A survey on boundary value problems for complex partial differential equations.

1 Introduction

The investigations on the boundary value problems for complex differential equations had a new starting point for complex differential equations in the middle of the nine-teenth century. Riemann has stated the problem

"Find a function w(z) = u + iv analytic in the domain [OMEGA], which satisfies at every boundarypoint the relation

F (u,v) = 0 (on [partial derivative][OMEGA]"

in his famous thesis . Later, this statement is known as Riemann problem. However, he stated only some general considerations regarding with the solvability of the problem. The generalization of Riemann problem to a linear first-order differential equation together with the linear boundary condition

[alpha]u + [beta]v = Re[[bar.[lambda](z)]w] = [gamma] on [partial derivative][OMEGA]

was considered by Hilbert . This new form of the problem is called as the generalized Riemann-Hilbert problem. Some extensions of this problem has been treated by many researchers [43,46,50,53,57,59].

The first-order linear complex partial differential equation

[W.sub.[bar.z]] = a(z)W(z) + b(z)[bar.w(z)]

has been considered by Vekua  and Bers  separately and simultaneously. Its solutions are called as generalized analytic (or pseudo-holomorphic) functions. The boundary value problems described above and their particular cases known as Schwarz, Dirichlet, Neumann and Robin problems are treated by many researchers.

Boundary value problems for higher-order linear complex partial differential equations gained attraction in the last twelve years. Dirichlet, Neumann, Robin, Schwarz and mixed boundary value problems for model equations, that is for the equations of the form

[[partial derivative].sup.m.sub.z] [[partial derivative].sup.n.sub.[bar.z]]w = f(z),

are introduced in the unit disc of the complex plane by Begehr . In this article we want to give a directed survey of the relevant literature on the boundary value problems of complex analysis, and reveal some problems which are still open.

2 Dirichlet Problem for Complex PDEs

2.1 Dirichlet Problem for Complex Model PDEs

2.1.1 Simply Connected Bounded Domain Case

Many authors have investigated the Dirichlet problem in simply connected domains. To give the explicit representations for the solutions of the problems, we will consider the particular case of the unit disc D of the complex plane. Let us start by giving the related harmonic and polyharmonic Green functions.

In D, the harmonic Green function is defined as

[G.sub.1](z,[zeta] = log [[absolute value of 1 - z[bar.[zeta]]/[zeta] - z].sup.2]

and its properties are given in [16,18]. A polyharmonic Green function [G.sub.n] is defined iteratively by

[G.sub.n](z,[zeta]) = - 1/[pi][integral][[integral].sub.D] [G.sub.1](z,[zeta]) [G.sub.n-1]([zeta],[zeta])d[xi]d[eta]

for n [greater than or equal to] 2, . The explicit expressions of [G.sub.n](z,[zeta]) for n = 2 and for n = 3 are given in [16,18] and in , respectively. [G.sub.n](z,[zeta]) are employed to solve the following n-Dirichlet problems for the n-Poisson equation, .

Theorem 2.1. The Dirichlet problem

[([[partial derivative].sub.z][[partial derivative].sub.[bar.z]]).sup.n] w = f in D, [([[partial derivative].sub.z][[partial derivative].sub.[bar.z]])sup.[mu]] w = [[gamma].sub.[mu]], 0 [less than or equal to][mu] [less than or equal to] n - 1 on [partial derivative]D

f [member of] [L.sup.1](D), [[gamma].sub.[mu]] [member of] C([partial derivative]D), 0 [less than or equal to] [mu] [less than or equal to] n - 1 is uniqualy solvable. The solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

The explicit forms of the solutions (2.1) are given in [15,16,18] in the case of n = 2. In , authors considered the problem given in Theorem 2.1 and they gave the explicit representation of the unique solution using the iterative sums for n = 2 and n = 3 are given. Begehr, Du and Wang  solved the Dirichlet problem for polyharmonic functions by using the decomposition of polyharmonic functions and transforming the problem into an equivalent Riemann boundary value problem for polyanalytic functions. In , authors solved the Dirichlet problem investigated in  by a new approach. The explicit expression of the unique solution for the Dirichlet problem of triharmonic functions in the unit disc is obtained by using the so-called weak decomposition of poly-harmonic functions and converting the problem into an equivalent Dirichlet boundary value problem for analytic functions. In contrast to the boundary condition according to , the requirement of smoothness for the given functions is reduced.

