Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 3, June 2023
Page(s) 185 - 191
DOI https://doi.org/10.1051/wujns/2023283185
Published online 13 July 2023

© Wuhan University 2023

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

Boundary value problems for (λ, k) bi-analytic functions are discussed in Refs.[1-6]. In Ref.[4], the author used the Cauchy-type integral of (λ, 1) bi-analytic functions to study a class of inverse Riemann problems of (λ, 1) bi-analytic functions. In this article, we study the general case by using the theory of boundary value problems for analytic functions.

Let Γ be a simple and closed smooth contour, oriented counter-clockwisely. D+(D-) denote respectively the interior (exterior) region bounded by Γ. And let O=(0, 0) in D+. Our inverse boundary value problem, simply called I-BVP, is to find functions ω(t), ϑ(t) and constants λ+, λ-, satisfying the following boundary conditions

{ f ( z ) z ¯ = λ ± - 1 4 λ ± ϕ ± ( z ) + λ ± + 1 4 λ ± ϕ ± ( z ) ¯ ,          z D ± , ϕ + ( z ) z ¯ = θ ( z ) ,             z D + , ϕ - ( z ) z ¯ = ϖ ( z )                   z D - , f + ( t ) = a i ( t ) f - ( t ) + b i ( t ) ω ( t ) , t Γ , i = 1,2 , ϕ + ( t ) = β i ( t ) ϕ - ( t ) + γ i ( t ) ϑ ( t ) ,     t Γ , i = 1,2 , f - ( t i ) = c i ,               t i Γ , i = 1 , 2 , (1)

where θ(t), ϖ(t), ai(t), bi(t), βi(t), γi(t)(i=1, 2)H(Γ) (Hölder continuous). ci (i=1, 2) are known constants.

1 Inverse Riemann Boundary Value Problems

In this section, we will consider a class of inverse Riemann boundary value problems, simply called I-RBVP. It is to find ϕ(z) and ϑ(t) meeting the following conditions

{ ϕ + ( z ) z ¯ = θ ( z ) , z D + ϕ - ( z ) z ¯ = ϖ ( z ) , z D - ϕ + ( t ) = β 1 ( t ) ϕ - ( t ) + γ 1 ( t ) ϑ ( t ) ,      t Γ ϕ + ( t ) = β 2 ( t ) ϕ - ( t ) + γ 2 ( t ) ϑ ( t ) ,      t Γ (2)

Suppose that γ1γ2 and β1γ2β2γ1, the solvable conditions and the representation of the solutions are obtained.

Assume that ΛHν(Γ)(0<ν1). Let the Cauchy integral operators

( S Γ [ Λ ] ) ( z ) = { 1 2 π i Γ Λ ( τ ) τ - z   d τ , z Γ 1 π i Γ Λ ( τ ) τ - t   d τ , z = t Γ (3)

and its projection operator

( S Γ ± [ Λ ] ) ( z ) = { 1 2 π i Γ Λ ( τ ) τ - z   d τ ,   z D ± 1 2 [ ( S Γ [ Λ ] ) ( t ) ± Λ ( t ) ] , z = t Γ (4)

Let

ϕ 1 ( z ) = { ϕ ( z ) - z ¯ θ ( z ) , z D + ϕ ( z ) - z ¯ ϖ ( z ) , z D - (5)

then we have

{ ϕ + ( t ) = ϕ 1 + ( t ) + t ¯ θ ( t )   ϕ - ( t ) = ϕ 1 - ( t ) + t ¯ ϖ ( t )   (6)

By (6) and (2), we obtain

{ ϕ 1 + ( t ) = G ( t ) ϕ 1 - ( t ) + t ¯ ( G ( t ) ϖ ( t ) - θ ( t ) ) ,     t Γ ϕ 1 ( z ) z ¯ = 0 ,    z Γ (7)

and

ϑ ( t ) = β 2 ( t ) - β 1 ( t ) γ 1 ( t ) - γ 2 ( t ) ( ϕ 1 - ( t ) + t ¯   ϖ ( t ) ) (8)

where

G ( t ) = β 1 ( t ) γ 2 ( t ) - β 2 ( t ) γ 1 ( t ) γ 2 ( t ) - γ 1 ( t ) , G ( t ) H ( Γ )   a n d   G ( t ) 0 (9)

Now we consider the Riemann boundary value problem, simply called RBVP. It is to find ϕ1(z) meeting the conditions given in (7). Denote κ=12π[argG(t)]Γ, then the index of RBVP (7) is κ. Suppose ϕ1()=0, thus from Refs.[7, 8], we have the following results.

