Carleman boundary value problems for polyanalytic functions.

1 Introduction

In , the following Carleman boundary value problem for analytic functions is given: Problem [C.sub.0]. Find a function [PHI](z) in the domain [D.sup.+] bounded by a simple and closed contour [GAMMA] that satisfies the boundary condition

[PHI][[alpha](t)] = G(t)[PHI](t) + g(t) (1.1)

where [alpha](t) satisfies the Carleman condition

[alpha][[alpha](t)]= t (1.2)

and G(t),g(t),[alpha]'(t) satisfy the Holder condition on [GAMMA] and G(t) [not equal to] 0, [alpha]'(t) [not equal to] 0.

The complete solution to Problem [C.sub.0] for a bounded and simple connected domain was given by D. A. Kveselava  by using the method of integral equations. In , the following theorem is proved.

Theorem 1.1. Let x = 1/2[pi][{arg G(t)}.sub.[GAMMA]] be an index of the boundary problem (1.1), and [m.sup.-] be a number of the fixed points of the movement [alpha](t). If x + [m.sup.-] [less than or greater than] 0, the number l of solutions of the homogeneous Carleman boundary value problem is

l = 1 -[ x + [m.sup.-]]/2,

and the corresponding nonhomogeneous problem is unconditionally solvable. If x + [m.sup.-] > 0, then the homogeneous problem has no nontrivial solutions (l = 0) and

p = [x + [m.sup.-]] / 2 -1

is the number of conditions that are necessary and sufficient for the solvability of the nonhomogeneous problem.

A corollary of this theorem is the case when x + [m.sup.-] = 2, when the nonhomogeneous problem (1.1) has a unique solution (l = p = 0).

Remark 1.2. In , an important special case of the Problem C0 is given: the homogeneous boundary value problem

[PHI][alpha(t)] = [lambda]*[PHI](t), [lambda] = [+ or -]1,

for [lambda] = 1 in [D.sup.+] has only a constant solution, but for [lambda] = -1, the constant is 0. Let us consider the elliptical system of partial equations

[u'.sub.x] - [v'.sub.y] = au - bv + c, [u'.sub.y] - [v'.sub.x] = bu + av + d, (1.3)

where a = a(x,y), b = b(x,y), c = c(x,y), d = d(x,y) are continuous functions in some closed domain D and w(z, [bar.z]) = u(x, y) + iv(x, y) is a determining function. This system plays an important role in different problems in mechanics and it leads to complex differential equation

[w'.sub.[bar.z]] + A (z, [bar.z])w + B(z, [bar.z]) = 0 (1.4)

where A(z, [bar.z]) = - [a + ib]/2, B(z, [bar.z]) = c + id/2. It is shown by S. Fempl  that the general solution of(1.4) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLA IN ASCII], (1.5)

where Q(z) is an arbitrary analytic function.

The regular solutions of the elliptic system of partial equations (1.3) define a class of so-called F-analytic functions that is an important connection between the set of analytic functions and the set of Vekua generalized analytic functions.

On the basis of (1.4), a differential operator of Fempl type can be introduced by

[F.sub.A,B]w = [w'.sub.[bar.z]] + A(z, [bar.z])w + B(z, [bar.z]), (1.6)

that for given continuous characteristics A(z, [bar.z]) and B(z, [bar.z]) to every differentiable complex function w corresponds to different continuous complex functions [PHI](z, [bar.z]).

Now an auxiliary Carleman boundary value problem for F-analytic functions can be formulated and solved, because its solution is necessary for solving the Carleman boundary value problem for F-polyanalytic functions.

Boundary value problem [F.sub.0]. Let the domain [D.sup.+] be bounded by a simple and smooth closed curve L and let G(t), g(t),[alpha](t) be functions from the contour (t [member of] L). Suppose that the functions G(t) and g(t) satisfy on L the Holder condition, the function [alpha](t) satisfies the Carleman condition (1.2) and [alpha]'(t) [not equal to] 0. Let A(z, [bar.z]) and B(z, [bar.z]) be given continuous functions on [D.sup.+] [intersection] L. The problem is to find a solution of the differential equation (1.4) that on L satisfies the Carleman boundary value condition

w[[alpha](t)] = G(t)w(t)+ g(t). (1.7)

We now find this solution. If we introduce the notations

[MATHEMATICAL EXPRESSION NNOT REPRODUCIBLE IN ASCII]

then according to (1.5), the solution of the equation (1.4) is

w = A(Q + B). (1.8)