Another polyharmonic kernel function is the so-called Green-Almansi function [[??].sub.n]  which is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.2)

Using [G.sub.n], the following theorem is proved in .

Theorem 2.2. The Dirichlet problem

[([[partial derivative].sub.z][[partial derivative].sub.[bar.z])].sup.n] w = f in D, f [member of] [L.sup.1](D) (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

is uniquelysolvable. The solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A variant of the problem (2.3), (2.4), (2.5) is discussed in the case of [[gamma].sub.[mu]] and [[??].sub.[mu]] are continuous on [partial derivative]D, [17,38]. The solution is attained by modifying the related Cauchy-Pompeiu representation with the help of the polyharmonic Green function.

The problem defined in Theorem 2.1 has also been solved using the Green-Almansi function [[??].sub.n] by Kumar and Prakash [47, 49].

The Green-m-Green Almansi-n function [G.sub.m,n](z,[zeta]) for m, n [member of] N (which is also called as a polyharmonic hybrid Green function) is defined [7,34] by the convolution of [G.sub.m] and [[??].sub.n] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.6)

Note that, [G.sub.m,1] (z,[zeta]) = [G.sub.m+1](z,[zeta]) for m [member of] N and we take [G.sub.0,n](z,[zeta]) = [[??].sub.n](z,[zeta]) for n [member of] N. Thus, [G.sub.0,1](z,[zeta]) = [G.sub.1] (z,[zeta]). Also, [G.sub.2,2](z,[zeta]) is defined in . Eq. (2.6) is employed in the following (m, n)-type Dirichlet problem, .

Theorem 2.3. The (m, n)-Dirichletproblem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

f or f [member of] [L.sup.1](D) [intersection]C(D) is uniquely solvable. The solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Begehr and Vaitekhovich  have considered the following similar problem with the inhomogeneous boundary conditions,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

They have given the solution by an iterative technique.

The inhomogeneous polyanalytic equation is studied by Begehr and Kumar  in D with Dirichlet conditions and the following result is obtained.

Theorem 2.4. The Dirichlet problem for the inhomogeneous polyanalytic equation in D

[[partial derivative].sup.n.sub.[bar.z]]w = f in D (2.7)

[[partial derivative].sup.n.sub.[bar.z]]w = [[gamma].sub.v] 0 [less than or equal to] v [less than or equal to] n - 1 on [partial derivative]D

is uniquely solvable for f [member of] [L.sup.1] (D;C), [[gamma].sub.v] [member of] C(D;C), 0 [less than or equal to] v [less than or equal to] n - 1 if and only if for 0 [less than or equal to] v [less than or equal to] n - 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If the problem is solvable, the solution is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The Dirichlet problem for the equation

[[partial derivative].sup.m+n.sub.[bar.z]]f + [alpha][bar.[[partial derivative].sup.n.sub.z] [partial derivative].sup.m.sub.[bar.z]]] f = 0 (2.8)

is investigated in  for 1 [less than or equal to] m, n [member of] N. (2.8) is known as bi-polyanalytic equation. In the case m =1, the solutions are called bi-polyanalytic functions.

In a half disc and a half ring of the complex plane, the Green function is given and the Dirichlet problem for the Poisson equation is explicitly solved by Begehr and Vaitekhovich, .

2.1.2 Simply Connected Unbounded Domain Case

In this subsection we give an overview of the problems defined in the upper half plane H = {z [member of] C : 0 < Im z} and in right upper quarter plane [Q.sub.1] = {z [member of] C : 0 < Re z, 0 < Im z}.

The polyharmonic Green function for H is given using the Almansi expansion :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Begehr and Gaertner  have proved the following theorem.

