Open Access

Boundary value problems associated with generalized Q-holomorphic functions

Boundary Value Problems20132013:33

DOI: 10.1186/1687-2770-2013-33

Received: 9 November 2012

Accepted: 31 January 2013

Published: 18 February 2013

Abstract

In this work, we discuss Riemann-Hilbert and its adjoint homogeneous problem associated with generalized Q-holomorphic functions and investigate the solvability of the Riemann-Hilbert problem.

Keywords

generalized Beltrami systems Q-holomorphic functions Riemann-Hilbert problem

Introduction

Douglis [1] and Bojarskiĭ [2] developed an analog of analytic functions for elliptic systems in the plane of the form
w z ¯ q w z = 0 ,
(1)
where w is an m × 1 vector and q is an m × m quasi-diagonal matrix. Also, Bojarskiĭ assumed that all eigenvalues of q are less than 1. Such systems are natural ones to consider because they arise from the reduction of general elliptic systems in the plane to a standard canonical form. Subsequently Douglis and Bojarkii’s theory has been used to study elliptic systems in the form
w z ¯ q w z = a w + b w ¯ + F

and the solutions of such equations were called generalized (or pseudo) hyperanalytic functions. Work in this direction appears in [35]. These results extend the generalized (or ‘pseudo’) analytic function theory of Vekua [6] and Bers [7]. Also, classical boundary value problems for analytic functions were extended to generalized hyperanalytic functions. A good survey of the methods encountered in a hyperanalytic case may be found in [8, 9], also see [10].

In [11], Hile noticed that what appears to be the essential property of elliptic systems in the plane for which one can obtain a useful extension of analytic function theory is the self-commuting property of the variable matrix Q, which means
Q ( z 1 ) Q ( z 2 ) = Q ( z 2 ) Q ( z 1 )
for any two points z 1 , z 2 in the domain G 0 of Q. Further, such a Q matrix cannot be brought into a quasi-diagonal form of Bojarskiĭ by a similarity transformation. So, Hile [11] attempted to extend the results of Douglis and Bojarskiĭ to a wider class of systems in the same form with equation (1). If Q ( z ) is self-commuting in G 0 and if Q ( z ) has no eigenvalues of magnitude 1 for each z in G 0 , then Hile called the system (1) the generalized Beltrami system and the solutions of such a system were called Q-holomorphic functions. Later in [12, 13], using Vekua and Bers techniques, a function theory is given for the equation
D w + A w + B w ¯ = 0 , where  D : = z ¯ Q ( z ) z ,
(2)

where the unknown w ( z ) = { w i j ( z ) } is an m × s complex matrix, Q ( z ) = { q i j ( z ) } is a self-commuting complex matrix with dimension m × m and q k , k 1 0 for k = 2 , , m . A = { a i j ( z ) } and B = { b i j ( z ) } are commuting with Q. Solutions of such an equation were called generalized Q-holomorphic functions.

In this work, as in a complex case, following Vekua (see [[6], pp.228-236]), we investigate the necessary and sufficient condition of solvability of the Riemann-Hilbert problem for equation (2).

Solvability of Riemann-Hilbert problems

In a regular domain G, we consider the problem
( A ) : { L [ w ] : = D w + A w + B w ¯ = F in  G , Re ( λ ¯ w ) = γ on  G .
(3)
We refer to this problem as boundary value problem (A). Where the unknown w ( z ) = { w i j ( z ) } is an m × s complex matrix-valued function, Q = { q i j ( z ) } is a Hölder-continuous function which is a self-commuting matrix with m × m and q k , k 1 0 for k = 2 , , m . A = { a i j ( z ) } and B = { b i j ( z ) } are commuting with Q, which is
Q ( z 1 ) A ( z 2 ) = A ( z 1 ) Q ( z 1 ) , Q ( z 1 ) B ( z 2 ) = B ( z 1 ) Q ( z 1 ) .

