# Boundary value problems associated with generalized *Q*-holomorphic functions

- Sezayi Hızlıyel
^{1}Email author

**2013**:33

**DOI: **10.1186/1687-2770-2013-33

© Hızlıyel; licensee Springer. 2013

**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

*w*is an $m\times 1$ vector and

*q*is an $m\times m$ 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

and the solutions of such equations were called generalized (or pseudo) hyperanalytic functions. Work in this direction appears in [3–5]. 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].

*Q*, which means

*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

where the unknown $w(z)=\{{w}_{ij}(z)\}$ is an $m\times s$ complex matrix, $Q(z)=\{{q}_{ij}(z)\}$ is a self-commuting complex matrix with dimension $m\times m$ and ${q}_{k,k-1}\ne 0$ for $k=2,\dots ,m$. $A=\{{a}_{ij}(z)\}$ and $B=\{{b}_{ij}(z)\}$ 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

*G*, we consider the problem

*A*). Where the unknown $w(z)=\{{w}_{ij}(z)\}$ is an $m\times s$ complex matrix-valued function, $Q=\{{q}_{ij}(z)\}$ is a Hölder-continuous function which is a self-commuting matrix with $m\times m$ and ${q}_{k,k-1}\ne 0$ for $k=2,\dots ,m$. $A=\{{a}_{ij}(z)\}$ and $B=\{{b}_{ij}(z)\}$ are commuting with

*Q*, which is

It is assumed, moreover, that *Q* is commuting with $\overline{Q}$ and $\lambda (z)\in {C}^{1}(\mathrm{\Gamma})$ is commuting with *Q*, where $\mathrm{\Gamma}=\partial G$, $\lambda \overline{\lambda}=I$. In respect of the data of problem (*A*), we also assume that *A*, *B* and $F\in {L}^{p,2}(\mathbb{C})$ and $\gamma \in {C}_{\alpha}(\mathrm{\Gamma})$. If $F\equiv 0$, $\gamma \equiv 0$, we have homogeneous problem (
).

*ϕ*is a generating solution for the generalized Beltrami system ([[11], p.109]), ${B}^{\ast}={\varphi}_{z}^{-1}\overline{{\varphi}_{z}}\overline{B}$, $\frac{d\varphi}{ds}:=\frac{\partial \varphi}{\partial z}\frac{dz}{ds}+\frac{\partial \varphi}{\partial \overline{z}}\frac{d\overline{z}}{ds}$ and

*ds*is the arc length differential. From the Green identity for

*Q*-holomorphic functions (see [[11], p.113]), we have

*Q*. For $L[w]=F$ and ${L}^{\mathrm{\prime}}[{w}^{\mathrm{\prime}}]=0$, this becomes

where *ϰ* is a real matrix commuting with *Q*.

*ϰ*as

*P*is a constant matrix defined by

*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

*k*and ${k}^{\mathrm{\prime}}$ are dimensions of null spaces of $\underset{\sim}{K}$ and ${\underset{\sim}{K}}^{\mathrm{\prime}}$ respectively. If $\{{\varkappa}_{1},\dots ,{\varkappa}_{k}\}$ 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 ${(\lambda \frac{d\varphi}{ds})}^{-1}\varkappa $ takes on the boundary values of a

*Q*-holomorphic function ${\psi}_{j}$ on each component of boundary contours ${\mathrm{\Gamma}}_{j}$ in ${C}^{1,\alpha}(\mathbb{C})$ which is, moreover,

*Q*-holomorphic in the domain ${G}_{j}$ bounded by the closed contour ${\mathrm{\Gamma}}_{j}$. Let $\{{\varkappa}_{1},\dots ,{\varkappa}_{{\ell}^{\mathrm{\prime}}}\}$ be solutions of equation (10) to which linearly independent solutions (see [15]) ${w}_{1}^{\mathrm{\prime}},\dots ,{w}_{{\ell}^{\mathrm{\prime}}}^{\mathrm{\prime}}$ of problem ( ) correspond, then the remaining solutions $\{{\varkappa}_{{\ell}^{\mathrm{\prime}}+1},\dots ,{\varkappa}_{k}\}$ satisfy the boundary condition of the form

*Q*-holomorphic functions outside of ${G}^{-}:=G+\mathrm{\Gamma}$ and $\mathrm{\Phi}(\mathrm{\infty})=0$. 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 ${\ell}_{\ast}$ be a number of linearly independent solutions of this problem. Obviously, ${\ell}^{\mathrm{\prime}}+{\ell}_{\ast}^{\mathrm{\prime}}=k$.

*A*), where we assume that $\varkappa =0$ in what follows. The solutions of this problem may be expressed in terms of the generalized Cauchy kernel as follows:

*μ*must satisfy the integral equation

*Q*-holomorphic outside $G+\mathrm{\Gamma}$ and $\mathrm{\Phi}(\mathrm{\infty})=0$. Denoting the numbers of linearly independent solutions of ( ) and ( ) by

*ℓ*and ${\ell}_{\ast}$ respectively, we have $k=\ell +{\ell}_{\ast}$. In order that (13) is solvable, it is necessary and sufficient that the nonhomogeneous data ${\gamma}_{0}$ satisfy the auxiliary conditions

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*, ${w}^{\mathrm{\prime}}$ *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*.

## Declarations

## Authors’ Affiliations

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## Copyright

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