Theorem 2.5. Forgiven f satisfying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. with the respective derivatives bounded, the Dirichlet problem

[([[partial derivative].sub.[bar.z]] [[partial derivative].sub.z]).sup.n]w = f in H

[([[partial derivative].sub.z] [[partial derivative].sub.[bar.z]]).sup.v]w = [[gamma].sub.v] for 0 [less than or equal to] 2v [less than or equal to] n-1

[[partial derivative].sup.v.sub.z] [[[partial derivative].sup.v+1.sub.[bar.z]] w = [[gamma].sub.v] for 0 [less than or equal to] 2v [less than or equal to] n-2 on R

is uniquely solvable in a weak sense by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where for 1 [less than or equal to] [alpha],

g[alpha](z, [zeta]) = 1 / [([zeta] - z).sup.[alpha]] + [(-1).sup.[alpha]] / [([zeta] - [z]).sup.[alpha]].

In [Q.sub.1] = {z [member of] C : 0 < Re z, 0 < Im z}, the following result for the Dirichlet boundary value problem is given for the inhomogeneous Cauchy-Riemann equation in .

Theorem 2.6. The Dirichlet problem

[w.sub.[bar.z]] = f in [Q.sub.1],

w = [[gamma].sub.a] for x [greater than or equal to] 0, y = 0, w = [[gamma].sub.2] for x = 0, y [greater than or equal to] 0

for f [member of] [L.sub.p,2]([Q.sub.1];C), 2 < p, [[gamma].sub.2], [[gamma].sub.2] [member of] C(R;C) such that [(1 + t).sup.[delta]] [[gamma].sub.1](t), [(1 + t).sup.[delta]] [[gamma].sub.2](t) are bounded for some [delta] > 0 and satisfying the compatibility condition [[gamma].sub.1](0) = [[gamma].sub.2](0) = 0 is uniquely weakly solvable in the class [C.sup.1] ([Q.sub.1]; C) [intersection] C ([[bar.Q].sub.1]; C) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The equation [[partial derivative].sup.n+1.sub.[bar.z]]f + [alpha][bar.[[partial derivative].sup.n.sub.z] [[partial derivative].sub.[bar.z]]f] = 0 which represents the bi-polyanalytic functions, is investigated in the upper half plane and different forms of boundary conditions leading to the well-known Schwarz, Dirichlet and Neumann problems in complex analysis are solved in the upper half plane in .

Open Problem 2.7. The results given for half disc, half ring and unbounded domains have not been extended to linear higher-order differential equations yet.

2.1.3 Multiply Connected Domain Case

The main contributions for boundary value problems are given by Begehr and Vaitekhovich [35,54,56]. They have considered the boundary value problems for inhomo-geneous Cauchy-Riemann equation and Poisson equation in concentric ring domains.

Open Problem 2.8. The Dirichlet problems for higher-order linear differential equations in multiply connected domains have not been solved yet.

2.2 Dirichlet Problem for Complex Linear Elliptic PDEs

Now we take the linear differential equations which have the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

where

[a.sub.kl], [b.sub.kl], f [member of] [L.sup.p] (D) (2.10)

and [q.sup.(1).sub.kl] and [q.sup.(2).sub.kl], are measurable bounded functions subject to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

The equation (2.9) is called a generalized higher-order Poisson equation. We pose the following problem  in the unit disc.

Dirichlet-(m, n) Problem. Find w [member of] [W.sup.2n,p](D) as a solution to equation (2.9) satisfying the Dirichlet condition

[([[partial derivative].sub.z][[partial derivative].sub.[bar.z]]).sup.[mu]]w = 0, 0 [less than or equal to] [mu] [less than or equal to] m - 1 in 2D (2.12)

[([[partial derivative].sub.z][[partial derivative].sub.[bar.z]]).sup.[mu]+m]w = 0, 0 [less than or equal to] 2[mu] [less than or equal to] n - m - 1 in 2D (2.13)

[[partial derivative].sub.vz][([[partial derivative].sub.z][[partial derivative].sub.[bar.z]]).sup.[mu]+m]w = 0, 0 [less than or equal to] 2[mu] [less than or equal to] n - m - 2 in 2D. (2.14)

We need some preparations to find the solutions.

