Selçuk J. Appl. Math. Selçuk Journal of Vol. 7. No. 1. pp. 9-15, 2006 Applied Mathematics
A Dirichlet Problem For Generalized Analytic Functions Murat Düz1 and Kerim Koca2
1Uluda¼g University, Faculty of Arts and Sciences, Department of Mathematics, Görükle/Bursa
Turkey.
e-mail:m duz7837@ yaho o.com
2K¬r¬kkale University, Faculty of Arts and Sciences, Department of Mathematics, 71450
Yah¸sihan /K¬r¬kkale, Turkey.
Received: August 08, 2005
Summary. For the existence of the solution for the Dirichlet Problem @w
@z = (Aw + Bw); z 2 D Rewj@D= g; g 2 C (@D)
Imw(z0) = c0; z02 D
in a domain having a smooth boundary D C, necessary conditions are studied. Here we assumed that z 2 D, g 2 C (@D), z02 D and A; B 2 C (D).
Key words: Generalized analytic functions, Dirichlet Problem, Contractive mapping, Boundary value problem.
1991 AMS Subject Classi…cation Codes (2000): 30G20. 1. Introduction
Various boundary value problems for Generalized Analytic Functions have been studied by many authors [3], [5]. For instance, in [3] the function space has been changed and in [5] the boundary values of the problem have been given in various forms. In [4] , Tutschke presented the existence of a Dirichlet problem using …xed point theorem. However, these problems have been considered in various meanings. The purpose of this paper is to study the boundary value
problems by putting special conditions on the coe¢ cients of the equations. We consider the system of non-homogenous real-valued partial di¤erential equation
ux vy+ au + bv = 0 (1)
uy+ vx+ cu + dv = 0
It can be easily shown that (1) is equal to the complex-valued partial di¤erential equation (2) @! @z = A ! + B ! with A = 1 4(a d + ic + ib); B = 1 4(a + d + ic ib); w = u + iv: 2. A Dirichlet Problem For Generalized Analytic Functions
In this section, we will study the solution of the complex partial di¤erential equation
(3) @w
@z + A(z)w + B(z)w = 0 in D
which belongs to the class of C (D), with the following boundary conditions:
(4) Re w(z) = g; z 2 @D
(5) Im w(z0) = c0; z02 D
where D is a bounded and simply connected domain with smooth boundary, c0 is a real constant, A; B 2 C (D), and g 2 C (@D) is Hölder continuous with Hölder constant H.
Now, by assuming h 2 C (D), let us consider the operator TD such that TD: C (D) ! C (D) h ! TDh(z) = 1Z D Z h( ) zd d where = + i .
Theorem 1: A function w 2 C1; (D) is a solution to the Dirichlet problem (3)-(5) if and only if w solves the integral equation
where ' 2 C (D) is holomorphic in a domain D satisfying the Dirichlet condi-tions
(7) Re'(z) = g ReTD[ (Aw + Bw)](z); z on @D
(8) Im'(z0) = c0 ImTD[ (Aw + Bw)](z0); z02 D
Proof: Assume w 2 C1; (D) is a solution of (3) satisfying the boundary con-ditions (4) and (5). We de…ne a function ' as follows:
'(z) = w(z) TD[ (Aw + Bw)]: Di¤erentiating ' with respect to z, we get
@' @z =
@w
@z + Aw + Bw = 0
at least in Sobolev’s sense. It follows from Weyl lemma that ' is a holomorphic function in D, hence the boundary conditions (7) and (8) implies:
Re'(z) = Re[w TD( Aw Bw)](z) = g ReTD( Aw Bw)(z) and
Im'(z0) = Imw[z0] ImTD( Aw Bw)](z0) = c0 ImTD( Aw Bw)(z0)
for the holomorphic function '. It is given that g 2 C (@D). Moreover, TD[ (Aw + Bw)] 2 C (D) for w 2 C (D). It follows that, ReTD[ (Aw + Bw)] 2 C (@D) in particular. Then ' 2 C (D) and therefore, we have shown that, w solves the integral equation
w(z) = '(z) + TD[ (Aw + Bw)]
where ' is holomorphic and satis…es the conditions (7) and (8).
