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Neumann and Dirichlet Problems for the Cauchy–Riemann and the Poisson Equations in the

Partial Eclipse Domain

Ali Darya∗, Nasir Taghizadeh
Faculty of Mathematics Sciences, University of Guilan, Rasht, 19141, Iran

alidarya@phd.guilan.ac.ir, taghizadeh@guilan.ac.ir
∗Correspondence: alidarya@phd.guilan.ac.ir

Abstract. In this paper, we consider the Neumann boundary value problem and the Dirichlet boundaryvalue problem for complex partial differential equations in the partial eclipse domain. First, By theparqueting–reflection principle and the Cauchy–Pompeiu formula, a modified integral representationformula in the partial eclipse domain is constructed. Then, we explicitly solve the Neumann problemfor the homogeneous equation and discuss the solvability conditions. Moreover, we investigate theDirichlet problem for the Poisson equation in the partial eclipse domain. In other words, with thehelp of the Green’s function, we provide a unique solution for the Dirichlet boundary value problemfor the Poisson equation and consider boundary behavior.

1. Introduction
Mathematical analysis is an active branch in mathematics which has grown significantly. It hasindeed flourished, playing a pivotal role in advancement of both pure and applied mathematics.Its growth can be seen in the development of new techniques for solving differential equations,advancements in complex analysis, and profound contributions to functional analysis. This expan-sion has not only deepened our theoretical understanding but has also paved the way for practicalapplications in fields like mathematical physics, fluid dynamics, engineering, etc [1, 3, 6, 9].The theory of boundary value problems for partial differential equations is a key area in math-ematical analysis and mathematical physics. The theory often focuses on conditions under whichsolutions exist and are unique. These conditions can depend on the properties of the differential op-erator, the domain, and the boundary conditions. Boundary value problems are critical in the studyof partial differential equations, as they involve finding a solution to a partial differential equationthat satisfies certain conditions at the boundaries of the domain. The most common boundary valueproblems are the Dirichlet, the Neumann and the Schwarz problems. In particular, the Dirichlet
Received: 2 Dec 2024.
Key words and phrases. boundary value problem; Neumann problem; Dirichlet problem; partial eclipse domain.1

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https://doi.org/10.28924/ada/ma.5.10


Eur. J. Math. Anal. 10.28924/ada/ma.5.10 2problem specifies the value of the function on the boundary and the Neumann problem specifiesthe values of the derivative (normal to the boundary).Several analytical and computational methods are used to solve boundary value problems, includ-ing integral representation formulas, the Green’s functions, etc. Integral representation formulasare crucial tools in solving complex boundary value problems, especially for analytic and harmonicfunctions. The Green’s functions are used to construct solutions for complex partial equation withboundary conditions. For the Dirichlet problem, the Green’s function represents the influence of apoint source on the boundary and is used to build the solution, see [1–9,16].In recent years, many mathematician have studied boundary value problems for complex partialdifferential equations and numerous results have been obtained, can be referenced in [1–16]. In2024, we introduced a new domain called partial eclipse and also investigated the Schwarz andthe Dirichlet boundary value problem for Cauchy–Riemann equations in the partial eclipse domain,( Ali Darya and Nasir Taghizadeh, “Schwarz and Dirichlet problems for complex partial differentialequations in the eclipse domain,” J Math Sci ) see [1].In the present paper, we consider the Neumann problem for first-order partial differential equa-tion and the Dirichlet problem for second-order partial differential equation in the partial eclipsedomain. In other words, we first solve the Neumann problem the homogeneous Cauchy–Riemannequation and discuss the solvability conditions. In the next step, With the help of the Green’s func-tion, we construct a unique solution for the Dirichlet problem for the Poisson equation in the partialeclipse domain and investigate the Dirichlet problem. In particular, we study the boundary behavior.
Let M be the partial eclipse domain in the complex plane C defined by [1]

M =
{
z ∈ C : |z − ai | <

√
2a, |z + ai | >

√
2a
}

where C1 =
{
z : |z − ai | =

√
2a
} and C2 =

{
z : |z + ai | =

√
2a
} are two circles with the sameradius and the boundary of M is denoted by ∂M . Due to the fact that the real number a has apositive arbitrary amount, the desired partial eclipse can be created by choosing a. So the lengthof the borders of the partial eclipse changes according to the number.

The parqueting–reflection principle is a technique used to solve boundary value problems for com-plex partial differential equations, particularly for domains with complex geometries. The methodinvolves reflecting the domain across its boundaries to simplify the problem. By doing so, theboundary conditions can be transformed into simpler forms, making it easier to find solutions. Byusing the reflections, one can construct solutions for complex boundary value problems such as theSchwarz, the Dirichlet and Neumann problems. This is particularly useful for domains composedof circular arcs and straight lines [1–4,6, 8–10].

https://doi.org/10.28924/ada/ma.5.10


Eur. J. Math. Anal. 10.28924/ada/ma.5.10 3In this section, using the parqueting–reflection method for the introduced domain M, we achievecoverage for the entire complex plane C. This coverage is obtained from reflections with threerepetitions.The reflection of z ∈ M at C1, is
|z − ai | =

√
2a⇒ (z − ai) (z̄ + ai) = 2a2 ⇒ z∗1 =

ai z̄ − a2 + 2a2

z̄ + ai
⇒ z∗1 =

ai z̄ + a2

z̄ + ai
.

