1
Professor Giorgi Mandjavidze
(6.V.1924 - 13.IV.1999)
Boundary Value Problems with Shift for Generalized Analytic Vectors
( In honor of Professor Giorgi Manjavidze)
G. Akhalaia, N.Manjavidze
The present work is devoted to boundary value problems of linear conjugation
with shift for generalized analytic vectors.
A great place in the works of Giorgi Manjavidze takes the investigation of boundary value problems of the theory of functions with shifts. In such problems the
boundary values of the desired functions are conjugating in the points which are
displaced to each other. The model problem is to find a function Φ(𝑧) holomorphic
on the complex plane 𝑧, cut along some simple closed curves, the boundary values
of which Φ+ (𝑡) and Φ− (𝑡) are satisfying the condition
Φ+ [𝛼(𝑡)] = 𝐺(𝑡)Φ− (𝑡) + 𝑔(𝑡), 𝑡 ∈ Γ
(*)
from both sides of Γ, where 𝐺(𝑡), 𝑔(𝑡) are given continuous functions on Γ,
𝛼(𝑡) is continuous function mapping Γ onto 𝛾 in one-to-one manner.
Later on result obtained by him in this direction became known as the theory of
conformal sewing. He also studied the problem with shift depended on parameter
2
and proved the invariance of partial indices for conformal mappings. A special
cycle of G. Manjavidze’s works is devoted to the investigation of boundary value
problems of the function theory by method of successive approximations by
means of which solution of the problems with shifts in case of several unknown
functions was considerably simplified.
G.Manjavidze obtained marked results for discontinuous boundary value problems
with shift for generalized analytic vectors. He studied the Riemann-Hilbert problem in domains with non-smooth boundaries, the Riemann-Hilbert-Poincare problem for generalized analytic functions, and the differential boundary value problem
of linear conjugation. In particular, he established both the solvability conditions
of these problems and the index formulas and discovered the connection between
the problems with shift for analytic and generalized analytic functions.
1. Generalized analytic vectors ( Definitions and Notations).
A vector 𝑤(𝑧) = (𝑤1 , ⋯ , 𝑤𝑛 ) is called generalized analytic in domain D if it is a
solution of the elliptic system
𝜕𝑧̅ 𝑤 − 𝑄𝜕𝑧 𝑤 + 𝐴𝑤 + 𝐵𝑤
̅ = 0,
(1.1)
where 𝐴(𝑧), 𝐵(𝑧) are given square matrices of order n of the class 𝐿𝑝0 (𝐷), 𝑝0 > 2,
and where 𝑄(𝑧) is a matrix of the special form: it is quasi-diagonal and every
𝑟
block 𝑄𝑟 = (𝑞𝑖𝑘
) is lower (upper) triangular matrix satisfying the conditions:
𝑟
𝑟
𝑞11
= ⋯ = 𝑞𝑚
= 𝑞 𝑟 , |𝑞 𝑟 | ≤ 𝑞0 < 1,
𝑟 ,𝑚𝑟
𝑟
𝑟
(𝑖 + 𝑠 ≤ 𝑛, 𝑘 + 𝑠 ≤ 𝑛).
𝑞𝑖𝑘
= 𝑞𝑖+𝑠,𝑘+𝑠
Moreover, 𝑄(𝑧) ∈ 𝑊𝑝1 (ℂ), 𝑝 > 2 and 𝑄(𝑧) = 0 outside of some circle. Under the
̅ ), the genesolution of (1.1) we mean so-called regular solution , i.e. 𝑤(𝑧) ∈ 𝐿2 (𝐷
ralized derivatives of which 𝑤𝑧̅, 𝑤𝑧 ∈ 𝐿𝜔 (𝐷′ ), 𝜔 > 2, where 𝐷′ is arbitrary closed
subset of 𝐷.
If 𝐴(𝑧) ≡ 𝐵(𝑧) ≡ 0 then
𝜕𝑧̅ 𝑤 − 𝑄𝜕𝑧 𝑤 = 0,
(1.2)
and the solutions of (1.2) are called 𝑄− holomorphic vectors.
