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Complex Variables, Theory and Application: An

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The classical and the modified dirichlet problem

for the inhomogeneous pluriholomorphic system

in polydiscs

Alip Mohammed a

a I.Math. Institute, Free University Berlin , Berlin, Germany

Published online: 29 May 2007.

To cite this article: Alip Mohammed (2001) The classical and the modified dirichlet problem for the inhomogeneous

pluriholomorphic system in polydiscs, Complex Variables, Theory and Application: An International Journal: An

International Journal, 45:3, 213-246

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The Classical and the Modified Dirichlet Problem for the Inhomogeneous Pluriholomorphic System in Polydiscs ALlP MOHAMMED*

I. Math. Institute, Free University Berlin, Berlin, Germany

Communicated by H . Begehr

(Received 7 July 1999)

The classical Dirichlet problem for the inhomogeneous pluriholomorphic system in a polydisc is studied. To get a unique solution the boundary condition is modified and its solvability conditions as well as the unique solution are given explicitly.

Keywords: Plwiholomorphic systems; Dirichlet problem; Modified Dirichlet problem; Polydisc

AMS Subject CIassifications: 35325, 32A07

1. INTRODUCTION

The Dirichlet problem for the inhomogeneous pluriharmonic system in polydiscs was studied in many papers to various extend, see [2] and [3]. However only in [I] the problem is solved in full scale: the solvability conditions and the unique solution are given explicitly. About the Dirichlet problem for the inhomogeneous pluriholomorphic system in polydiscs there is no such rich result, see again [2] and [3].

Let Dn be the unit polydisc {z : z = (zl,. . . , zn) E Cn, lzkl < 1, 1 I k I n} and fkl, 70 be given functions with fkl2,. E L I (1(D") n Ca(IID"),

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E C(doDn), a 2 112. Consider the following inhomogeneous system of n(n + 1)/2 independent equations

with given right-hand sides, satisfying the compatibility conditions

Problem D Find a ca(D")-solution of system (I), satisfying the Dirichlet condition

It is known that any solution to (1) can be represented as, see [I],

where pk(z)(k = 0, 1, . . . , n) are arbitrary analytic functions, in Dn, uo(z) is a special solution to (1) and has to be found. For this purpose we quote a theorem from [I].

THEOREM 1 Let D" := XtZ1Dk, D" := X'&lE where Dk is a smooth boundedplane domain in @, 1 5 k < n. Let w have mixed derivatives with respect to each variable of Jirst order in LI (D"). Then w = p + wo where cp is analytic in Dn and

Remark 1 Tk, 1 5 k 5 n is the Pompeiu operator, given by

see [5].

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DIRICHLET PROBLEM

2. SPECIAL SOLUTION

Applying once this theorem to the system (1) one can easily obtain the special solution

For solvability of (5) we need the compatibility conditions

to be satisfied, which is actually equivalent to fM =fek. Repeating once more the above procedure we get a particular solution to (1)

Carefully combining (5) with (6), the final explicit form of uo can be found.

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216 A. MOHAMMED

Since

it follows that

Let a+/? and a , / ? ~ { l ,..., n ) . Then

where the condition (2) is used. Paying attention to the fact that a , /? E {1, . . . , n) and that for every (a, /?)-term there is one and only one (/?, a)-term and they are equal to each other by the condition (2) it is easy to see that

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DIRICHLET PROBLEM

Thus

Further by means of the compatibility condition (2) the special solution uo can be written as

3. THE CLASSICAL PROBLEM

Now we specify the general solution (4) with the Dirichlet boundary condition (3):

The equality (9) holds if and only if

holds for 1 5 k 5 n. Since the left-hand side is the boundary value of a holomorphic function, the right-hand side is too. Thus from the

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218 A. MOHAMMED

Cauchy formula it follows that (10) holds if and only if

for any k, h ~ ( 1 , ..., n}, i.e.,

Evidently, if the right-hand side of (10) is the boundary value of an analytic function in D", then from (1 1) it follows that

This means (12) is a necessary solvability condition of the system (1) for the boundary condition (3). However, it is not sufficient for the above problem to be solvable. This can be shown by a simple example:

Let n = 3 and fke=O, %(c) =C?G"G", t i€N , i=1,2,3. Clearly the necessary condition (12) is satisfied for this example, but the condition (11) is not satisfied for c2 and c3. The reason is that yo(c)-uo(c) is not the boundary value of a function, which is holomorphic in D3.