It is assumed, moreover, that Q is commuting with Q ¯ and λ ( z ) C 1 ( Γ ) is commuting with Q, where Γ = G , λ λ ¯ = I . In respect of the data of problem (A), we also assume that A, B and F L p , 2 ( C ) and γ C α ( Γ ) . If F 0 , γ 0 , we have homogeneous problem ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq26_HTML.gif ).

We refer to the adjoint, homogeneous problem (A) as ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq27_HTML.gif ); it is given by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_Equ4_HTML.gif
(4)
where ϕ is a generating solution for the generalized Beltrami system ([[11], p.109]), B = ϕ z 1 ϕ z ¯ B ¯ , d ϕ d s : = ϕ z d z d s + ϕ z ¯ d z ¯ d s and ds is the arc length differential. From the Green identity for Q-holomorphic functions (see [[11], p.113]), we have
Re [ 1 2 i Γ d ϕ w w G ϕ z ( w L [ w ] L [ w ] w ) d x d y ] = 0 ,
(5)
where w is commuting with Q. For L [ w ] = F and L [ w ] = 0 , this becomes
1 2 i Γ d ϕ ( z ) w ( z ) λ ( z ) γ ( z ) Re ( G ϕ z ( z ) w ( z ) F ( z ) d x d y ) = 0 .
(6)
Since w satisfies the boundary condition
Re ( d ϕ d s λ w ) = 0 ,
(7)
we have
w = i λ 1 ( d ϕ d s ) 1 ϰ ,
(8)

where ϰ is a real matrix commuting with Q.

The solutions to problem ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq27_HTML.gif ) may be represented by means of fundamental kernels in terms of a real, matrix density ϰ as
w ( z ) = P 1 Γ ( d ϕ ( ζ ) Ω ( 1 ) ( z , ζ ) w ( ζ ) d ϕ ( ζ ) ¯ Ω ( 2 ) ( z , ζ ) w ( ζ ) ¯ ) = i P 1 Γ ( Ω ( 1 ) ( z , ζ ) λ 1 ( ζ ) + Ω ( 2 ) ( z , ζ ) λ 1 ( ζ ) ¯ ) ϰ ( ζ ) d s ,
(9)
see ([[14], p.543]). In (9), P is a constant matrix defined by
P ( z ) = | z | = 1 ( z I + z ¯ Q ) 1 ( I d z + Q d z ¯ )
called P-value for the generalized Beltrami system [11]. Since ϰ is a real matrix commuting with Q, inserting the expression (9) into the boundary condition (7), we have
Γ K 1 ( ζ , z ) ϰ ( ζ ) d s ζ = 0 , z , ζ = ζ ( s ) Γ ,
(10)
where
K 1 ( ζ , z ) = Re [ i P 1 λ ( z ) d ϕ ( z ) d s ( Ω ( 1 ) ( z , ζ ) λ 1 ( z ) + Ω ( 2 ) ( z , ζ ) λ 1 ( z ) ¯ ) ] .
The integral in (10) is to be taken in the Cauchy principal value sense. If we denote this equation in an operator form by K ϰ = 0 and its adjoint by K f = 0 , then it may be easily demonstrated that the index of (10) is κ = k k = 0 . Here k and k are dimensions of null spaces of K and K respectively. If { ϰ 1 , , ϰ k } is a complete system of solutions of (10), putting each of this into (9), we obtain the solutions of problem ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq27_HTML.gif ). However, it is possible that some of these solutions may turn out to be trivial solutions, which occurs when ( λ d ϕ d s ) 1 ϰ takes on the boundary values of a Q-holomorphic function ψ j on each component of boundary contours Γ j in C 1 , α ( C ) which is, moreover, Q-holomorphic in the domain G j bounded by the closed contour Γ j . Let { ϰ 1 , , ϰ } be solutions of equation (10) to which linearly independent solutions (see [15]) w 1 , , w of problem ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq27_HTML.gif ) correspond, then the remaining solutions { ϰ + 1 , , ϰ k } satisfy the boundary condition of the form
ϰ ( z ) = i λ ( z ) d ϕ d s Φ ( z ) on  Γ .
(11)
Here Φ are meant to be Q-holomorphic functions outside of G : = G + Γ and Φ ( ) = 0 . Hence the Q-holomorphic functions satisfy the homogeneous boundary conditions
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_Equ12_HTML.gif
(12)

In a complex case, Vekua refers to problems of this type as being concomitant to ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq27_HTML.gif ) and denotes them by ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq52_HTML.gif ). Let be a number of linearly independent solutions of this problem. Obviously, + = k .