The equation (1.2) has a solution of the form
𝜁(𝑧) = 𝑧𝐼 + 𝑇𝜔,
where 𝐼 is unit matrix and 𝜔(𝑧) is a solution of the equation
(1.3)
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𝜔(𝑧) − 𝑄(𝑧)Π𝜔 = 𝑄(𝑧)
belonging to 𝐿𝑝 (ℂ), 𝑝 > 2. 𝑇, Π are well-known integral operators. The solution
(1.3) is analogous of the fundamental homeomorphism of Beltrami equation. The
matrix
𝑉(𝑡, 𝑧) = 𝜕𝑡 𝜁(𝑡)[𝜁(𝑡) − 𝜁(𝑧)]−1
(1.4)
is called generalized Cauchy kernel for the equation (1.2), and consider Cauchy
type generalized integral
Φ(𝑧) =
1
∫ 𝑉(𝑡, 𝑧)𝑑𝑄 𝑡𝜇(𝑡),
2𝜋𝑖 Γ
(1.5)
where Γ is closed simple smooth curve, 𝜇(𝑡) ∈ 𝐿1 (Γ) and
𝑑𝑄 𝑡 = 𝐼𝑑𝑡 + 𝑄(𝑡)𝑑𝑡̅.
If the density 𝜇(𝑡) in (1.5) is Holder-continuous on Γ then (1.5) is Holder-conti+ and 𝐷
− ; the boundary values of Φ on Γ are given by
̅̅̅̅
̅̅̅̅
nuous in 𝐷
1
Φ± (𝑡) = ± 𝜇(𝑡) +
2
1
∫ 𝑉(𝜏, 𝑧)𝑑𝑄 𝜏𝜇(𝜏).
2𝜋𝑖 Γ
(1.6)
If 𝜇(𝑡) ∈ 𝐿𝑝 (Γ), p > 1, then (1.6) are fulfilled almost everywhere on Γ, provided
that Φ± (𝑡) are angular boundary values of the vector Φ(𝑧). Here very important
role play the analogous integral operators
̃ )[𝑧] = − 1 ∫ 𝑉(𝑡, 𝑧)𝑓(𝑡)𝑑𝜎𝑡,
(𝑇𝑓
𝐷
𝜋
(1.7)
1
̃ )[𝑧] = − ∫ 𝜕𝑧 𝑉(𝑡, 𝑧)𝑓(𝑡)𝑑𝜎𝑡.
(Π𝑓
𝜋 𝐷
̃ ) is completely continuous operator from 𝐿𝑝 (D
̅ ), p > 2,
If 𝑄 ∈ 𝐻𝛼0 (ℂ) then (𝑇𝑓
̃ is linear bounded
onto 𝐻𝛼 (𝐷), 𝛼 = 𝑚𝑖𝑛{𝛼0 , (𝑝 − 2)/𝑝} , moreover the operator Π
̅ ) to 𝐿𝑝 (D
̅ ), and
operator from 𝐿𝑝 (D
̃ =Π
̃ 𝑓.
(𝜕𝑧̅ − 𝑄𝜕𝑧 ) ̃
𝑇𝑓 = 𝑓, 𝜕𝑧 𝑇𝑓
(1.8)
Using 𝑄 − holomorphic vectors the generalized analytic vectors can be represented
as follows (Bojarski B. Theory of Generalized Analytic Vector.)
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𝑤(𝑧) = Φ(𝑧) + ∬ Γ1 (𝑧, 𝑡) Φ(𝑡) 𝑑𝜎𝑡, +
𝐷
(1.9)
𝑁
̅̅̅̅̅̅ 𝑑𝜎𝑡, + ∑ 𝑐𝑘 𝑤𝑘 (𝑧),
+ ∬ Γ2 (𝑧, 𝑡) Φ(𝑡)
𝐷
𝑘=1
where Φ(𝑧) is 𝑄 − holomorphic vector, 𝑤𝑘 (𝑧), (𝑘 = 1, ⋯ , 𝑁) is a complete system of linearly independent solutions of Fredholm equation
1
̅̅̅̅̅̅] 𝑑𝜎𝑡 = 0,
𝐾𝑤 ≡ 𝑤(𝑡) − ∬𝐷 𝑉(𝑡, 𝑧)[𝐴(𝑡)𝑤(𝑡) + 𝐵(𝑡)𝑤(𝑡)
𝜋
(1.10)
𝑤𝑘 (𝑧) turn to be continuous vectors in whole plane vanishing at infinity, 𝑐𝑘 are
arbitrary real constants, the kernels Γ1 (𝑧, 𝑡), Γ2 (𝑧, 𝑡) satisfy integral equations in
turn.