THEOREM 2 Let W(boDn) be the Wiener algebra on b0Dn and I? E W(boDn). Then the condition

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is necessary and together with

is necessary and suficient for I? to be the boundary values of a holomorphic function in Dn.

Proof By the definition of analytic functions we know that the function I' is analytic in Dn if and only if

Suppose condition (15) holds. That means

Thus the case k = 1 is proved. For 6') from (15) it follows that

Further by the previous equality we have

1 dq* 1

i.e., condition (14) is true for the case k = 2.

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220 A. MOHAMMED

We assume condition (14) is true for the case k = n- 1, i.e.,

Applying (1 5) for p-') we have

Hence, from the assumption for k = n - 1 it follows that

So the case k = n is proved. Next from (13) and (14) we derive (15), i.e., that the function r(C) is

analytic for every Ck, 1 5 k 5 n, lCkl 5 1. Assume condition (13) and (14) hold, i.e.,

1 - dq* 1 (274' L o k r(q) = 3 ~ ( ~ ( k ) ) 7

q = ($,ql), c ( ~ ) = ( C , $ ) E W " ; C , ~ * E W ~ ,

Then

and

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DIRICHLET PROBLEM

But since

it follows that

i.e., function r(r]) is analytic for q,, lqnl 5 1. The rest can be proved in the same way.

This theorem can be proved also by applying the properties of the Wiener algebra and Fourier series method.

Remark 2 C(doD") W(aoDn), cu 2 112, see [4].

LEMMA 1 Equivalently to ( l l ) , condition (12) together with

becomes necessary and suflcient for the Problem D to be solvable.

Interestingly in the case n = 1 condition (16) vanishes automatically and therefore condition (12) alone is necessary and sufficient for the problem to be solvable.

Now by (lo), if condition (11) holds, it is obvious that

Substituting it into (4) we have

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THEOREM 3 Let fke (2) E c'+" (D") n L, ([[D"), 112 < a and satisfy the compatibility conditions (2). A particular solution to the inhomogeneous system (1) is given by (7). If the Problem D is solvable then the condition (12) must be satisfied. The Problem D is solvable i f and only if the condition (11) holds. The solution is given by (17). The corresponding homogeneous problem has an infinite number of nontrivial solutions. The problem is not well-posed.

In order to get a unique solution we may introduce a proper boundary condition.

4. THE MODIFIED PROBLEM

Problem M Find a cl+"(D")-solution of system (I), which satisfies condition (3) and

Re(gradp,C)=r~(C), C ~ d o D ~ , I m u ( o ) = C o , (18)

as well.

The representation (4) gives

So by the first part of the modified boundary condition (18) we have

Taking the real part of (8) and combining it with (19) leads to

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DIRICHLET PROBLEM 223

This is a simple Schwarz problem for analytic functions in Dn and this problem is solvable if and only if, see [I],

is satisfied. For any real C then

is analytic in Dn and satisfies (20). Having the second part of the modified boundary condition M in

mind, by (17) and (22) we have got the unique solution of the system (I), i.e.,

Next we simplify the solvability condition (21) and the solution (23). Let

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224

Since

A. MOHAMMED

see [I], it is easy to see

Substituting - r l - C

T 2 f K ) = y f ( r l ) z d % d n . = D

we have

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DIRICHLET PROBLEM

By simple repeatition of the Pompeiu formula for one variable it is easy to derive

where f is defined and properly differentiable in Dn, continuous even on nn. Direct application of the formula for n - 1 variables shows that

The second term we have to simplify is

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226 A. MOHAMMED

Applying (21) and its special case (z=O), we have

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DIRICHLET PROBLEM

Since the second term turns out to be

it candles together with the third term.