Conversely, suppose that w is a solution of the equation (6) where ' is a holo-morphic function satisfying (7) and (8). Di¤erentiating (6) with respect to z we obtain
@w
@z = 0 (Aw + Bw)
and obviously, w satis…es (7) and (8). This shows that w 2 C1; (D) is a solution of (3)-(4).
Let us consider a function w 2 C1; (D) and de…ne an operator
w ! P (w) = '(w)+ TD( Aw Bw)
where '(w) is a holomorphic function in D and is uniquely determined by the boundary conditions
(10) Re'(w)(z) = g ReTD[ (Aw + Bw)] z on @D
(11) Im'(w)(z0) = c0 ImTD[ (Aw + Bw)](z0); z02 (D):
Hence, P (w) satis…es the boundary condition (7),(8) and if w is a …xed point of the operator P , that is
w = '(w)+ TD[ (Aw + Bw)] then w is a solution of (3)-(4).
Theorem 2: If A and B belongs to C (D) then the operator P : C (D) ! C (D)
de…ned by (10) is contractive if
kAkC (D)+ kBkC (D)<
1
(K + 1)kTDkC (D) where K is a constant depending on only.
Proof:Let us choose w1 and w2in C1; (D). Then we have P (w1) = '(w1)+ TD[ (Aw1+ Bw1)]
P (w2) = '(w2)+ TD[ (Aw2+ Bw2)]
where the holomorphic functions '(w1) and '(w2) are uniquely determined by the boundary conditions
(12) Re'(w1)(z) = g ReTD[ (Aw1+ Bw1)](z); z on @D
(13) Im'(w1)(z0) = c0 ImTD[ (Aw1+ Bw1)](z0); z02 D and
(15) Im'(w2)(z0) = c0 ImTD[ (Aw2+ Bw2)](z0); z02 D respectively. Therefore we obtain
(16)
P (w1) P (w2) = ('(w1) '(w2)) + TD[ (Aw1+ Bw1) + (Aw2+ Bw2)]
= ('(w1) '(w2)) + TD[ A(w1 w2) B(w1 w2)] where '(w1) '(w2) have the boundary values
Re('(w1) '(w2))(z) = ReTD[ (Aw1+ Bw1) + (Aw2+ Bw2)](z)
= ReTD[ A(w1 w2) B(w1 w2)]; z on @D Im('(w1) '(w2))(z0) = ImTD[ A(w1 w2) B(w1 w2)](z0); z02 D In order to show that P is contractive, we will compare the distance between the elements w1; w2 2 C1; (D) and their corresponding images P (w1) and P (w2). Thus we will need the corresponding estimates for the norms of
('(w1) '(w2)) and TD[ A(w1 w2) B( w1 w2)]: We have kTD[ A(w1 w2) B(w1 w2)]kC (D) kTDkC (D)kA(w1 w2)+B(w1 w2)kC (D) kTDkC (D)[kA(w1 w2)kC (D)+ kB(w1 w2)kC (D)] = kTDkC (D)[kAkC (D)kw1 w2kC (D)+ kBkC (D)kw1 w2kC (D)] = kTDkC (D)[kAkC (D)+ kBkC (D)]kw1 w2kC (D): We know k'(w1) '(w2)kC (D) = max 8 < :SupD 'w1 'w2 ; Supz16=z2 ('(w1) '(w2))(z1) ('(w1) '(w2))(z2) jz1 z2j 9 = ; Now we consider Dirichlet Problem de…ned for '(w1) '(w2)and investigate the behaviour of the real part of the function in @D .