Finally the vector Φ(𝑧) has to satisfy the following conditions
𝑅𝑒 ∬𝐷 Φ(𝑧)𝑣𝑘 (𝑧) 𝑑𝜎𝑡 = 0, 𝑘 = 1, ⋯ , 𝑁,
(1.11)
̅ )(k = 1, ⋯ , N)form system of linearly independent solutions
where 𝑣𝑘 (𝑧) ∈ 𝐿𝑝 (D
of Fredholm integral equation
′ (𝑧)
̅̅̅̅̅̅̅
𝐴
𝐵′ (𝑧)
̅̅̅̅̅̅̅̅̅
̅̅̅̅̅̅𝑑𝜎𝑡 = 0.
𝑣(𝑧) +
∬ 𝑉(𝑧, 𝑡)𝑣(𝑡) 𝑑𝜎𝑡 +
∬ 𝑉 ′ (𝑧, 𝑡)𝑣(𝑡)
𝜋
𝜋
𝐷
𝐷
It should be mentioned that, generally speaking, the Liouville theorem is not true
for the solutions of (1.1). This explains the appearance of the constants 𝑐𝑘 in (1.9)
and that the condition (1.11) has to be satisfied.
2. Relation between BVP of Linear Conjugation and Generalized Analytic
Functions
Now we show the connection of linear conjugation problem with shift and the
theory of generalized analytic functions. This gives the possibility to consider the
problem of linear conjugation in somewhat different formulation.
Let Γ1 and Γ2 be Liapunov curves, 𝛼(𝑡) is function mapping Γ1 onto Γ2 in one-toone manner preserving orientation, 𝛼(𝑡(𝑠)) is absolutely continuous function, 𝑀 ≥
5
|𝛼 ′ (𝑡)| ≥ 𝑚 > 0 (𝑀, 𝑚 are constants), 𝑎(𝑡), 𝑏(𝑡) are given matrices of the class
1
𝐻𝜇 (Γ1 ) (𝜇 > ) , 𝑎(𝑡) is nonsingular square matrix of order 𝑛, 𝑏(𝑡) is (𝑛 ×
2
𝑙) −matrix.
Find piecewise-holomorphic matrix 𝜑(𝑧) having finite order at infinity,
𝜑 + (𝑡), 𝜑 − (𝑡) ∈ 𝐻(Γ) and satisfying the boundary condition
𝜑 + [𝛼(𝑡)] = 𝑎(𝑡)𝜑 − (𝑡) + 𝑏(𝑡), 𝑡 ∈ Γ1 .
(2.1)
We call the piecewise-holomorphic matrix 𝜒(𝑧) with finite order at infinity the
canonical matrix of the problem (2.1) if 𝑑𝑒𝑡 𝜒(𝑧) ≠ 0 everywhere except perhaps
at the point 𝑧 = ∞; 𝜒(𝑧) has a normal form at infinity with respect to columns and
𝜒 + [𝛼(𝑡)] = 𝑎(𝑡)𝜒 − (𝑡), 𝑡 ∈ Γ1 .
Mapping conformally 𝐷2+ and 𝐷1− into interior and exterior parts of the unit circle Γ
respectively we get the same problem (2.1), where 𝛼(𝑡) maps Γ onto Γ; the matrices 𝑎(𝑡), 𝑏(𝑡) have the same properties. Consider the problem in case Γ1 = Γ2 = Γ.
After proving some useful propositions we get the following
Theorem 1. All solutions of the problem (2.1) are given by the formulas
𝜑[𝛼(𝑧)] = 𝜒𝛼 [𝛼(𝑧)][𝑇𝑓 + ℎ(𝑧) + 𝑃(𝑤0 (𝑧))],
𝑧 ∈ 𝐷+ ,
(2.2)
𝜑(𝑧) = 𝜒𝛼 (𝑧)[𝑇𝑓 + ℎ(𝑧) + 𝑃(𝑤0 (𝑧))],
−
𝑧∈𝐷 ,
+ ), 𝑝 > 2, is a
̅̅̅̅
where 𝑃(𝑧) is arbitrary polynomial vector and the vector 𝑓 ∈ 𝐿𝑝 (𝐷
solution (unique) of the equation
𝐾𝑓 =: 𝑓(𝑧) − 𝑞(𝑧)Π𝑓 = 𝑔(𝑧);
−1
1
[𝜒𝛼+ (𝛼(𝑡))] 𝑏(𝑡)
ℎ(𝑧) =
∫
𝑑𝑡,
2𝜋𝑖 𝛾
𝑡−𝑧
+ ).