Lemma 1   If κ0, then the solutions of RBVP (7) are

ϕ 1 ( z ) = X ( z ) ( S Γ [ t ¯ ( G ϖ - θ ) X + ] ) ( z ) + P κ - 1 ( z ) X ( z ) , z Γ (10)

where

X ( z ) = { X + ( z ) = e x p { ( S Γ [ l o g ( ( t - z 0 ) - κ G ) ] ) ( z ) } ,   z D + X - ( z ) = ( z - z 0 ) - κ e x p { ( S Γ [ l o g ( ( t - z 0 ) - κ G ) ] ) ( z ) } , z 0 D + ,   z D - (11)

X ( z ) is the canonical, Pκ-1(z) is arbitrary polynomial whose degree does not exceed κ-1.

If κ<0, then the RBVP (7) has a unique solution

ϕ 1 ( z ) = X ( z ) ( S Γ [ t ¯ ( G ϖ - θ ) X + ] ) ( z ) , z Γ (12)

if and only if

Γ t j t ¯ ( G ( t ) ϖ ( t ) - θ ( t ) ) X + ( t ) d t = 0 , j = 0 ,   1 ,     ,   - κ - 1 (13)

Hence, we obtain

Theorem 1   Under the requirement ϕ()=¯ ϖ(), when κ0, the solutions of I-RBVP (2) are

ϕ ( z ) = { X ( z ) ( S Γ [ t ¯ ( G ϖ - θ ) X + ] ) ( z ) + P κ - 1 ( z ) X ( z ) + z ¯ θ ( z ) , z D + X ( z ) ( S Γ [ t ¯ ( G ϖ - θ ) X + ] ) ( z ) + P κ - 1 ( z ) X ( z ) + z ¯ ϖ ( z ) , z D - (14)

where X(z) is given in (11), Pκ-1(z) is arbitrary polynomial whose degree does not exceed κ-1, and

ϑ ( t ) = β 2 ( t ) - β 1 ( t ) γ 1 ( t ) - γ 2 ( t ) { X - ( t ) ( S Γ - [ τ ¯ ( G ϖ - θ ) X + ] ) ( t ) + P κ - 1 ( t ) X - ( t ) + t ¯ ϖ ( t ) } (15)

When κ<0, the solutions of I-RBVP (2) are

ϕ ( z ) = { X ( z ) ( S Γ [ t ¯ ( G ϖ - θ ) X + ] ) ( z ) + z ¯ θ ( z ) , z D + X ( z ) ( S Γ [ t ¯ ( G ϖ - θ ) X + ] ) ( z ) + z ¯ ϖ ( z ) , z D - (16)

if and only if

Γ t j t ¯ ( G ( t ) ϖ ( t ) - θ ( t ) ) X + ( t ) d t = 0 , j = 0 ,   1 ,     ,   - κ - 1 (17)

and

ϑ ( t ) = β 2 ( t ) - β 1 ( t ) γ 1 ( t ) - γ 2 ( t ) { X - ( t ) ( S Γ - [ τ ¯ ( G ϖ - θ ) X + ] ) ( t ) + t ¯ ϖ ( t ) } (18)

2 Inverse Boundary Value Problems for (λ, 1) Bi-Analytic Functions

In this section, we will consider the I-BVP for (λ, 1) bi-analytic functions. It is to find ω(t) and constants λ± meeting the following conditions

{ f ( z ) z ¯ = λ ± - 1 4 λ ± ϕ ( z ) + λ ± + 1 4 λ ± ϕ ( z ) ¯ ,   z D ± f + ( t ) = a 1 ( t ) f - ( t ) + b 1 ( t ) ω ( t ) , t Γ f + ( t ) = a 2 ( t ) f - ( t ) + b 2 ( t ) ω ( t ) , t Γ f - ( t i ) = c i ,     t i Γ ,   i = 1 ,   2 (19)

where ai(t),bi(t)(i=1, 2)H(Γ), constants ci(i=1, 2) are known.