2.2.1 A Class of Integral Operators Related to Dirichlet Problems

In this section, using [G.sub.m,n](z, [zeta]) and its derivatives with respect to z and Z as the kernels, we define a class of integral operators related to (m, n)-type Dirichlet problems.

Definition 2.9. For m, k, l [member of] [N.sub.0], n [member of] N with (k,l) [not equal to] (n, n) and k + l [less than or equal to] 2n, we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for a suitable complex valued function f given in D.

It is easy to observe that the operators [G.sup.k,l.sub.m,n] are weakly singular for k + l < 2n and strongly singular for k + l = 2n.

Using Definition 2.9, we can obtain the following operators by some particular choices of n, k, l:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

One should observe that, [G.sup.0,0.sub.0,1], [G.sup.1,0.sub.0,1]) and [G.sup.2,0.sub.0,1] are the operators [[PI].sub.0], [[PI].sub.1] and [[PI].sub.2] respectively, which were investigated by Vekua . It is easy to show that these operators satisfy

[[partial derivative].sub.z] [G.sup.0,0.sub.0,1]f = [G.sup.1,0.sub.0,1]f and [[partial derivative].sup.2.sub.z] [G.sup.0,0.sub.0,1]f = [G.sup.2,0.sub.0,1]f (2.15)

for f [member of] [L.sup.p](D), p > 2, in Sobolev's sense. The other properties of these operators can be found in . We will mention just one of them:

For k [member of] N, if f [member of] [W.sup.k,p](D), then

[[partial derivative].sup.k-1.sub.z] [G.sup.2,0.sub.0,1]f(z) = [G.sup.1,0.sub.0,1]([(D - D*).sup.k]f(z)) (2.16)

where Df(z) = [[partial derivative].sub.z] = [partial derivative]f(z), D * f(z) = [[partial derivative].sub.[bar.z]]([[bar.z].sup.2]f(z)).

(2.16) is very important for the solution of the boundary value problem for linear partial differential equations. Our main result is given by the following theorem.

Theorem 2.10. The equation (2.9) with the conditions (2.12), (2.13) and (2.14) is solvable if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.17)

and a solution is of the form w(z) = [G.sup.0,0.sub.m,n-m]g(z) where g [member of] [L.sup.p](D), p > 2, is a solution of the singular integral equation

(I + D + K)g = f (2.18)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3 Neumann Problem for Complex PDEs

3.1 Neumann Problem for Complex Model PDEs

3.1.1 Simply Connected Bounded Domain Case

The harmonic Neumann function for the domain D is given by

N1(z, [zeta]) = log [[absolute value of ([zeta - z])(1 - z[bar.[zeta]])].sup.2] (3.1)

for z, [zeta] [member of] D, . (3.1) satisfies

[[partial derivative].sub.[v.sub.z]][N.sub.1](z, [zeta]) = (z[[partial derivative].sub.z] + [bar.z][[pzrtial derivative].sub.[bar.z]])[N.sub.1](z, [zeta]) = 2 (3.2)

for z [member of] [partial derivative]D, [zeta] [member of] D. But the higher-order Neumann functions are not easy to find in their explicit forms. They may be defined iteratively for n [member of] N where n [greater than or equal to] 2, as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.3)

For the explicit form in the case of n = 2 and n = 3, see [18, 37, 41]. By the aid of (3.3), the higher-order Poisson equation is investigated under the Neumann conditions and the following result is obtained .

Theorem 3.1. The Neumann-n problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

is solvable if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.4)

Here [[alpha].sub.1] = 2 and for 3 [less than or equal to] k

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.5)

The solution is unique and given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Particularly, for the inhomogeneous biharmonic equation, analogous results are presented in [16,18]. We may consult with  for the solutions of Bitsadze equation under Neumann conditions.

The inhomogeneous polyanalytic equation (2.7) with the half-Neumann conditions

z[[partial derivative].sup.v.sub.[bar.z]] [[partial derivative].sub.z]w = [[gamma].sub.v] on [partial derivative]D, [[partial derivative].sup.v.sub.[bar.z]]w(0) = [c.sub.v]

is uniquely solved with some solvability conditions in .