̅̅̅̅
𝑔(𝑧) = (𝑔1 , ⋯ , 𝑔𝑛 ) = 𝑞(𝑧)ℎ′ (𝑧) ∈ 𝐿𝑝 (𝐷
The solutions vanishing at infinity are given by the formulas (2.2) where
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𝑃(𝑧) = (𝑃𝜘1−1 , ⋯ , 𝑃𝜘𝑛 −1 ),
𝜘1 ≥ ⋯ ≥ 𝜘𝑛
are the partial indices of the problem (2.1), 𝑃𝑗 (𝑧) is arbitrary polynomial of order
𝑗(𝑃𝑗 (𝑧) = 0 if 𝑗 < 0); if 0 ≥ 𝜘𝑠+1 ≥ ⋯ ≥ 𝜘𝑛 then the vector 𝑏(𝑡) has to satisfy
the following conditions
−1
2𝑖 ∬ 𝑔𝑗 (𝜁)𝐿(𝜁 𝑘 )𝑑𝜁𝑑𝜂 = ∫ 𝑡 𝑘 {[𝜒𝛼+ (𝛼(𝑡))] 𝑏(𝑡)} 𝑑𝑡,
𝐷
𝛾
𝑗
𝑗 = 𝑠 + 1, ⋯ , 𝑛; 𝑘 = 0, ⋯ , |𝜘𝑗 | − 1,
where 𝐿𝑓 = 𝑓(𝑧) − Π(𝑞𝑓).
3. BVP of Linear Conjugation with Shift for Generalized Analytic Vectors.
Define the classes for 𝑄-holomorphic vectors. Let 𝐷+ , (𝐷− ) be finite (infinite)
domain which is bounded by a simple closed Liapunov smooth curve Γ.
Denote by 𝐸𝑠,𝑝 (𝐷, 𝑄), 𝑠 ≥ 0, 𝑝 ≥ 1, 𝑄(𝑧) = (𝑞𝑖𝑘 ) ∈ 𝑊𝑝𝑠0 (ℂ), 𝑝 > 2 (D is one of
the domains 𝐷+ , 𝐷− , 𝑊𝑝𝑠0 (ℂ) is a Sobolev space) the class of 𝑄− holomorphic
vectors Φ(𝑧) = ( Φ1 , ⋯ , Φn ) in the domain 𝐷 satisfying the following conditions
∫𝛿
𝑘𝑟
|
𝜕 𝑠 Φ𝑘 𝑝
𝜕𝑧 𝑠
| |𝑑𝑧| ≤ 𝐶, 𝑘 = 1, ⋯ , 𝑛,
(3.1)
where 𝐶 is a constant, 𝛿𝑘𝑟 is an image of the circle |𝜉| = 𝑟, 𝑟 < 1, while quasiconformal mapping 𝜉 = 𝜔𝑘 (𝑆𝑘 (𝑧)) of the unit circle |𝜉| < 1 onto 𝐷, 𝜔𝑘 is an analytic
function in the domain 𝑆𝑘 (𝐷), 𝑆𝑘 is a fundamental homeomorphism of the Beltrami equation
𝜕𝑧̅ 𝑆 − 𝑞𝑘𝑘 (𝑧)𝜕𝑧 𝑆 = 0.
If 𝐷 is infinite domain , then for the simplicity of notation we suppose that
𝑊(∞) = 0 (remind that 𝑄 − holomorphic vectors are the analytic functions in vicinity of the point z = ∞, because 𝑄 = 0 at infinity). By 𝐸𝑠,𝑝 (𝐷, 𝑄, 𝑆) denote the
class of the vectors Φ , belonging to the class 𝐸𝑠,𝜆 (𝐷, 𝑄) for some 𝜆 > 1, for
which the angular boundary values are belonging to 𝐿𝑝 (Γ, 𝜌),
𝜌(𝑡) = ∏𝑟𝑘=1|𝑡 − 𝑡𝑘 |𝜈𝑘 , 𝑡𝑘 ∈ Γ, −1 < 𝜈𝑘 < 𝑝 − 1, 𝑝 > 1.
(3.2)
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Le𝑡 𝑄+ (𝑧) and 𝑄 − (𝑧) are two given matrices, satisfying the conditions 𝑄+ ∈
(ℂ), 𝑙, 𝑚 ≥ 0, 𝑝0 > 2. ( 𝑄+ , 𝑄− ∈ 𝑊𝑝10 (ℂ) when 𝑙 = 𝑚 = 0).