In the following discussion, we assume that b1(t)b2(t) and a1(t)b2(t)a2(t)b1(t). By (5), we get

ϕ ( z ) = { ϕ 1 ( z ) + z ¯ θ ( z ) , z D + ϕ 1 ( z ) + z ¯ ϖ ( z ) , z D - (20)

Substituting (20) into the first equation in (19) and letting

f 1 ( z ) = { f ( z ) - λ + - 1 8 λ + ( z ¯ ) 2 θ ( z ) - λ + + 1 4 λ + z   Θ ( z ) ¯ , z D + f ( z ) - λ - - 1 8 λ - ( z ¯ ) 2 ϖ ( z ) - λ - + 1 4 λ - z   Ω ( z ) ¯ , z D - (21)

where Θ'(z)=θ(z), Ω'(z)=ϖ(z), we obtain

f 1 ( z ) z ¯ = λ ± - 1 4 λ ± ϕ 1 ( z ) + λ ± + 1 4 λ ± ϕ 1 ( z ) ¯ ,   z D ± (22)

(22) and the second equation in (7) imply f1(z) is a (λ, 1) bi-analytic function with associate function ϕ1(z). By (22), we obtain

f 1 ( z ) = λ ± - 1 4 λ ± ϕ 1 ( z ) z ¯ + λ ± + 1 4 λ ± Φ 1 ( z ) ¯ + ψ ( z ) ,     z D ± (23)

where Φ1'(z)=ϕ1(z). Let zt from the inner domain D+ and the outer domain D-, respectively, one get

f 1 + ( t ) = f + ( t ) - λ + - 1 8 λ + ( t ¯ ) 2 θ ( t ) - λ + + 1 4 λ + t   Θ ( t ) ¯ = λ + - 1 4 λ + ϕ 1 + ( t ) t ¯ + λ + + 1 4 λ + Φ 1 + ( t ) ¯ + ψ + ( t ) ,    t Γ (24)

and

f 1 - ( t ) = f - ( t ) - λ - - 1 8 λ - ( t ¯ ) 2 ϖ ( t ) - λ - + 1 4 λ - t   Ω ( t ) ¯ = λ - - 1 4 λ - ϕ 1 - ( t ) t ¯ + λ - + 1 4 λ - Φ 1 - ( t ) ¯ + ψ - ( t ) ,    t Γ (25)

respectively.

By the second equation and the third equation in (19), we get

| 1 b 1 ( t ) 1 b 2 ( t ) | f + ( t ) = | a 1 ( t ) b 1 ( t ) a 2 ( t ) b 2 ( t ) | f - ( t ) , t Γ (26)

and

ω ( t ) = a 2 ( t ) - a 1 ( t ) b 1 ( t ) - b 2 ( t ) f - ( t ) (27)

Submitting f+(t) in (24) and f-(t) in (25) into (26), we obtain

ψ + ( t ) = | a 1 ( t ) b 1 ( t ) a 2 ( t ) b 2 ( t ) | | 1 b 1 ( t ) 1 b 2 ( t ) | ψ - ( t ) + Ξ ( t ) , t Γ (28)

where

Ξ ( t ) = | a 1 ( t ) b 1 ( t ) a 2 ( t ) b 2 ( t ) | | 1 b 1 ( t ) 1 b 2 ( t ) | { λ - - 1 4 λ - [ ϕ 1 - ( t ) t ¯ + 1 2 ( t ¯ ) 2 ϖ ( t ) ] + λ - + 1 4 λ - [ Φ 1 - ( t ) ¯ + t Ω ( t ) ¯ ] } - { λ + - 1 4 λ + [ ϕ 1 + ( t ) t ¯ + 1 2 ( t ¯ ) 2 θ ( t ) ] + λ + + 1 4 λ + [ Φ 1 + ( t ) ¯ + t Θ ( t ) ¯ ] } (29)

From (23) and (22), obviously,

ψ ( z ) z ¯ = 0 , z Γ (30)

Now we consider the following RBVP. It is to find ψ(z) meeting the conditions (28) and (30), namely,

{ ψ + ( t ) = G * ( t ) ψ - ( t ) + Ξ ( t ) , t Γ ψ ( z ) z ¯ = 0 , z Γ (31)

where

G * ( t ) = a 1 ( t ) b 2 ( t ) - a 2 ( t ) b 1 ( t ) b 2 ( t ) - b 1 ( t ) (32)

G * ( t ) H ( Γ ) and G*(t)0. κ*=12π[argG*(t)]Γ is the index of RBVP (31). If ψ() is to be finite, then we have the following results from Refs.[7, 8].