3.1.2 Unbounded Domain Case

The Neumann boundary value problem is considered for the inhomogeneous Cauchy-Riemann equation in a quarter plane and the solvability conditions and solutions are given in explicit form in .

Neumann Problem. Let f [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Find w [member of] [C.sup.1]([[bar.Q].sub.1]; C) satisfying

[w.sub.z] = f in [Q.sub.1], [[partial derivative].sub.y]w = [[gamma].sub.1] for 0 < x, y = 0, [[partial derivative].sub.x]w = [[gamma].sub.2] for 0 < y, x = 0, w(0) = c.

Theorem 3.2. The Neumann problem is uniquely solvable in the weak sense if and only if for any z [not member of] [[bar.Q].sub.1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

holds. The solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

Also, in the upper half plane the Neumann problem is considered for the inhomogeneous Cauchy-Riemann equation and Poisson equation, .

3.1.3 Multiply Connected Domain Case

The Neumann problem for analytic functions, more generally for the inhomogeneous Cauchy-Riemann equation and Poisson equation are investigated in a circular ring domain; the representations to the solutions and solvability conditions are given in an explicit form by Vaitekhovich [54-56].

3.2 Neumann Problem for Complex Linear Elliptic PDEs

For n [member of] N, k, l [member of] [N.sub.0] with (k, l) [not equal to] (n, n) and k + l [less than or equal to] 2n, the operators given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

for a suitable complex valued function f given in D, are the operators related to Neumann problem for generalized n-Poisson equations. In , these operators are shown to be uniformly bounded and uniformly continuous for the case n [less than or equal to] 2 and f [member of] [L.sup.p](D) for p > 2 and bounded in [L.sup.p](D) for f [member of] [L.sup.p](D) and n > 1. Using these operators and a property similar to (2.16), the following problem is investigated.

Neumann Problem. Find w [member of] [W.sup.2n,p](D) as a solution of the linear complex partial differential equation (generalized n-Poisson equation)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.8)

where

[a.sub.kl], [b.sup.kl], f [member of] [L.sup.p](D), (3.9)

and [q.sup.(1).sub.kl] and [q.sup.(2).sub.kl], are measurable bounded functions satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.10)

with Neumann conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.11)

satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.12)

The solvability of this problem is given in the following theorem.

Theorem 3.3. If the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.13)

holds for some K [member of] K([L.sup.p](D)), 0 < p - 2 < [member of], then the equation (3.8) with the boundary conditions (3.11) and normalization conditions (3.12) has a solution of the form w(z) = [S.sub.n,0,0]g(z) + [psi](z), where g [member of] [L.sup.p](D) is a solution of the singular integral equation

(I + N + K)g = f , (3.14)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to the solvability conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.15)

where 0 [less than or equal to] [sigma] [less than or equal to] n - 1 , [[alpha].sub.1] = 2 and for 3 [less than or equal to] k

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (316)

Open Problem 3 4. The Neumann problems for higher-order linear complex partial differential equations are not considered in half plane, quarter plane and concentric rings. These problems can be handled after some studies of the corresponding integral operators

4 Robin Problem for Complex PDEs

4.1 Robin Problem for Complex Model PDEs

4.1.1 Simply Connected Domain Case

Begehr and Harutyunyan  obtained the following result for the Robin problem in D.

Theorem 4.1. The Robin problem for the inhomogeneous Cauchy-Riemann equation in the unit disc

[w.sub.[bar.z]] = f in D, w + [[partial derivative].sub.v]w = [gamma] on D

is uniquely solvable for given f [member of] [L.sup.1](D) [intersection] C ([partial derivative]D), [gamma] [member of] C ([partial derivative]D) if and only if for all [absolute value of z] < 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and the solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the same article, they have also investigated the problem

[w.sup.n.sub.[bar.z]] = f in D,

[[partial derivative].sup.v-1.sub.[bar.z]]w + z[[partial derivative].sup.v-1.sub.[bar.z]]w + [bar.z][[partial derivative].sup.v.sub.[bar.z]]w = [[gamma].sub.v] on D, v = 1,..., n

and obtained the representation of the solution with the corresponding solvability conditions by converting the problem into an equivalent system of n Robin problems for the Cauchy-Riemann operator.