𝑊𝑝𝑙0 (ℂ), 𝑄 − ∈ 𝑊𝑝𝑚
0
±
By 𝐸𝑙,𝑚,𝑝
(Γ, 𝑄± , 𝜌± ) we denote the class of the vectors defined on cut along Γ
plane, belonging to the class 𝐸𝑙,𝑝 (𝐷+ , 𝑄+ , 𝜌+ ) (𝐸𝑚,𝑝 (𝐷− , 𝑄− , 𝜌− )), in the domain
𝐷+ (𝐷− ), 𝐸0,𝑝 (𝐷, 𝑄) ≡ 𝐸𝑝 (𝐷, 𝑄). Introduce the classes of the generalized analytic
vectors, satisfying the equation of the form
𝑀𝑤 ≡ 𝜕𝑧̅ 𝑤 − 𝑄𝜕𝑧 𝑤 + 𝐴𝑤 + 𝐵𝑤
̅ = 0;
(3.3)
in case of infinite domain we suppose, that 𝑄, 𝐴, 𝐵 are equal to zero at infinity.
By 𝐸𝑙,𝑝 (𝐷, 𝑀), 𝑙 ≥ 0, 𝑝 ≥ 1, denote the class of the solutions of the equation
(3.3) satisfying the conditions
∫𝛿
𝑘𝑟
|
𝜕 𝑙 𝑤𝑘
𝜕𝑧 𝑙
𝑝
𝜕𝑠 𝑤
𝑝
| |𝑑𝑧| ≤ 𝐶, ∫𝛿 | 𝑠𝑘| |𝑑𝑧| ≤ 𝐶, 𝑘 = 1, ⋯ , 𝑛, 𝑠 = 1, ⋯ , 𝑙 − 1,
𝜕𝑧
𝑘𝑟
the curve 𝛿𝑘𝑟 is defined above, 𝐶 is a constant; if 𝐷 is infinite domain, then
𝑤(∞) = 0. Here we consider, that Γ ∈ 𝐻𝛼𝑚 , 𝑄(𝛼) ∈ 𝑊𝑝𝑙0 (ℂ), 𝑝0 > 2 (when 𝑙 = 0,
̅ )) , 𝑄 ∈ 𝑊𝑝1 (ℂ), 𝐴, 𝐵 ∈ 𝐿∞ (𝐷
̅ )), 𝐴(𝑧), 𝐵(𝑧) ∈
𝑄 ∈ 𝑊𝑝10 (ℂ), 𝐴, 𝐵 ∈ 𝐿∞ (𝐷
0
̅ ).
𝐻𝛼𝑙−1 (𝐷
By 𝐸𝑙,𝑝 (𝐷, 𝑀, 𝜌), 𝑙 ≥ 0, 𝑝 > 1, 𝜌(t) is a function of the form (3.2), denote the
class of the vectors 𝑤(𝑧), belonging to the class 𝐸𝑙,𝜆 (𝐷, 𝑀), for some λ > 1, for
which the angular boundary values of the vector
𝜕𝑙 𝑤
𝜕𝑧 𝑙
∈ 𝐿𝑝 (Γ, 𝜌).
±
𝐸𝑙,𝑚,𝑝
(Γ, 𝑀± , 𝜌± ) denotes the class of the vectors defined on the plane cut along Γ
and belonging to the class 𝐸𝑙,𝑝 (𝐷+ , 𝑀+ , 𝜌+ )[ (𝐸𝑚,𝑝 (𝐷− , 𝑀− , 𝜌− ) ] in 𝐷+ (𝐷− ).
Consider the following boundary value problem:
±
Find a vector Φ(𝑧) = ( Φ1 , ⋯ , Φn ) of the class 𝐸𝑙,𝑚,𝑝
(Γ, 𝑄± , 𝜌± ) satisfying boundary condition
− (𝑡) + 𝑓(𝑡), 𝑡 ∈ Γ.
̅̅̅̅̅̅̅̅
Φ+ [𝛼(𝑡)] = 𝑎(𝑡)Φ− (𝑡) + 𝑏(𝑡)Φ
(3.4)
Γ is simple closed Liapunov curve, 𝛼(𝑡) is function mapping Γ onto Γ in one-toone manner, preserving the orientation , 𝑎(𝑡), 𝑏(𝑡) are given piecewise-continuous
8
(𝑛 × 𝑛) −matrices on Γ, 𝑖𝑛𝑓|det 𝑎(𝑡)| > 0, 𝑓(𝑡) is given vector of the class
𝐿𝑝 (Γ, ρ), ρ = ρ− (t) is a function (3.2), ρ+ (t) = ∏rk=1|t − α(t k )|νk .
The following proposition holds.