Let us now return to the discussion of problem (A), where we assume that ϰ = 0 in what follows. The solutions of this problem may be expressed in terms of the generalized Cauchy kernel as follows:
w ( z ) = w 1 ( z ) + w 2 ( z ) = C [ λ γ ] ( z ) + C [ i λ μ ] ( z ) ,
where
C [ Φ ] = P 1 Γ d ϕ ( ζ ) Ω ( 1 ) ( z , ζ ) Φ ( ζ ) d ϕ ( ζ ) ¯ Ω ( 2 ) ( z , ζ ) Φ ( ζ ) ¯
(see [[14], p.543]). From the Plemelj formulas, it is seen that the density μ must satisfy the integral equation
γ 0 = Γ K 1 ( ζ , z ) μ ( z ) d s z ,
(13)
where
γ 0 ( ζ ) = γ ( ζ ) Re [ λ ( ζ ) ¯ w 1 + ( ζ ) ] = Re [ λ ( ζ ) ¯ w 1 ( ζ ) ] .
(14)
Problem ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq56_HTML.gif ) concomitant to problem ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq57_HTML.gif ) has the boundary condition Re [ λ 1 ( z ) Φ ( z ) ] = 0 on Γ, where Φ is Q-holomorphic outside G + Γ and Φ ( ) = 0 . Denoting the numbers of linearly independent solutions of ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq57_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq60_HTML.gif ) by and respectively, we have k = + . In order that (13) is solvable, it is necessary and sufficient that the nonhomogeneous data γ 0 satisfy the auxiliary conditions
Γ ϰ j ( ζ ) γ 0 ( ζ ) d s ζ = 0 ( j = 1 , , k ) ,
(15)
where ϰ j are solutions to integral equation (10). These solutions may be broken up into two groups { ϰ 1 , , ϰ } and { ϰ + 1 , , ϰ k } such that ϰ j = i λ ( z ) d ϕ d s w j ( z ) for j = 1 , , and ϰ j = i λ ( z ) d ϕ d s Φ j for j = + 1 , , k , where z Γ . Here w j and Φ j are solutions of problems ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq27_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq52_HTML.gif ) respectively. The condition (15) for γ 0 given by (14) becomes
Γ ϰ j ( ζ ) γ 0 ( ζ ) d s ζ = i Γ d ϕ ( ζ ) λ ( ζ ) w j ( ζ ) γ ( ζ ) + Re [ i Γ d ϕ ( ζ ) w j ( ζ ) w 1 + ( ζ ) ]
for j = 1 , , , whereas for j = + 1 , , k , we have
Γ ϰ j ( ζ ) γ 0 ( ζ ) d s = Re [ i Γ d ϕ ( ζ ) Φ j ( ζ ) w 1 ( ζ ) ] = 0 .

Consequently, the conditions (15) are seen to hold if (6) (with F = 0 ) holds. From the above discussion, one obtains a Fredholm-type theorem for problem (A).

Theorem 1 Non-homogeneous boundary problem (A) is solvable if and only if the condition (6) is satisfied, w being an arbitrary solution of adjoint homogeneous boundary problem ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq27_HTML.gif ).

This theorem immediately implies the following.

Theorem 2 Non-homogeneous boundary problem (A) is solvable for an arbitrary right-hand side if and only if adjoint homogeneous problem ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-33/MediaObjects/13661_2012_Article_284_IEq27_HTML.gif ) has no solution.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Art and Science, Uludağ University

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Copyright

© Hızlıyel; licensee Springer. 2013

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