The Robin function for harmonic operator is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and for polyharmonic operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The Robin problem forinhomogeneous harmonic equation is treated in [16,18,36]. For the higher-order Poisson operators the problem is defined by

[([[partial derivative].sub.z] [[partial derivative].sub.z]).sup.n]w = f in D

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], v = 1,... n on [partial derivative]D

This problem is studied by Begehr and Harutyunyan . In the cases n =1 and n = 2, the explicit solutions are given for the corresponding problems.

4.1.2 Unbounded Domain Case

The following Robin boundary value problem is investigated in [Q.sub.1] for the inhomogeneous Cauchy-Riemann equation in .

Robin Problem. Let f [member of] [L.sub.P,2]([Q.sub.1]; C) [intersection] [C.sup.[alpha]] ([[bar.Q].sub.1]; C) for 2 < p, 0 < [alpha] < 1,[[gamma].sub.1], [[gamma].sub.2] [member of] C([R.sup.+]; C) such that for some 0 < [delta] the functions [(1 + t)[delta]][[gamma].sub.2], [(1 + t).sup.[delta]][[gamma].sub.2](t), [(1 + t).sup.[delta]] f(t), [(1 + t).sup.[delta]] f(it) are bounded on [R.sup.+], c [member of] C. Find w [member of] [C.sup.1]([[bar.Q].sub.1];C) satisfying

[w.sub.z] = f in [Q.sub.1], w(0) = c,

w - i[[partial derivative].sub.[gamma]]w = [[gamma].sub.1] for 0 < x, y = 0,

w + [[partial derivative].sub.x]w = [[gamma].sub.2] for 0 < y, x = 0.

Theorem 4.2. This particular Robin problem is uniquely solvable in the weak sense if and only if for [not member of] [[bar.Q].sub.1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

The solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

In Theorem 4.2,

Tf (z) = - [1/[pi]] [integral][[integral].sub.[Q.sub.1]] f([zeta]) [d[xi]d[eta] / [zeta] - z].

4.1.3 Multiply Connected Domain Case

Explicit Robin functions are given for Poisson equation for a circular ring in the complex plane by Begehr and Vaitekhovich . Robin boundary value problem for analytic functions and for the inhomogeneous Cauchy-Riemann equation are investigated in ring domains .

4.2 Open Problems with Robin Conditions

For the higher-order linear differential equations, Robin boundary value problem is not considered particularly. Just in the last section of this survey, it will be considered as a part of a mixed problem in the unit disc. The Robin problem is not considered for higher-order model and linear differential equations in the case of unbounded domains and multiply connected domains. The corresponding Robin functions are not known yet.

5 Schwarz Problem for Complex PDEs

5.1 Schwarz Problem for Complex Model PDEs

5.1.1 Simply Connected Domain Case

The first article in Schwarz problem for analytic functions is given in . The following theorem gives the unique solution of the Schwarz problem for inhomogeneous polyanalytic equation, [10,14,31].

Theorem 5.1. The Schwarz problem for the homogeneous polyanalytic equation in the unit disc D defined by

[[partial derivative].sup.k.sub.[bar.z]]w = f in D , Re [[partial derivative].sup.l.sub.[bar.z]] = 0 on [partial derivative]D , Im [[partial derivative].sup.l.sub.[bar.z]]w(0) = 0 , 0 [less than or equal to] l [less than or equal to] n - 1 ,

is uniquely solvable for f [member of] [L,sup.1](D). The solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Previously the cases of k =1 and k = 2 have been studied .

5.1.2 Unbounded Domain Case

In the upper right quarter plane, the following problem is defined and solved by Abdymanapov et al .

Schwarz Problem. Let f [member of] [L.sub.1]([Q.sub.1]; C),[[gamma].sub.1],[[gamma].sub.2] [member of] C([R.sup.+]; R) be bounded on [R.sup.+] = (0, + [infinity]). Find a solution of

[w.sub.[bar.z]] = f in [Q.sub.1] satisfying

Re w = [[gamma].sub.1] on 0 < x, y = 0,

Im w = [[gamma].sub.2] on 0 < y, x = 0.