Similarly, the reflections of z∗1 and z at C2, are the points,
z∗2 =

a2

z
, z∗3 =

−ai z̄ + a2

z̄ − ai .

Those reflections produce a parqueting of the entire complex plane and those points will alsobe needed for constructing the integral representation formula for M . To solve boundary valueproblems for partial differential equations, the integral formula should be appropriately modifiedaccording to the type of problem and partial differential equation, [1–4,7, 8, 10].The Neumann boundary value problem is a classical in the field of complex analysis and partialdifferential equations. It involves finding a function that satisfies complex partial differential equa-tions within a given domain and whose normal derivative on the boundary of the domain matchesa specified function. The Neumann boundary value problem is significant, because it is a gatewayto understanding the deeper intricacies of complex functions and broader field of complex analysis.The insights gained from solving these problems can lead to advancement in both mathematicaltheory and practical applications, see [2, 6, 7, 9, 11].In this section, we investigate the Neumann boundary value problem for the homogeneousCauchy– Riemann equation. The fundamental tool for complex boundary value problems is theintegral representation formula which just has to be properly modified.
Theorem 1.1. Any ω ∈ C1(M;C)

⋂
C(M;C) can be represented as

ω(z) =
1

2πi

∫
∂M

ω(ζ)

[
1

ζ − z +
z

ζz − a2

]
dζ

−
1

π

∫
M

ωζ̄(ζ)

[
1

ζ − z +
z

ζz − a2

]
dξdη, (1.1)

where ζ = ξ + iη.

Proof. The Cauchy–Pompieu formula
1

2πi

∫
∂M

ω(ζ)
dζ

ζ − z −
1

π

∫
M

ωζ̄(ζ)
dξdη

ζ − z =

{
ω(z) z ∈ M,

0 z /∈ M,

applied to z ∈ M and z∗2 /∈ M, respectively, gives the following equalities:
ω(z) =

1

2πi

∫
∂M

ω(ζ)
dζ

ζ − z −
1

π

∫
M

ωζ̄(ζ)
dξdη

ζ − z , (1.2)

https://doi.org/10.28924/ada/ma.5.10


Eur. J. Math. Anal. 10.28924/ada/ma.5.10 4

0 =
1

2πi

∫
∂M

ω(ζ)
zdζ

ζz − a2
−

1

π

∫
M

ωζ̄(ζ)
zdξdη

ζz − a2
, (1.3)

adding the resulting above relations, leads to claimed the integral representation formula. �

Next, we state the Neumann boundary value problem for Cauchy–Riemann equation in the par-tial eclipse domain as follows.
Neumann boundary value problem: Find an analytic function in the partial eclipse domain, i.e. asolution to Cauchy–Riemann equation, satisfying,

∂vzω = γ, on ∂M, γ ∈ C(∂M;C).

The classical Neumann problem involves finding a function that satisfies Cauchy–Riemann equa-tion in a domain, along with prescribed values of its normal derivative on the boundary. When ad-justed for analytic functions, the Neumann condition is often reformulated to align with the conceptof analyticity, ensuring the function meets the criteria for complex differentiability [2,6,7,9,11,16].To formulate the Neumann boundary value problem, we need to define the outward normalderivative at the boundary of M. The normal derivative on the boundary of is given by the formulas,
∂vzω =

 ( z−ai√
2a

)ωz , on C1,

( z+ai√
2a

)ωz , on C2.

Now, according to the above definition, we state and prove the following theorem.
Theorem 1.2. The Neumann boundary value problem

ωz̄ = 0, z ∈ M,

∂vzω = γ, z ∈ ∂M, (1.4)
ω(t) = c,

where t ∈ M, c ∈ C and

∂vzω =

 ( z−ai√
2a

)ωz , on C1,

( z+ai√
2a

)ωz , on C2,

is solvable, if and only if, for z ∈ M,

1

2πi

∫
∂M

γ(ζ)

[
z̄ − ai

ζ(z̄ − ai) + ai z̄ − a2
+

z̄ + ai

ζ(z̄ + ai)− ai z̄ − a2

]
dζ = 0,

and its solution is

ω(z) =
1

2πi

∫
∂M

γ(ζ)

[
z − t
ζ
− log(

ζ − z
ζ − t ) +

a2

ζ2
log(

ζz − a2

ζt − a2
)

]
dζ,

where ζ = ξ + iη.