Lemma. If the vector Φ(𝑧) ∈ 𝐸𝑝± (Γ, 𝑄± , 𝜌± ) then it is uniquely representable in
the form
1
Φ(𝑧) =
∫ 𝑉+ (𝜏, 𝑧)𝑑𝑄+ 𝜏𝜇[𝛽(𝜏)]′ ,
Γ
{2𝜋𝑖 1
∫ 𝑉− (𝜏, 𝑧)𝑑𝑄− 𝜏𝜇(𝜏),
2𝜋𝑖 Γ
𝑧 ∈ 𝐷+
𝑧∈𝐷
(3.5)
−
where 𝜇(𝑡) ∈ 𝐿𝑝 (Γ, ρ) is a solution of Fredholm integral equation
𝑁𝜇 ≡ 𝜇(𝑡) +
1
∫ [𝑉 (𝛼(𝜏), 𝛼(𝑡))𝑑𝑄+ 𝛼(𝜏) − 𝑉− (𝜏, 𝑡)𝑑𝑄− 𝜏𝜇(𝜏)] =
2𝜋𝑖 Γ +
(3.6)
Φ
+ [𝛼(𝑡)]
−Φ
− (𝑡),
𝛽(𝑡) is inverse of 𝛼(𝑡).
Substituting the representation (3.5) in the boundary condition (3.4) for the desired
vector 𝜇(𝑡) we obtain the singular integral equation
̅̅̅̅̅̅ + 1 ∫ 𝑀1 (𝜏, 𝑡) 𝜇(𝜏)𝑑𝜏
ℒ𝜇 ≡ [𝐼 + 𝑎(𝑡1 )]𝜇(𝑡) + 𝑏(𝑡)𝜇(𝑡)
𝜋𝑖 Γ
1
̅̅̅̅̅̅𝑑𝜏 = 2𝑓(𝑡),
+ ∫ 𝑀2 (𝜏, 𝑡)𝜇(𝜏)
𝜋𝑖 Γ
(3.7)
𝑀1 (𝜏, 𝑡)𝜇(𝜏)𝑑𝜏 = [𝑉+ (𝛼(𝜏), 𝛼(𝑡))𝑑𝑄+ 𝛼(𝜏) − 𝑎(𝑡)𝑉− (𝜏, 𝑡)𝑑𝑄− 𝜏]𝜇(𝜏),
̅̅̅̅̅̅𝑑𝜏 = 𝑏(𝑡)𝑉
̅̅̅̅̅̅̅̅̅
̅̅̅̅̅̅̅
𝑀2 (𝜏, 𝑡)𝜇(𝜏)
− (𝜏, 𝑡)𝑑𝑄− 𝜏𝜇(𝜏).
The following result holds.
Theorem 2. If the equation (3.7) is Noetherian in the space 𝐿𝑝 (Γ, ρ) then the
boundary problem (3.4) is Noetherian in
𝐸𝑝± (Γ, 𝑄± , 𝜌± ); the necessary and
sufficient solvability conditions have the form
𝑅𝑒 ∫Γ 𝑓(𝑡)𝜓𝑘 (𝑡)𝑑𝑡 = 0,
𝑘 = 1, ⋯ , 𝑙 ′ ,
(3.8)
9
where 𝜓𝑘 (𝑡) is a complete system of linearly independent solutions of conjugate
p
homogeneous equation ℒ ′ 𝜓 = 0 of the class 𝐿𝑞 (Γ, ρ1−q ), q =
; the index of the
1+p
problem (3.4) of the class
of the class 𝐿𝑝 (Γ, ρ).
𝐸𝑝± (Γ, 𝑄± , 𝜌± )
is equal to the index of the equation (3.7)
Consider now the problem (3.4) for generalized analytic vectors satisfying the
equations of the form (3.3):
Find the vector 𝑤(𝑧) ∈ 𝐸𝑝± (Γ, 𝑀± , 𝜌± ) satisfying the boundary condition
− (𝑡) + 𝑓(𝑡), 𝑡 ∈ Γ.
