Abstract
In this work, we discuss RiemannHilbert and its adjoint homogeneous problem associated with generalized Qholomorphic functions and investigate the solvability of the RiemannHilbert problem.
Keywords:
generalized Beltrami systems; Qholomorphic functions; RiemannHilbert problemIntroduction
Douglis [1] and Bojarskiĭ [2] developed an analog of analytic functions for elliptic systems in the plane of the form
where w is an
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 selfcommuting property of the variable matrix Q, which means
for any two points
where the unknown
In this work, as in a complex case, following Vekua (see [[6], pp.228236]), we investigate the necessary and sufficient condition of solvability of the RiemannHilbert problem for equation (2).
Solvability of RiemannHilbert problems
In a regular domain G, we consider the problem
We refer to this problem as boundary value problem (A). Where the unknown
It is assumed, moreover, that Q is commuting with
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]),
where
Since
we have
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
see ([[14], p.543]). In (9), P is a constant matrix defined by
called Pvalue 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
where
The integral in (10) is to be taken in the Cauchy principal value sense. If we denote
this equation in an operator form by
Here
In a complex case, Vekua refers to problems of this type as being concomitant to
() and denotes them by (). Let
Let us now return to the discussion of problem (A), where we assume that
where
(see [[14], p.543]). From the Plemelj formulas, it is seen that the density μ must satisfy the integral equation
where
Problem () concomitant to problem () has the boundary condition
where
for
Consequently, the conditions (15) are seen to hold if (6) (with
Theorem 1Nonhomogeneous boundary problem (A) is solvable if and only if the condition (6) is satisfied,
This theorem immediately implies the following.
Theorem 2Nonhomogeneous boundary problem (A) is solvable for an arbitrary righthand side if and only if adjoint homogeneous problem () has no solution.
Competing interests
The author declares that they have no competing interests.
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