By the substitution of (1.8) in (1.7), we obtain

A[[alpha](t)] {Q[ [alpha](t)] + B [[alpha](t)]} = G(t)A(t)[Q(t) + B(t)] + g(t),

and after some arranging

[MATHEMATICAL EXPRESSION NNOT REPRODUCIBLE IN ASCII] (1.9)

The relation (1.9) is the Carleman boundary value problem for determining an analytic function Q(z). When such a function is determined through the procedure from , by its substitution in (1.8), the solution of the boundary problem (1.7) is obtained. Let the index of boundary value problem (1.9) be denoted as

[MATHEMATICAL EXPRESSION NNOT REPRODUCIBLE IN ASCII].

Because of Theorem 1.1, in the case when x + [m.sup.-] = 2, the boundary value problem [F.sub.0] has a unique solution.

2 On F-Polyanalytic Functions

The polyanalytic function of the n-th order is defined by Teodorescu  as a solution of the polyanalytic differential equation

[[partial derivative].sup.n]w/[partial derivative][[bar.z].sup.n] = 0

It was shown by Teodorescu that the general solution of (2.1) is

[MATHEMATICAL EXPRESSION NNOT REPRODUCIBLE IN ASCII] (2.2)

where [[phi].sub.k](z) are arbitrary analytic functions in some closed domain D. It is important to mention that Balk  developed a theory of polyanalytic functions by defining n generalized polyanalytic functions as solutions of the differential equation

[[D.sup.n].sub.A]f = 0 ([D.sub.A]f = [f'.sub.[bar.z]] + Af), (2.3)

Where A(z, [bar.z]) is a given n-times differentiable function in some domain D. Gabrinovich and Sokolov  as well gave a survey on different generalizations of polyanalytic functions and these boundary value problems, and Canak  defined p-polyanalytic functions and solved the corresponding boundary value problem of the Riemann type.

In this article, a new generalization of ordinary polyanalytic functions is defined via F-polyanalytic functions. Let us consider the differential equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

where A(z, [bar.z]), B(z, [bar.z]) are given n-times differentiable functions in some closed domain D. For n = 1, the equation (2.4) becomes

[w'.sub.[bar.z]] + A(z, [bar.z])w + B (z, [bar.z]) = 0. (2.5)

Because of (1.5), its general solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

and Q(z) is an arbitrary analytic function. For n = 2, the equation (2.4) becomes [F.sup.(1.2).sub.A,B]w = 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

The general solution of (2.8) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

By using mathematical induction, the general solution of a polyanalytic differential equation of the Fempl type is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

where [Q.sub.1](z), [Q.sub.2](z), ... , [Q.sub.n](z) are arbitrary polyanalytic functions and the function [S.sub.n-1] (z, [bar.z]) satisfies the recurrence formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12)

The regular solutions of the differential equation (2.4) will be called F-polyanalytic functions of the n-th order.

Remark 2.1. In the case when B(z, [bar.z]) = 0, we have [S.sub.1] = [S.sub.2] = ... = [S.sub.n] = 0, and the equation (2.4) is the Balk polyanalytic differential equation (2.3). When A(z, [bar.z]) = A(z) and B(z, [bar.z]) = B(z), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.13)

and the solution (2.11) is called an areolar exponential equation of the Fempl type . Finally, when A(z, [bar.z]) = B(z, [bar.z]) = 0, the ordinary polyanalytic functions are obtained.

3 Carleman Boundary Value Problem

Let L be a simple, smooth and closed contour that bounds the finite domain [D.sup.+] and let [G.sub.k] (t), [g.sub.k] (t) (k = 0,1,..., n - 1) and [alpha](t) be given functions on the contour L. Let us suppose that the functions [G.sub.k] (t), [g.sub.k] (t) satisfy the Holder condition, the function [alpha](t) satisfies the Carleman condition (1.2) and [alpha]'(t) [not equal to] 0. Let A(z, [bar.z]) and B(z, [bar.z]) be given continuous functions on [D.sup.+] [union] L. The problem is to find a regular solution of the differential equation (2.4) that on L satisfies n boundary conditions of the Carleman type

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.1)