Theorem 5.2. The Schwarz problem is uniquely weakly solvable. The solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.1)

In the case of upper half plane H, the following result is obtained in .

Theorem 5.3. Let f [member of] [L.sub.p,2](H; C), 2 < p, [gamma] [member of] C(R), c [member of] R such that [gamma] is bounded on R. Then the Schwarz problem

[w.sub.[bar.z]] = f in H

Re w = [gamma] on H, Im w(i) = c

is uniquely solvable in the weak sense. The solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

5.1.3 Multiply Connected Domain Case

Schwarz problems are solved for the inhomogeneous Cauchy-Riemann equation and Poisson equation in a concentric ring domain (concentric annulus) by Vaitekhovich [54-56]

5.2 Schwarz Problem for Complex Linear Elliptic PDEs

Begehr [10, 31] considered the Schwarz problem for some higher-order equations and proved the solvability of the problem. Schwarz problem for a general linear elliptic complex partial differential equation whose leading term is the polyanalytic operator is discussed in [3,5].

Schwarz Problem. Find w G Wk)P(D) as a solution to the k-th order complex differential equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5.2)

where

[a.sub.ml]. [b.sub.ml] [member of] [L.sup.p](D),f [member of] [L.sup.p](D), (5.3)

and [q.sub.ij] and [q.sub.2j], j = 1,..., k, are measurable bounded functions satisfying

[k.summation over (j=1)] ([absolute value of [q.sub.1j](z)] + [absolute value of [q.dub.2j](z)]) [less than or equal to] [q.sub.0] < 1 (5.4)

satisfying the nonhomogeneous Schwarz boundary conditions

Re [[partial derivative].sup.l]w / [partial derivative][[bar.z].sup.l] = [[gamma].sub.l] on [partial derivative]D, Im [[partial derivative].sup.l]w / [partial derivative][[bar.z].sup.l] (0) = [c.sub.l], 0 [less than or equal to] l [less than or equal to] k - 1 , (5.5)

where [[gamma].sub.1] [member of] C([partial derivative]D; R), [c.sub.l] [member of] R, 0 [less than or equal to] l [less than or equal to] k - 1.

Theorem 5.4. If the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.6)

is satisfied for some [K.sub.1] [member or] K ([L.sup.p](D)), 0 < p - 2 < [member of], then equation (5.2) with the boundary conditions (5.5) has a solution of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5.7)

where [g.sub.1] [member of] [L.sup.p](D), p > 2, is a solution of the singular integral equation

(I + [??] +[??])[g.sub.1] = f, (5.8)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.9)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.10)

In Theorem 5.4, the operators [[??].sub.k] are defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for k [member of] N with [T.sub.0]f (z) = f (z), see [11, 12,27]. [[partial derivative].sup.l.sub.z] [T.sub.k] are weakly singular integral operators for 0 [less than or equal to] l [less than or equal to] k - 1 , while

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.11)

is a Calderon-Zygmund type strongly singular integral operator. [[PI].sub.k] are shown to be bounded in the space [L.sup.p] for 1 < p < [infinity] and in particular their [L.sup.2] norms are estimated in . These operators are investigated by decomposing them into two parts as [[PI].sub.k] = [T.sub.-k,k] + [P.sub.k], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5.12)

which is investigated extensively in [21, 26].

In , the Schwarz problem for the Beltrami equation

[w.sub.[bar.z]] + [cw.sub.z] = f in D

Re w = [gamma], Im w(0) = c on D

is solved as a particular form of the above problem.

The Schwarz problem for Poisson equation is explicitly solved in .

Open Problem 5.5. The Schwarz problem is not considered for higher-order Poisson equations in D, in unbounded domains and in multiply connected domains.

