## Boundary Value Problems

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# Boundary value problems associated with generalized Q-holomorphic functions

Boundary Value Problems20132013:33

DOI: 10.1186/1687-2770-2013-33

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
${w}_{\overline{z}}-q{w}_{z}=0,$
(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
${w}_{\overline{z}}-q{w}_{z}=aw+b\overline{w}+F$

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
$Q\left({z}_{1}\right)Q\left({z}_{2}\right)=Q\left({z}_{2}\right)Q\left({z}_{1}\right)$
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 $Q\left(z\right)$ is self-commuting in ${G}_{0}$ and if $Q\left(z\right)$ 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
(2)

where the unknown $w\left(z\right)=\left\{{w}_{ij}\left(z\right)\right\}$ is an $m×s$ complex matrix, $Q\left(z\right)=\left\{{q}_{ij}\left(z\right)\right\}$ is a self-commuting complex matrix with dimension $m×m$ and ${q}_{k,k-1}\ne 0$ for $k=2,\dots ,m$. $A=\left\{{a}_{ij}\left(z\right)\right\}$ and $B=\left\{{b}_{ij}\left(z\right)\right\}$ 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
(3)
We refer to this problem as boundary value problem (A). Where the unknown $w\left(z\right)=\left\{{w}_{ij}\left(z\right)\right\}$ is an $m×s$ complex matrix-valued function, $Q=\left\{{q}_{ij}\left(z\right)\right\}$ is a Hölder-continuous function which is a self-commuting matrix with $m×m$ and ${q}_{k,k-1}\ne 0$ for $k=2,\dots ,m$. $A=\left\{{a}_{ij}\left(z\right)\right\}$ and $B=\left\{{b}_{ij}\left(z\right)\right\}$ are commuting with Q, which is
$Q\left({z}_{1}\right)A\left({z}_{2}\right)=A\left({z}_{1}\right)Q\left({z}_{1}\right),\phantom{\rule{2em}{0ex}}Q\left({z}_{1}\right)B\left({z}_{2}\right)=B\left({z}_{1}\right)Q\left({z}_{1}\right).$

It is assumed, moreover, that Q is commuting with $\overline{Q}$ and $\lambda \left(z\right)\in {C}^{1}\left(\mathrm{\Gamma }\right)$ 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 $F\in {L}^{p,2}\left(\mathbb{C}\right)$ and $\gamma \in {C}_{\alpha }\left(\mathrm{\Gamma }\right)$. If $F\equiv 0$, $\gamma \equiv 0$, we have homogeneous problem ( ).

We refer to the adjoint, homogeneous problem (A) as ( ); it is given by
(4)
where 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
$Re\left[\frac{1}{2i}{\int }_{\mathrm{\Gamma }}\phantom{\rule{0.2em}{0ex}}d\varphi {w}^{\mathrm{\prime }}w-{\iint }_{G}{\varphi }_{z}\left({w}^{\mathrm{\prime }}L\left[w\right]-{L}^{\mathrm{\prime }}\left[{w}^{\mathrm{\prime }}\right]w\right)\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy\right]=0,$
(5)
where ${w}^{\mathrm{\prime }}$ is commuting with Q. For $L\left[w\right]=F$ and ${L}^{\mathrm{\prime }}\left[{w}^{\mathrm{\prime }}\right]=0$, this becomes
$\frac{1}{2i}{\int }_{\mathrm{\Gamma }}\phantom{\rule{0.2em}{0ex}}d\varphi \left(z\right){w}^{\mathrm{\prime }}\left(z\right)\lambda \left(z\right)\gamma \left(z\right)-Re\left({\iint }_{G}{\varphi }_{z}\left(z\right){w}^{\mathrm{\prime }}\left(z\right)F\left(z\right)\phantom{\rule{0.2em}{0ex}}dx\phantom{\rule{0.2em}{0ex}}dy\right)=0.$
(6)
Since ${w}^{\mathrm{\prime }}$ satisfies the boundary condition
$Re\left(\frac{d\varphi }{ds}\lambda {w}^{\mathrm{\prime }}\right)=0,$
(7)
we have
${w}^{\mathrm{\prime }}=i{\lambda }^{-1}{\left(\frac{d\varphi }{ds}\right)}^{-1}\varkappa ,$
(8)