THEOREM 4 Let f E Ll ( D) . Then

where S = T ~ .

Proof Using (25) we have

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228 A. MOHAMMED

Applying (27) and

see [I], we obtain

With (26) then

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DIRICHLET PROBLEM 229

Switching the summation index from p to X:=p+y the integral above can be written as

For the second term with some ax,, we have

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230 A. MOHAMMED

So adding the first term with the second one we get

The third term to be calculated is

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DIRICHLET PROBLEM

From (24) it follows that

By direct application of (24), we have Zs3a = 0 and therefore Zs3 =

Hence from Zs3a(~) 0, z E Dn, i.e., Zs3a(0) = 0 and

it is obvious that

Further (28) gives

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232 A. MOHAMMED

Applying (26) once again leads to

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DIRICHLET PROBLEM 233

By switching the summation index from v to A:= v+p and taking 1 I v < n, 1 I p I n - v into account it is easily seen that

Changing the summation order of the second and the fourth term for some a , ~ gives

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234 A. MOHAMMED

Adding up the first term with the second and the third term with the fourth we reach

Thus

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Next we simplify the solvability conditions (21). Since the test kernel of (21) is real, the condition is actually equivalent to

The remaining is to clarify the second and the third term. Applying (5) and (6) we find

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A. MOHAMMED

where

is used. Since

Using

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DIIUCHLET PROBLEM

and (30) again, we see

Applying the formula,

f E c1 (D")

see [l], we get

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DIRICHLET PROBLEM 239

Replacing the summation from 1 to a := v+L and changing the order of summation gives for the last two terms

Since

for some ax with 1 5 X 5 v - 1, we have

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Similarly from

for some aAp with 1 < X < u - 1 , 1 S p S a - u it can be easily shown that

a-2a-A-1 a-p

A=l fl=1 v=A+l v - X

where the rule of changing orders of summation is just the same as of multiple integrals. Thus we get the sum simplified, i.e.,

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n=l a=3X=l p 1 I 5 h, < ... < h, 5 n 1 5 h,, < ... < h*" 5 n I 5 h,, < ... < 6 . 5 n

namely Z(z) is simplified one step further,

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242 A. MOHAMMED

From the first and the second term it follows that

Adding this term to the last term of Z(z) in (35) leads to

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DIRICHLET PROBLEM 243

As (29) is just needed on &Dn, if we consider (36) for z E &Dn instead of z E Dn and take

into account, then it can be written as

further via ,B := X + p

But from

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244 A. MOHAMMED

where aa are some definite terms with 1 5 8 < v- 1, it follows that

Finally we have got explicitly the solvability conditions as well as the unique solution, i.e.,

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DIRICHLET PROBLEM

1 R~F? c ,J "=2X=l l < k , < ... < k d W aDn [TO (5) - n (01

1 5 k*, < ... < k, s "

where the integral is understood as Cauchy principal value,

Ckl - Zkl

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THEOREM 5 Let fke(z) E c'+~ (D") n L1 (w) and satisfy the compat- ibility conditions (2). Aparticular solution to the inhomogeneous system (1) is given by (7). If the Problem M is solvable then condition (12) must be satisfied. The modified Problem M is solvable if and only if condition (1 1) and (38) or (39) hold. Then the unique solution is given by (40). The corresponding homogeneous problem has no any nontrivial solutions. The modged Problem M is well-posed.

References

Begehr, H. and Dzhuraev, A., An introduction to several complex variables and partial differential equations, Pitman monographs and surveys in pure and applied mathematics 88, Addison Wesley Longman, Harlow, 1997. Begehr, H. and Wen, G. C., Nonlinear Elliptic Boundary Value Problems and Their Application., Addison Wesley Longman, Harlow, 1996. Cheng, J. and Li, M. Z., Boundary value problems for complex overdetermined partial differential equations of second order, ?Journal, 1990. Kufner, A. and Kadlec, J., Fourier series, London, ILIFFE BOOKS, Akademia, Prague, 1971. Vekua, I. N., Generalized Analytic Functions. Moscow, 1959; Pergamon Press, London, 1962.

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