding boundary value problems for analytic functions with finite set of turbulences.
We study the customary statement of the Riemann-Hilbert problem in the form
Re[e-ivW$(t)] =0, t = ei0, t = t., j = (1)
Here G(t) = a(t) — ib(t), and a2(t) + b2(t) = 0 on L, and the function v(0) = argG(t), t = ei0, satisfies the Holder condition everywhere on L except vicinities of points tj, where it has essential discontinuities. We assume that
vj
v(0) = E Vj W + Vj(e)=< j=l
| sin((e - 6j)/2)b
vj
| sin((e - 6j)/2)|pj
, o < e < ej,
, ej <e < 2n,
(2)
where , pj, are known values 0 < pj < 1, and the function 7(0)
satisfies the Holder condition on L. We seek an analytic and bounded in the unit disk D function $(z) satisfying the boundary condition (1).
2. Solution of the homogeneous problem. In order to isolate the essential discontinuities (2) at the points tj, we introduce in the unit disk D the function
/.eiaj _
Pj (z) + iQj (z) = Tj-^-, j = 1,n.
(z - tj )
Its boundary values on the circle L are
Pj (ei6 ) + iQj (ei6 )
j cos(aj — Yj (e)pj) + i sin(aj — Yj (e)pj)
where
Yj (e) :=arg(ei6 - ei6, )
2Pj | sin((e - ej )/2)|Pj
(3n + e + ej-)/2, o < e < ej
, (3)
Jn + 0 + 0y)/2, 0y <0 < 2n. Let us fix the values j > 0 and ay G [0,2n] such that the condition
j / 3n Û \ 2pj cos («j - — Pj - ejPj) = Vj
jcos (aj - 2Pj - ejPj) = j
2P
fulfils. We separate its imaginary and real parts of formula (3), and obtain the representations
Pj (ei6) =
+ • I 6,-6 vj sin '
2T-Pj) + vj sin((n - 6j-6)Pj
I • 6-6, |Pj • / \ I sin 2 j 1 j sin(nPj )
Vj sin I
in( ^ Pj) + vj sin((n - )Pj)
I • 6-6, I Pj . , N
1 sin 2 1 j sin(nPj)
o < e < ej-,
ej- < e < 2n,
Vj cos| (n
((n — ^ )pi) — V+ cos( ^ Pj j
Qj (elti ) =
I • 6-6- I Pj •
Isin —1 J
sin(npj )
, 0 < 0 <8j,
- cos( 6"2- pj) — Vt coS( (n — 8—- )pj) H 6-6j I Pj . , N
|sin 2 j 1 j sin(npj )
, 0j <0 < 2n.
These relations imply the asymptotic formulas
Vj
Pj (e1 )=<
| sin(0 — 0j)/2|pj
Vj
| sin(0 — 0j)/2|Pj
+ o(|0 — 0j|1-j, 0 ^ 0
^ 0j — 0,
+ o(|0 — 0j|1-j, 0 ^ 0j +0,
(4)
V- cos(npj ) — vj+
Qj (ei6 ) =
| sin(0 — 0j)/2|P; sin(npj)
V- — V+ cos(npj ) „ | sin(0 — 0j)/2|Pj sin(npj)
^ + , |0 — 0j|i-p,, 0 ^ 0j — 0,
+ o( |0 — 0j |1-Pj), 0 ^ 0j +0.
The function >(er 6) = v(el6) — ^"=1 Pj (el6) is continuous by virtue of (4). We introduce the Cauchy type integral
2n
e6 + z
d0,
and rewrite the boundary condition of homogeneous problem (1), (2) as
Re
,-ir{t)
—i
(t — tj )pj
$(t)
(5)
Consider the analytic in the disk D function
F (z) = ie
-iT(z)
nexH —i (z—
j=1
j eia;
(z — tj )pj
$(z).
Its boundary values, by virtue of (5), satisfy the relation Im F(t) = 0, t € L, t = tj.
(6)
V
e4 6 - z
I, e
j
0
We express the desired function from equation (6):
n ( /.piOj ^
$(z) = -ie*r(z) J]exP| i (;/_t,)p.} F (z). (8)
If function (8) is a bounded solution of problem (1), then, clearly, there exists an analytic in unit disc D function (6) growing near tj no faster than CieC2/|ij, 0 < pj < 1, i.e., it satisfies the inequalities
|F(z)| < CieC2/|ij, z ^ tj, z e D, 0 < pj < 1, j = 1~n, (9) and its boundary values satisfy condition (7) and the inequality
|F(t)| < CexpjQj(t) j, C = const, t G L.
lj=i J
(10)
In order to prove the converse proposition, we need the following version of the maximum principle for analytical functions ( [4], pp.456, 457). One can find its proof in [10].
Lemma 1. Iîa regular in the disk D function F(z) satisfies the condition HmjF(z)| < C, z G D, t G dD,
at all boundary points of boundary excluding a finite set of points tj, where it grows no faster than CieC2/|z-ij 1 j, 0 < pj < 1, near tj, then |F(z)| ^ Co, Co = const, in the whole disc D.
The following inequality
n
ÏÏm|F(z)| n e-Qj(z) < C, j=i
is valid everywhere on L by virtue of (10). Hence, function (8) is bounded on D by virtue of the maximum principle. Thus, we have proved
Theorem 1. The solution $(z) of the homogeneous boundary-value problem is bounded in D if and only if the function F(z) in the formula of the general solution (8) satisfies the restriction on its growth (9) in the disk D and conditions (7) and (10) on its boundary.
