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 ,

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 345. 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 89, 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 G0 and if Q(z) has no eigenvalues of magnitude 1 for each z in G0, then Hile called the system (1) the generalized Beltrami system and the solutions of such a system were called Q-holomorphic functions. Later in 1213, 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 ,

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).

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 .

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 qk,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 ().

We refer to the adjoint, homogeneous problem (A) as (); it is given by

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 ,

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 .

Since w′ satisfies the boundary condition

Re ( d ϕ d s λ w ′ ) = 0 ,

we have

w ′ = i λ − 1 ( d ϕ d s ) − 1 ϰ ,

where ϰ is a real matrix commuting with Q.

The solutions to problem () 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 ,

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 ,

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 (). 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 ℓ ′ } be solutions of equation (10) to which linearly independent solutions (see 15) w 1 ′ , … , w ℓ ′ ′ of problem () correspond, then the remaining solutions { ϰ ℓ ′ + 1 , … , ϰ k } satisfy the boundary condition of the form

ϰ ( z ) = i λ ( z ) d ϕ d s Φ − ( z ) on  Γ .

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

In a complex case, Vekua refers to problems of this type as being concomitant to () and denotes them by (). Let ℓ ∗ be a number of linearly independent solutions of this problem. Obviously, ℓ ′ + ℓ ∗ ′ = 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 ,

where

γ 0 ( ζ ) = γ ( ζ ) − Re [ λ ( ζ ) ¯ w 1 + ( ζ ) ] = − Re [ λ ( ζ ) ¯ w 1 − ( ζ ) ] .

Problem () concomitant to problem () 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 () and () by ℓ and ℓ∗ respectively, we have k = ℓ + ℓ ∗ . 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 ) ,

where ϰ j are solutions to integral equation (10). These solutions may be broken up into two groups { ϰ 1 , … , ϰ ℓ ′ } and { ϰ ℓ ′ + 1 , … , ϰ k } such that ϰ j = i λ ( z ) d ϕ d s w j ′ ( z ) for j = 1 , … , ℓ ′ and ϰ j = i λ ( z ) d ϕ d s Φ j for j = ℓ ′ + 1 , … , k , where z ∈ Γ . Here w j ′ and Φ j are solutions of problems () and () 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,…,ℓ′, whereas for j = ℓ ′ + 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 ().

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 () has no solution.

The author declares that they have no competing interests.