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
$\begin{array}{rcl}{w}^{\mathrm{\prime }}\left(z\right)& =& {P}^{-1}{\int }_{\mathrm{\Gamma }}\left(d\varphi \left(\zeta \right){\mathrm{\Omega }}^{\left(1\right)}\left(z,\zeta \right){w}^{\mathrm{\prime }}\left(\zeta \right)-d\overline{\varphi \left(\zeta \right)}{\mathrm{\Omega }}^{\left(2\right)}\left(z,\zeta \right)\overline{{w}^{\mathrm{\prime }}\left(\zeta \right)}\right)\\ =& i{P}^{-1}{\int }_{\mathrm{\Gamma }}\left({\mathrm{\Omega }}^{\left(1\right)}\left(z,\zeta \right){\lambda }^{-1}\left(\zeta \right)+\overline{{\mathrm{\Omega }}^{\left(2\right)}\left(z,\zeta \right){\lambda }^{-1}\left(\zeta \right)}\right)\varkappa \left(\zeta \right)\phantom{\rule{0.2em}{0ex}}ds,\end{array}$
(9)
see ([[14], p.543]). In (9), P is a constant matrix defined by
$P\left(z\right)={\int }_{|z|=1}{\left(zI+\overline{z}Q\right)}^{-1}\left(I\phantom{\rule{0.2em}{0ex}}dz+Q\phantom{\rule{0.2em}{0ex}}d\overline{z}\right)$
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
${\int }_{\mathrm{\Gamma }}{K}_{1}\left(\zeta ,z\right)\varkappa \left(\zeta \right)\phantom{\rule{0.2em}{0ex}}d{s}_{\zeta }=0,\phantom{\rule{1em}{0ex}}z,\zeta =\zeta \left(s\right)\in \mathrm{\Gamma },$
(10)
where
${K}_{1}\left(\zeta ,z\right)=-Re\left[i{P}^{-1}\lambda \left(z\right)\frac{d\varphi \left(z\right)}{ds}\left({\mathrm{\Omega }}^{\left(1\right)}\left(z,\zeta \right){\lambda }^{-1}\left(z\right)+\overline{{\mathrm{\Omega }}^{\left(2\right)}\left(z,\zeta \right){\lambda }^{-1}\left(z\right)}\right)\right].$
The integral in (10) is to be taken in the Cauchy principal value sense. If we denote this equation in an operator form by $\underset{\sim }{K}\varkappa =0$ and its adjoint by ${\underset{\sim }{K}}^{\mathrm{\prime }}f=0$, then it may be easily demonstrated that the index of (10) is $\kappa =k-{k}^{\mathrm{\prime }}=0$. Here k and ${k}^{\mathrm{\prime }}$ are dimensions of null spaces of $\underset{\sim }{K}$ and ${\underset{\sim }{K}}^{\mathrm{\prime }}$ respectively. If $\left\{{\varkappa }_{1},\dots ,{\varkappa }_{k}\right\}$ 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 ${\left(\lambda \frac{d\varphi }{ds}\right)}^{-1}\varkappa$ takes on the boundary values of a Q-holomorphic function ${\psi }_{j}$ on each component of boundary contours ${\mathrm{\Gamma }}_{j}$ in ${C}^{1,\alpha }\left(\mathbb{C}\right)$ which is, moreover, Q-holomorphic in the domain ${G}_{j}$ bounded by the closed contour ${\mathrm{\Gamma }}_{j}$. Let $\left\{{\varkappa }_{1},\dots ,{\varkappa }_{{\ell }^{\mathrm{\prime }}}\right\}$ 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 $\left\{{\varkappa }_{{\ell }^{\mathrm{\prime }}+1},\dots ,{\varkappa }_{k}\right\}$ satisfy the boundary condition of the form
(11)
Here ${\mathrm{\Phi }}^{-}$ are meant to be Q-holomorphic functions outside of ${G}^{-}:=G+\mathrm{\Gamma }$ and $\mathrm{\Phi }\left(\mathrm{\infty }\right)=0$. Hence the Q-holomorphic functions satisfy the homogeneous boundary conditions
(12)

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, ${\ell }^{\mathrm{\prime }}+{\ell }_{\ast }^{\mathrm{\prime }}=k$.