j ReTD[ A(w1 w2) B(w1 w2)](z1) + ReTD[ A(w1 w2) B(w1 w2)](z2)j j TD[ A(w1 w2) B ((w1) (w2))](z1)+TD[ A(w1 w2) B((w1) (w2))](z2)j
kTD[ A(w1 w2) B(w1 w2)]kC (D)jz1 z2j
kTDkC (D)k[ A(w1 w2) B(w1 w2)]kC (D)jz1 z2j :
Therefore, the real part is Hölder continuous with the Hölder constant being not larger than
kTDkC (D)[kAkC (D)+ kBkC (D)]kw1 w2kC (D): Then
j('(w1) '(w2))(z1) ('(w1) '(w2))(z2)j kkTDkC (D)[kAkC (D)
(17)
+kBkC (D)]kw1 w2kC (D)jz1 z2j ; where k is a constant de…ned by
k = 2 +3 cos( 2 ) 2 (1 + 2 ) + 1 : And …nally (18) j('(w1) '(w2)(z)j 2 kkTDkC (D)[kAkC (D)+ kBkC (D)]kw1 w2kC (D) +Sup@Dj ReTD[ A(w1 w2) B(w1 w2)]j +j ImTD[ A(w1 w2) B(w1 w2)](z0)j Here
Sup@Dj ReTD[ A(w1 w2) B(w1 w2)]j
Sup@DjTD[ A(w1 w2) B(w1 w2)]j SupDjTD[ A(w1 w2) B(w1 w2)]j kTD[ A(w1 w2) B(w1 w2)]kC (D) kTDkC (D)(kAkC (D)+ kBkC (D))kw1 w2kC (D) and j ImTD[ A(w1 w2) B(w1 w2)]j jTD[ A(w1 w2) B(w1 w2)]j SupDjTD[ A(w1 w2) B(w1 w2)]j kTD[ A(w1 w2) B(w1 w2)]kC (D) kTDkC (D)[kAkC (D)+ kBkC (D)]kw1 w2kC (D): Thus we obtain
j('(w1) '(w2)(z)j (2 k + 2)kTDkC (D)[kAkC (D)
(19)
+ kBkC (D)]kw1 w2kC (D): Consequently, using (17) and (19) we have the following estimate
k'w1 'w2kC (D) (2 k+2)kTDkC (D)[kAkC (D)+kBkC (D)]kw1 w2kC (D): If we call K = 2 k + 2, we get kP (w1) P (w2)kC (D) k'(w1) '(w2)kC (D)+ kTD[ A(w1 w2) B(w1 w2]kC (D) (2 k + 2)kTDkC (D)[kAkC (D)+ kBkC (D)]kw1 w2kC (D) = (K + 1)kTDkC (D)[kAkC (D)+ kBkC (D)]kw1 w2kC (D): Consequently, the operator (9) is contractive if
(K + 1)kTDkC (D)[kAkC (D)+ kBkC (D)] < 1 [kAkC (D)+ kBkC (D)] < 1 (K + 1)kTDkC (D) : References
1. U. Akin (2002): On a Boundary Value Problem for Generalized Analytic Functions, Master Thesis, METU.
2. W. Tutschke (1977): Partielle komplexe Di¤ erentialgleichungen in einer und in mehrenen komplexen Variablen, VEB Deutscher Verlag der Wissenschaften.
3. W. Tutschke (1983): Partielle Di¤ erentialgleichungen, klassische funktionalana-lytische und komplexe Methoden, TEUBNER TEXTE Zur Math., Band 27, Leibzig. 4. W. Tutschke (1976): Losung nichtlinearer partieller Di¤ erentialgleichungssysteme erster Ordnung in der Ebene durch Verwendung einer komplexen Normalform, Math. Nachr., 75, 283–298.
5. I. N. Vekua (1963): Verallgemeinerte analytische Funktionen, Akademie Verlag, 1959, Berlin, Ubersetzung aus dem Russischen von Dr. W. Schmidt.