Lemma 2   1) If κ*+10, then the solutions of RBVP (31) are

ψ ( z ) = X * ( z ) ( S Γ [ Ξ X * + ] ) ( z ) + P κ * ( z ) X * ( z ) , z Γ (33)

where Pκ*(z) is arbitrary polynomial whose degree does not exceed κ*,

X * ( z ) = { X * + ( z ) = e x p { ( S Γ [ l o g ( ( t - z * ) - κ * G * ) ] ) ( z ) } ,   z D + X * - ( z ) = ( z - z * ) - κ * e x p { ( S Γ [ l o g ( ( t - z * ) - κ * G * ) ] ) ( z ) } , z * D + ,   z D - (34)

2) If κ*+1<0, then the RBVP (31) has a unique solution (33) (Pκ*0) if and only if

Γ t j Ξ ( t ) X * + ( t ) d t = 0 , j = 0 ,   1 ,     ,   - κ * - 2 (35)

3) ω(t) is written as following

ω ( t ) = a 2 ( t ) - a 1 ( t ) b 1 ( t ) - b 2 ( t ) { λ - - 1 4 λ - [ ϕ 1 - ( t ) t ¯ + 1 2 ( t ¯ ) 2 ϖ ( t ) ] + λ - + 1 4 λ - [ Φ 1 - ( t ) ¯ + t   Ω ( t ) ¯ ] + ψ - ( t ) } , t Γ (36)

(26) can be changed to

f - ( t ) = b 2 ( t ) - b 1 ( t ) a 1 ( t ) b 2 ( t ) - a 2 ( t ) b 1 ( t ) f + ( t ) (37)

Submitting f+(t) in (24) into (37), one get

f - ( t ) = b 2 ( t ) - b 1 ( t ) a 1 ( t ) b 2 ( t ) - a 2 ( t ) b 1 ( t ) { λ + - 1 4 λ + [ ϕ 1 + ( t ) t ¯ + 1 2 ( t ¯ ) 2 θ ( t ) ] + λ + + 1 4 λ + [ Φ 1 + ( t ) ¯ + t   Θ ( t ) ¯ ] + ψ + ( t ) } (38)

where ψ+(t) is from ψ(z) (let zt from the inner domain D+) given in Lemma 2, namely,

ψ + ( t ) = X * + ( t ) ( S Γ + [ Ξ X * + ] ) ( t ) + P κ * ( t ) X * + ( t ) (39)

so, we get

A ( t ) 1 λ + + B ( t ) 1 λ - = C ( t ) (40)

where

A ( t ) = b 2 ( t ) - b 1 ( t ) a 1 ( t ) b 2 ( t ) - a 2 ( t ) b 1 ( t ) { ϕ 1 + ( t ) t ¯ + 1 2 ( t ¯ ) 2 θ ( t ) - Φ 1 + ( t ) ¯ - t   Θ ( t ) ¯ }   + b 2 ( t ) - b 1 ( t ) a 1 ( t ) b 2 ( t ) - a 2 ( t ) b 1 ( t ) X * + ( t ) ( S Γ + [ - ϕ 1 + τ ¯ - 1 2 ( τ ¯ ) 2 θ + Φ 1 + ¯ + τ   Θ ¯ X * + ] ) ( t ) (41)

B ( t ) = X * + ( t ) ( S Γ + [ ϕ 1 - τ ¯ + 1 2 ( τ ¯ ) 2 ϖ - Φ 1 - ¯ - τ   Ω ¯ X * + ] ) ( t ) (42)