The general solution of the equation (2.4) is given via (2.11), from where we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By substitution of (3.2) in (2.11) and (3.1), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

where A(z, [bar.z]) = [e.sup.[integral] A-(z, [bar.z])d[bar.z])]. The last condition (3.3) is the Carleman boundary value problem for determining the analytic function [Q.sub.n] (z). It can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

that corresponds to the boundary condition (1.9). If the sum of indexes of the boundary value problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the number of fixed points [m.sup.-] of the Carleman movement is [x.sub.n-1] + [m.sup.-] = 2, then this problem has a unique solution [Q.sub.n] (z) that can be determined by the technique developed by Litvinchuk. By the substitution of this functions [Q.sub.n](z) in (3.3), the Carleman boundary value problem is obtained as well for determining the function [Q.sub.n-1](z). This problem has a unique solution if its coefficients satisfy the Holder condition, the free coefficient is not annulated and [x.sub.n-1] + [m.sup.-] = 2 holds. By repeating this procedure n times, all the unknown functions [Q.sub.n](z),[Q.sub.n-1] (z), ... , [Q.sub.1](z) are determined, and by these substitutions in (2.11), the solution of the boundary problem is determined. Hence the following result holds.

Theorem 3.1 (Existence and uniqueness of solutions of the Carleman boundary value problem). If the indexes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

of the boundary value problem (2.4), (3.1) satisfy the condition

[x.sub.i] + [m.sup.-] = 2

and if all the free coefficients of the successive boundary value problems are not zero, then the boundary value problem (2.4), (3.1) has a unique solution.

Example 3.2. Find an F-bianalytic function with characteristics A =1, B = 1 that on the unit circle [bar.t] = 1/t satisfies the Carleman boundary value conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

Solution. By (2.11), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

By substitution of (3.7) in (3.5) and observing that [bar.t] = 1/t, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

The second relation from (3.8) is the Carleman boundary value problem for determining the analytic function [Q.sub.2](z). As the index ofthe boundary value problem x = 0 and the number of fixed points [m.sup.-] = 2 for the Carleman movement [alpha](t) = 1/t, we have x + [m.sup.-] = 2, and the problem has a unique solution. By using Litvinchuk's technique and after lengthy calculations, we obtain that the unique solution is [Q.sub.2](z) = z. If this is substituted in the first boundary condition in (3.8), then

[Q.sub.1](1/t) = [Q.sub.1](t) (3.9)

is obtained. The relation (3.9) is the Carleman homogeneous boundary value problem that has only a constant solution [Q.sub.1](z) = c. So, the general solution of the problem (3.5) is

w(z, [bar.z]) = [e.sup.-[bar.z]][c + z[bar.z]]- 2. (3.10)

This solves the problem []

References

 M. B. Balk. Polyanalytic functions. In Complex analysis, volume 61 of Math. Lehrbucher Monogr. II. Abt. Math. Monogr., pages 68-84. Akademie-Verlag, Berlin, 1983.

 Milos Canak. Randwertaufgabe von Riemann-Typus fur die p-polyanalytischen Funktionen auf der spiralformigen Kontur. Mat. Vesnik, 40(3-4):197-203, 1988. Third International Symposium on Complex Analysis and Applications (Herceg novi, 1988).

 Stanimir Fempl. Regulare Losungen eines Systems partieller Differentialgleichungen. Publ. Inst. Math. (Beograd) (N.S.), 4 (18):115-120, 1964.

 Stanimir Fempl. Areolare Exponentialfunktion als Losung einer Klasse Differentialgleichungen. Publ. Inst. Math. (Beograd) (N.S.), 8 (22):138-142, 1968.

 V. A. Gabrinovich and I. A. Sokolov. Studies in boundary value problems for polyanalytic functions. In Proceedings ofa commemorative seminar on boundaryvalue problems (Russian) (Minsk, 1981), pages 43-47, Minsk, 1985. "Universitet-skoe".

 D. A. Kveselava. Resenie odnoi granicnoi zadaci T. Karlemana. Dokl. Akad. Nauk SSSR, 55(8):683-686, 1947.

 G. S. Litvinchuk. Kraevye zadachi i singulyarnye integralnye uravneniya so sdvigom. Izdat. "Nauka", Moscow, 1977.

 N. Teodorescu. La derivee areolaire et ses applications a la physique mathematique. PhD thesis, Sorbonne, Paris.

Liljana Stefanovska

Faculty of Technology and Metallurgy

Skopje, Macedonia

liljana@tmf.ukim.edu.mk

Milos Canak

University of Novi Pazar, Serbia

miloscanak12@yahoo.com

Received September 30, 2010; Accepted January 20, 2011

Communicated by Malisa Zizovic