3. Solvability. Theorem 1 means that existence and number of solutions of problem (1), (2) depend on existence and number of analytic functions satisfying conditions (7), (10), (9). We denote by U(Q1,Q2,... ,Qn) the set (maybe, empty) of all functions that are analytic everywhere except the points t1,t2,... ,tn, where they admit essential singularities satisfying conditions (7), (10), (9).
Together with the Hilbert boundary value problem (1), (2) with finite set of turbulences, we consider n boundary value problems on L with a single turbulence for each.
For every j = l,n we consider the homogeneous Hilbert boundary value problem on the circle L with a single point of two-sided turbulence
Vj ( 6U,(t)]=0 t = e46
Re[e-l^j(6>$j(t)] =0, t = ex 6, t = tj, (11)
Vj
Pj (0) = Vj (0) + Vj (0),Vj (0) =
| sin((0 — 0j)/2)|p;
Vj
| sin((0 — 0j)/2)|pj
, 0 < 0 <0j,
, 0j <0 < 2n,
(12)
where Vj(0) satisfies the Holder condition on the circle L.
The problem (11), (12) is studied for 0j = n, in the article [2]. According to the results of this paper, we rewrite the general solution (z) for our case in the form
( l .pio-j 1
^(z) = —ieiT;(z) exp< -— If,(z). (13)
exp< i ■
P (z — tj )pj
Here rj (z) is a Cauchy type integral on L with density Vj (0); a.j ,lj, tj, pj are the same values as above, and analytical in D function Fj (z) satisfies the condition
F(z)| < C1e°2/\tj-z|Pj, z € D, 0 <pj < 1, C1,C2 = const,
(14)
|Fj(t)| < CeQj(t),C = const, ImF(t) = 0, t € L.
We denote by U(Qj) the set of analytic on the whole complex plane except the point tj functions satisfying (14) in the disk D. Then the following theorem is true.
Theorem 2. The homogeneous boundary value problem (1), (2) is solvable if and only if each of n problems (11), (12) is solvable.
Proof. Let problem (11), (12) be solvable for any j = 1,n. This means that for any j there exists an analytic in D function Fj (z) satisfying condition (14) and continuous on the boundary except the point tj, where its growth does not exceed C1eC2/|ij. We introduce the function
n
F(z) := n Fj(z). j=i
Clearly, by virtue of (14)), the introduced function F(z) satisfies conditions (7), (10), (9). Consequently, the formula (8) is meaningful, and the problem (1), (2) has a solution.
Now let us consider a case where the problem (1), (2) has a solution, i.e., there exists a function F(z) ee U(Q1, Q2,...,Qn). We prove the necessity first for n = 2. Assume that the problem (11), (12) has a solution for j = 1 and ha not have one for j = 2, i. e., there exist functions F(z) e e U(Qi,Q2), Fi(z) e U(Qi). But then we have F(z) - Fi(z) e U(Q2), and the problem (11), (12) is solvable, too. Let us suppose now that the function F(z) e U(Qi,Q2) exists, but the problem (11), (12) is not solvable neither for j = 1, nor for j = 2. We denote by Gi(z) the main part of expansion of the function F(z) in the Laurent series in powers of 1/(z — ti). Then the function Ri(z) := F(z) — Gi(z) is regular in D, and Im Ri(t) = — Im Gi(t), t e L. Let Si(z) be a solution of the Schwartz problem in the unit disc for a continuous function Im Gi(t). Obviously, then the function F(z) —Gi(z)+iSi(z) = F2(z) belongs to U(Q2), and the problem (11), (12) is solvable for j = 2; this contradicts our assumption. Thus, Theorem 2 is proved for n = 2; the case n > 2 can be justified by means of mathematical induction. □
The complete picture of solvability of the problem (11), (12) in the class of bounded in the unit disc analytic functions is described by the following two theorems [2].
Theorem 3. Let pj < 1/2. Then the homogeneous boundary value problem (11), (12)
a) has no non-trivial bounded solutions if either v— cos(npj) — v+ < 0 or v+ cos(npj) — v- > 0;
b) has a unique solution (z) = —iAer(z)e (z 1)j , A = const, Im A = 0, if
( v- cos(npj) — v+ =0, J cos(npj) — V- = 0,
I V+ cos(npj ) — V- < 0, I V- cos(npj) — V+ > 0;
c) has infinite set of solutions of the form (13), where Fj(z) is arbitrary analytic in D function satisfying there inequality (14), if v- cos(np) — - v + > 0 and v + cos(np) — v- < 0.
Theorem 4. If l > p ^ 1/2 and either v- < 0 or v + > 0, then the homogeneous boundary-value problem has only null solution; if 1 > p > 1/2 and conditions v- > 0 and v + < 0 hold, then the homogeneous boundary value problem has infinite set of solutions (13), where F(z) is arbitrary analytic in D function satisfying condition (14).
Acknowledgment. This work was supported by RFFI (project 17-01-00282-a).
References
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Received June 10, 2018.
In revised form, September 19, 2018. Accepted September 17, 2018. Published online September 25, 2018.
A. Kh. Fatykhov
Kazan State University of Architecture and Engineering
Kazan, Russia
E-mail: vitofat@gmail.com
P. L. Shabalin
Kazan State University of Architecture and Engineering
Kazan, Russia
E-mail: shabalin@ksaba.ru