̅̅̅̅̅̅̅̅
w + [𝛼(𝑡)] = 𝑎(𝑡)w − (𝑡) + 𝑏(𝑡)𝑤
(3.9)
We seek the solution of (3.9) by the formula (1.9) in the following form
± (𝑡) 𝑑𝜎 +
̅̅̅̅̅̅̅̅
𝑤 ± (𝑧) = Φ± (𝑧) + ∬𝐷 Γ1± (𝑧, 𝑡)Φ± (𝑡) 𝑑𝜎𝑡, + ∬𝐷 Γ2± (𝑧, 𝑡)Φ
𝑡,
(3.10)
±
± ±
+ ∑𝑁
𝑘=1 𝑐𝑘 𝑤𝑘 (𝑧),
𝐸𝑝± (Γ, 𝑄 ± , 𝜌± ), 𝑐𝑘± (𝑘 = 1, ⋯ , 𝑁 ± )
where Φ± (𝑧) ∈
are desired real numbers,
𝑤𝑘± (𝑧) - solutions of the corresponding integral equations. The vectors Φ± (𝑡) have
to satisfy the conditions
𝐼𝑚 ∫ Φ± (𝑡)𝑑𝑄± 𝑡𝜓𝑘± (𝑡) = 0, 𝑘 = 1, ⋯ , 𝑁 ± ,
𝜓𝑘±
Γ
where
are the complete systems of the homogeneous conjugate equations. With
respect to the vector Φ± (𝑧) we obtain the following boundary problem
− (𝑡 ) + ℒ Φ+ + ℒ Φ− + 𝑓 (𝑡 ),
̅̅̅̅̅̅̅̅̅
Φ+ [𝛼(𝑡0 )] = 𝑎(𝑡0 )Φ− (𝑡0 ) + 𝑏(𝑡0 )Φ
0
+
−
0 0
(3.12)
𝑁−
𝑁+
+
−
̅̅̅̅̅̅̅̅
𝑓0 (𝑡) = 𝑓(𝑡) + ∑ 𝑐𝑘− [𝑎𝑘 (𝑡)𝑤𝑘− (𝑡) + 𝑏𝑘 (𝑡)𝑤
𝑘 (𝑡)] − ∑ 𝑐𝑘 𝑤𝑘 [𝛼(𝑡)];
𝑘=1
𝑘=1
the operators ℒ+ and ℒ− are defined as follows
+ (𝑡)]𝑑𝜎 ,
̅̅̅̅̅̅̅̅
ℒ+ Φ+ = − ∬𝐷+[ Γ1+ (𝛼(𝑡0 , 𝑡)Φ+ (𝑡) + Γ2+ (𝛼(𝑡0, 𝑡)Φ
𝑡
−
̅̅̅̅̅̅̅
ℒ− Φ = 𝛼(𝑡0 )𝐹(𝑡0 ) + 𝑏(𝑡0 )𝐹(𝑡
0 ),
− (𝑡)]𝑑𝜎 .
̅̅̅̅̅̅̅̅
𝐹(𝑡0 ) = ∬ −[ Γ1− (𝑡0 , 𝑡)Φ− (𝑡) + Γ2− (𝑡0, 𝑡)Φ
𝑡
𝐷
10
Substituting the representations (3.5) first in these formulas we obtain that the operators ℒ+ and ℒ− are completely continuous operators in 𝐿𝑝 (Γ, ρ+ ), 𝐿𝑝 (Γ, ρ− ) with
respect to the angular boundary values Φ+ (𝜏), Φ− (τ) and then in (3.12) we get
singular integral equation with respect to the vector 𝜇(𝑡)
̅̅̅̅̅̅ + ∑𝑁
Ω𝜇 ≡ (Ω0 + Ω1 )𝜇 + Ω2 𝜇̅ = 2𝑓(𝑡)
𝑘=1 𝑑𝑘 𝜂𝑘 (𝑡),
(3.13)
where Ω0 is completely continuous operator, Ω1 and Ω2 are singular integral operators
𝑏𝑘 (𝑡)
𝜇(𝜏)𝑑𝜏
Ω𝑘 𝜇 ≡ 𝑎𝑘 (𝑡)𝜇(𝑡) +
∫
,
𝜋𝑖 Γ 𝜏 − 𝑡
𝑎1 = 𝐼 + 𝑎, 𝑏1 = 𝐼 − 𝑎, 𝑎2 = 𝑏2 = 𝑏,
𝜂𝑘 (𝑡) are linearly independent continuous vectors, represented by 𝑤𝑘± (𝑡), 𝑑𝑘 (𝑘 =
1, ⋯ , 𝑁, 𝑁 ≤ 𝑁 + + 𝑁 − ) are desired real constants.
Besides (3.13) the vector 𝜇 has to satisfy the conditions
𝐼𝑚 ∫Γ 𝜇(𝑡)𝜔𝑘 (𝑡)𝑑𝑡 = 0, 𝑘 = 1, ⋯ , 𝑁,
(3.14)
where 𝜔𝑘 (𝑡), (𝑘 = 1, ⋯ , 𝑁) are linearly independent vectors, represented by
𝜓𝑘± (𝑡).