Let us now return to the discussion of problem (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:
$w\left(z\right)={w}_{1}\left(z\right)+{w}_{2}\left(z\right)=\underset{\sim }{C}\left[\lambda \gamma \right]\left(z\right)+\underset{\sim }{C}\left[i\lambda \mu \right]\left(z\right),$
where
$\underset{\sim }{C}\left[\mathrm{\Phi }\right]={P}^{-1}{\int }_{\mathrm{\Gamma }}\phantom{\rule{0.2em}{0ex}}d\varphi \left(\zeta \right){\mathrm{\Omega }}^{\left(1\right)}\left(z,\zeta \right)\mathrm{\Phi }\left(\zeta \right)-d\overline{\varphi \left(\zeta \right)}{\mathrm{\Omega }}^{\left(2\right)}\left(z,\zeta \right)\overline{\mathrm{\Phi }\left(\zeta \right)}$
(see [[14], p.543]). From the Plemelj formulas, it is seen that the density μ must satisfy the integral equation
${\gamma }_{0}={\int }_{\mathrm{\Gamma }}{K}_{1}\left(\zeta ,z\right)\mu \left(z\right)\phantom{\rule{0.2em}{0ex}}d{s}_{z},$
(13)
where
${\gamma }_{0}\left(\zeta \right)=\gamma \left(\zeta \right)-Re\left[\overline{\lambda \left(\zeta \right)}{w}_{1}^{+}\left(\zeta \right)\right]=-Re\left[\overline{\lambda \left(\zeta \right)}{w}_{1}^{-}\left(\zeta \right)\right].$
(14)
Problem ( ) concomitant to problem ( ) has the boundary condition $Re\left[{\lambda }^{-1}\left(z\right){\mathrm{\Phi }}^{-}\left(z\right)\right]=0$ on Γ, where Φ is Q-holomorphic outside $G+\mathrm{\Gamma }$ and $\mathrm{\Phi }\left(\mathrm{\infty }\right)=0$. Denoting the numbers of linearly independent solutions of ( ) and ( ) by 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
${\int }_{\mathrm{\Gamma }}{\varkappa }_{j}\left(\zeta \right){\gamma }_{0}\left(\zeta \right)\phantom{\rule{0.2em}{0ex}}d{s}_{\zeta }=0\phantom{\rule{1em}{0ex}}\left(j=1,\dots ,k\right),$
(15)
where ${\varkappa }_{j}$ are solutions to integral equation (10). These solutions may be broken up into two groups $\left\{{\varkappa }_{1},\dots ,{\varkappa }_{{\ell }^{\mathrm{\prime }}}\right\}$ and $\left\{{\varkappa }_{{\ell }^{\mathrm{\prime }}+1},\dots ,{\varkappa }_{k}\right\}$ such that ${\varkappa }_{j}=i\lambda \left(z\right)\frac{d\varphi }{ds}{w}_{j}^{\mathrm{\prime }}\left(z\right)$ for $j=1,\dots ,{\ell }^{\mathrm{\prime }}$ and ${\varkappa }_{j}=i\lambda \left(z\right)\frac{d\varphi }{ds}{\mathrm{\Phi }}_{j}$ for $j={\ell }^{\mathrm{\prime }}+1,\dots ,k$, where $z\in \mathrm{\Gamma }$. Here ${w}_{j}^{\mathrm{\prime }}$ and ${\mathrm{\Phi }}_{j}$ are solutions of problems ( ) and ( ) respectively. The condition (15) for ${\gamma }_{0}$ given by (14) becomes
$-{\int }_{\mathrm{\Gamma }}{\varkappa }_{j}\left(\zeta \right){\gamma }_{0}\left(\zeta \right)\phantom{\rule{0.2em}{0ex}}d{s}_{\zeta }=-i{\int }_{\mathrm{\Gamma }}\phantom{\rule{0.2em}{0ex}}d\varphi \left(\zeta \right)\lambda \left(\zeta \right){w}_{j}^{\mathrm{\prime }}\left(\zeta \right)\gamma \left(\zeta \right)+Re\left[i{\int }_{\mathrm{\Gamma }}\phantom{\rule{0.2em}{0ex}}d\varphi \left(\zeta \right){w}_{j}^{\mathrm{\prime }}\left(\zeta \right){w}_{1}^{+}\left(\zeta \right)\right]$
for $j=1,\dots ,{\ell }^{\mathrm{\prime }}$, whereas for $j={\ell }^{\mathrm{\prime }}+1,\dots ,k$, we have
${\int }_{\mathrm{\Gamma }}{\varkappa }_{j}\left(\zeta \right){\gamma }_{0}\left(\zeta \right)\phantom{\rule{0.2em}{0ex}}ds=Re\left[i{\int }_{\mathrm{\Gamma }}\phantom{\rule{0.2em}{0ex}}d\varphi \left(\zeta \right){\mathrm{\Phi }}_{j}^{-}\left(\zeta \right){w}_{1}^{-}\left(\zeta \right)\right]=0.$

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

## Authors’ Affiliations

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

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