C ( t ) = b 2 ( t ) - b 1 ( t ) a 1 ( t ) b 2 ( t ) - a 2 ( t ) b 1 ( t ) { ϕ 1 + ( t ) t ¯ + 1 2 ( t ¯ ) 2 θ ( t ) + Φ 1 + ( t ) ¯ + t   Θ ( t ) ¯ + 4 P κ * ( t ) X * + ( t ) }

    + b 2 ( t ) - b 1 ( t ) a 1 ( t ) b 2 ( t ) - a 2 ( t ) b 1 ( t ) X * + ( t ) ( S Γ + [ - ϕ 1 + τ ¯ - 1 2 ( τ ¯ ) 2 θ - Φ 1 + ¯ - τ   Θ ¯ X * + ] ) ( t )

  + X * + ( t ) ( S Γ + [ ϕ 1 - τ ¯ + 1 2 ( τ ¯ ) 2 ϖ + Φ 1 - ¯ + τ   Ω ¯ X * + ] ) ( t ) - 4 f - ( t ) (43)

By the fourth equations in (19), we have the following system of equations

{ A ( t 1 ) 1 λ + + B ( t 1 ) 1 λ - = C ( t 1 ) A ( t 2 ) 1 λ + + B ( t 2 ) 1 λ - = C ( t 2 ) (44)

Furthermore, we obtain

{ λ + ( t ) = A ( t 1 ) B ( t 2 ) - B ( t 1 ) A ( t 2 ) C ( t 1 ) B ( t 2 ) - B ( t 1 ) C ( t 2 ) λ - ( t ) = A ( t 1 ) B ( t 2 ) - B ( t 1 ) A ( t 2 ) A ( t 1 ) C ( t 2 ) - C ( t 1 ) A ( t 2 ) (45)

Remark 1   Take =1 while κ*+10. Take =0 while κ*+1<0.

Summarizing the above discussion, we have the following result.

Theorem 2   For the I-BVP of (λ, 1) bi-analytic functions, the solutions are given in Theorem 1, (36) and (45), namely

ϑ ( t ) = β 2 ( t ) - β 1 ( t ) γ 1 ( t ) - γ 2 ( t ) { X + ( t ) ( S Γ + [ τ ¯   ( G ϖ - θ ) X + ] ) ( t ) + P κ - 1 ( t ) X + ( t ) + t ¯   ϖ ( t ) } ,    t Γ ,

(=1 while κ0, =0 while κ<0.)

ω ( t ) = a 2 ( t ) - a 1 ( t ) b 1 ( t ) - b 2 ( t ) { λ - - 1 4 λ - [ ϕ 1 - ( t ) t ¯ + 1 2 ( t ¯ ) 2 ϖ ( t ) ] + λ - + 1 4 λ - [ Φ 1 - ( t ) ¯ + t   Ω ( t ) ¯ ] + ψ - ( t ) } , t Γ ,

λ + ( t ) = A ( t 1 ) B ( t 2 ) - B ( t 1 ) A ( t 2 ) C ( t 1 ) B ( t 2 ) - B ( t 1 ) C ( t 2 )

λ - ( t ) = A ( t 1 ) B ( t 2 ) - B ( t 1 ) A ( t 2 ) A ( t 1 ) C ( t 2 ) - C ( t 1 ) A ( t 2 )

where A(t), B(t), C(t) are given in (41), (42), (43), respectively. ϕ1±(t) is from ϕ1(z) given in Lemma 1. ψ-(t) is from ψ(z) given in Lemma 2. For ϕ1(z) and ψ(z), there exist four cases.

1) When κ0 and κ*+10, ϕ1(z) and ψ(z) are given in (10) and (33), respectively.

2) When κ0 and κ*+1<0, ϕ1(z) is given in (10) and ψ(z) is given in (33)(Pκ*0) if and only if the condition (35) is satisfied.

3) When κ<0 and κ*+10, ϕ1(z) is given in (12) if and only if the condition (13) is satisfied and ψ(z) is given in (33).

4) When κ<0 and κ*+1<0, ϕ1(z) and ψ(z) are given in (12) and (33)(Pκ*0) if and only if the conditions (13) and (35) are satisfied, respectively.

Remark 2   The upper conclusions may be applied to interface problems of the elastic system in plane, for instance, the welding problems and the quasi-static system of thermoelasticity, etc.Refs.[9-14].

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