The necessary and sufficient solvability conditions of the problem (3.9) in the class
𝐸𝑝± (Γ, 𝑀± , 𝜌± ) have the form (Prossdorf S. Some Classes of Singular Equations.)
𝑅𝑒 ∫Γ 𝑓(𝑡)Υ𝑘 (𝑡)𝑑𝑡 = 0, 𝑘 = 1, ⋯ , 𝑅,
(3.15)
where the linearly independent vectors Υ𝑘 (𝑡) (𝑘 = 1, ⋯ , 𝑅) belonging to the class
𝐿𝑝 (Γ, ρ1−q ), are representable by 𝜓 + , 𝜓 − and by the vectors composing the basis
of subspace of the solutions of adjoint homogeneous equation Ω′ 𝑣 = 0. The index
of the problem (3.9) is equal to
𝜘 + 𝑁 − 𝑅,
(3.16)
where 𝜘 is the index of the operator Ω of the class 𝐿𝑝 (Γ, ρ). Actually 𝑁 = 𝑅 in the
formula (3.16).
Let 𝑋 ± be sets of the vectors 𝑤 ± (𝑧) defined in the domains 𝐷± representable in
the form
𝑤 ± (𝑧) = Φ± (𝑧) + ℎ± (𝑧),
(3.17)
11
Φ± (𝑧) ∈ 𝐸𝑝± (Γ, 𝑄 ± , 𝜌± ), ℎ± (𝑧) ∈ 𝐻𝛼 (𝐷± ).
This pair of sets coincides with the class 𝐸𝑝± (Γ, 𝑀± , 𝜌± ). Introduce the norms
|𝑤 ± |𝑋 ± = 𝑖𝑛𝑓 {|Φ± |𝐿𝑝(Γ,ρ±) , |ℎ± |𝐻 𝛼(𝐷±) },
(3.18)
𝑋 ± are Banach spaces. Let 𝑋 = (𝑋 + , 𝑋 − ) be new Banach space with the norm
|𝑤|𝑋 = 𝑚𝑎𝑥[|𝑤 + |𝑋 +, |𝑤 − |𝑋 − ]. It is evident, that this norm in 𝐸𝑝± (Γ, 𝑀± , 𝜌± ) is
independent of 𝐴± , 𝐵± . Consider the set of the operators
𝑀𝜆± 𝑤 ± = 𝜕𝑧̅ 𝑤 ± − 𝑄 ± 𝜕𝑧 𝑤 ± + 𝜆[𝐴± 𝑤 ± + 𝐵± ̅̅̅̅
𝑤 ± ],
where 𝜆 ∈ [0,1]. In order to calculate the index of the problem (3.9) we may take
the differential operators of the form 𝜕𝑧̅ 𝑤 ± − 𝑄 ± 𝜕𝑧 𝑤 ± and for such operators the
numbers 𝑁 + , 𝑁 − are equal to zero and hence 𝑁 = 𝑅 in (3.16).
Theorem 3.The necessary and sufficient solvability conditions of the problem
(3.9) in the class 𝐸𝑝± (Γ, 𝑀± , 𝜌± ) are the conditions (3.15); the index of this problem
is equal to the index 𝜘 of the operator Ω.
Note that if the matrices 𝑎(𝑡) and 𝑏(𝑡) are continuous then the index of any class is
given by the formula
1
𝜘 = [arg 𝑑𝑒𝑡𝑎(𝑡)]Γ .
𝜋
References
[1] G. Akhalaia, G. Manjavidze, Boundary value problems of the theory of generalized analytic vectors. Complex Methods for Partial Differential Equations, vol. 6,
Kluwer Academic Publishers, Dordecht/Boston/London, 1999, pp. 57-97.
[2] N. Manjavidze. Boundary value problems for analytic and generalized analytic
functions on a cut plane, Memoirs on Differential Equations and Mathematical
Physics, vol 33, Publishing House GCI, Tbilisi, 2004, pp.121-154.
[3] G. Manjavidze, Boundary value problems for analytic and generalized functions. Some Topics on Value Distribution and Differentiability in Complex and Padic Analysis, vol 11, Science Press, Beijing, 2008, pp.499-631.
[4]G. Manjavidze, N. Manjavidze, Boundary value problems for generalized analytic functions, Journal of Math. Sciences, vol.160, 6, Springer, 2009, pp. 745-821.