6 Mixed Type Problems for Complex PDEs

6.1 Mixed Type Problems for Complex Model PDEs

6.1.1 Simply Connected Domain Case

In order to state and solve the mixed problems containing Schwarz, Neumann, Dirichlet and Robin problems, we define the following polyharmonic hybrid Green type functions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which are obtained by convoluting Green, Neumann and Robin functions iteratively, 

The integral operators

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are defined in relation to the mixed problems, . The higher-order model differential equation with mixed boundary conditions is discussed in the following theorem.

Theorem 6.1. The mixed problem for model equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

for f [member of] [L.sup.p](D), is uniquely solvable iff

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.1)

for a suitable [??]. The solution is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.2)

Remark 6.2. The problem given in Theorem 6.1 covers some mixed problems given in [16,18] for bi-Poisson equation and in [30,48] for inhomogeneous polyanalytic and polyharmonic equations with homogeneous boundary conditions cases.

6.2 Mixed Type Problems for Complex Elliptic Linear PDEs

We consider the following mixed problem for higher-order complex differential equation of arbitrary order .

Problem M. Find w [member of] [W.sup.m+n,P](D) as a solution to the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.3)

satisfying boundary conditions

Re [[partial derivative].sup.[mu].sub.[bar.z]]w = 0 on [partial derivative]D,

Im [[partial derivative].sup.[mu].sub.[bar.z]]w(0) = 0, 0 [less than or equal to] [mu] [less than or equal to] n - m - 1 (6.4)

[[partial derivative].sup.[mu].sub.z][[partial derivative].sup.[mu]+n-m.sub.[bar.z]]w = 0, 0 [less than or equal to] [mu] [less than or equal to] a - 1 on [partial derivative]D (6.5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6.7)

where

[a.sub.kl], [b.sub.kl], f [member of] [L.sup.p](D) (6.8)

and [q.sup.(1).sub.kl] and [q.sup(2).sub.kl], are measurable bounded functions

[summation over (k+l = m+n (k,l) [not equal to] (m,n))] ([absolute value of [q.sup.(1).sub.kl](z)] + [absolute value of [q.sup.(2).sub.kl](z)]) [less than or equal to] [q.sub.0] < 1 (6.9)

and l < n - m for k + l [greater than or equal to] n - m, k + l < m + n for l [greater than or equal to] n - m, k + l = m + n for l [greater than or equal to] n - m .

We transform the Problem M to a singular integral equation.

Lemma 6.3. The mixed problem (6.3), (6.4), (6.5), (6.6) and (6.7) is equivalent to the singular integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.10)

Solvability of the problem is given in the next theorem.

Theorem 6.4. If the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.11)

is satisfied, then equation (6.3) with the conditions (6.4), (6.5), (6.6) and (6.7) has a solution of the form w = [B.sup.0,0.sub.a,b,m-a-b,n-m]g, where g [member of] [L.sup.p](D) is a solution of the singular integral equation (6.10) with p > 2 and g satisfies the solvability condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6.12)

where [([[partial derivative].sub.z][[partial derivative].sub.z]).sup.m-a-b]g = g satisfy the conditions

[[partial derivative].sup.[mu]+a+b.sub.z] [[partial derivative].sup.[mu]+n-m+a+b.sub.[bar.z]]g + [[partial derivative].sub.[v.sub.z]] ([[partial derivative].sup.[mu]+a+b.sub.z] [[partial derivative].sup.[mu]+n-m+a+b.sub.[bar.z]])g = 0, (6.13)

0 [less than or equal to] [mu] [less than or equal to] m - a - b - 1 on [partial derivative]D.

Open Problem 6.5. On unbounded domains and multiply connected domains, mixed type problems are not studied for higher-order linear equations and model equations.

Acknowledgment

The authors would like to thank the referees for the careful reviews and the valuable comments.

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Umit Aksoy

Atilim University

Department of Mathematics

06836 Incek, Ankara, Turkey

uaksoy@atilim.edu.tr

A. Okay Celebi

Yeditepe University

Department of Mathematics

34755 Kayisdagi, Istanbul, Turkey

acelebi@yeditepe.edu.tr

Received April 7, 2010; Accepted May 6, 2010 Communicated by Agacik Zafer
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