https://doi.org/10.28924/ada/ma.5.10


Eur. J. Math. Anal. 10.28924/ada/ma.5.10 5

Proof. Suppose ω is a solution to the Neumann problem. Introducing a new function φ = ωz , since
φ is an analytic function, φ = ωz is a solution to the following problem,

φz̄ = 0, in M,

φ = ωz , on ∂M, (1.5)where ωz on ∂M is represented by,
ωz(z) =

 ( z̄−ai√
2a

)γ, on C1,

( z̄+ai√
2a

)γ, on C2.Equation (5) is equivalent to the Dirichlet boundary value problem for the homogeneous Cauchy–Riemann equation. By Theorem 3.2 in [1], the above Dirichlet problem is solvable if and only if
1

2πi

∫
∂M

γ(ζ)

[
z̄ − ai

ζ(z̄ − ai) + ai z̄ − a2
+

z̄ + ai

ζ(z̄ + ai)− ai z̄ − a2

]
dζ = 0,

then, the unique solution is given by,
ωz(z) =

1

2πi

∫
∂M

γ(ζ)

[
1

ζ − z +
z

ζz − a2

]
dζ. (1.6)

The primitive of the function in (6) is
ω(z) =

1

2πi

∫
∂M

γ(ζ)

[
z

ζ
− log(ζ − z) +

a2

ζ2
log(ζz − a2)

]
dζ + c.

Define c as
c = −

1

2πi

∫
∂M

γ(ζ)

[
t

ζ
− log(ζ − t) +

a2

ζ2
log(ζt − a2)

]
dζ.

This completes the proof. �

2. The Dirichlet Problem for M
In this section, we consider the Dirichlet problem for the Poisson equation in the partial eclipsedomain. In order to treat the Dirichlet boundary value problem for second order complex partialdifferential equations some special kernel functions, the Green functions, have to be constructed.It is essential to construct the Green’s functions tailored to the specific domain. These Green’sfunctions serve as fundamental tools, transforming the differential equation into an integral formthat can be more easily analyzed and solved, see [5–9]. The harmonic Green function for the partialeclipse domain M is

G1(z, ζ) = log

∣∣∣∣ ζ̄(z + ai)− aiz − a2

ζ − z
ζ̄(z − aiz) + aiz − a2

ζz − a2

∣∣∣∣2 .

https://doi.org/10.28924/ada/ma.5.10


Eur. J. Math. Anal. 10.28924/ada/ma.5.10 6The outward normal derivative of the boundary ∂M is given by
For z ∈ ∂M ∩ C1, that is, |z − ai | =

√
2a, we have

∂vzG1(z, ζ) =

(
(
z − ai√

2a
)∂z + (

z̄ + ai√
2a

)∂z̄

)
G1(z, ζ),

and for z ∈ ∂M ∩ C2, that is, |z + ai | =
√

2a, we have
∂vzG1(z, ζ) =

(
(
z + ai√

2a
)∂z + (

z̄ − ai√
2a

)∂z̄

)
G1(z, ζ).

The Harmonic Green functions play an essential role in solving the Dirichlet boundary valueproblem for second order complex partial differential equations. The next theorem contains arepresentation formula for a class of functions via the Green function, which is used to solve theDirichlet problem for Poisson equation (see [5, 7, 9]).
Theorem 2.1. Let Ω ⊂ C be a regular domain, and let G1 be the harmonic Green function for Ω

Then any ω ∈ C2(Ω;C) ∩ C1(Ω;C) can be represented as follows:

ω(z) = −
1

4π

∫
∂Ω

ω(ζ)∂vζG1(z, ζ)dtζ −
1

π

∫
Ω

ωζζ̄(ζ)G1(z, ζ)dξdη,

where v is the outward normal derivative on ∂Ω and t is the arc length parameter [5, 7, 9].

Therefore, based on Theorem 3, the explicit form of the Green representation formula for thepartial eclipse domain is as following:
ω(z) =

1

2πi

∫
∂M

⋂
C1

ω(ζ)

(
ζ + ai

ζ − z +
ζ̄ − ai
ζ̄ − z̄

− 1

+
z(ζ − ai)
ζz − a2

+
z̄(ζ̄ + ai)

ζ̄z̄ − a2
− 1

)
dζ

ζ − ai

+
1

2πi

∫
∂M

⋂
C2

ω(ζ)

(
ζ + ai

ζ − z +
ζ̄ − ai
ζ̄ − z̄

− 1

+
z(ζ + ai)

ζz − a2
+
z̄(ζ̄ − ai)
ζ̄z̄ − a2

− 1

)
dζ

ζ + ai

−
1

π

∫
M

ωζζ̄(ζ)G1(z, ζ)dξdη. (2.1)
In fact formula (7) provides a solution to the Dirichlet problem for the Poisson equation in M .
Theorem 2.2. The Dirichlet problem for the Poisson equation in M

ωzz̄ = f , z ∈ M, f ∈ C(M;C),

ω = γ, on ∂M, γ ∈ C(∂M;C), (2.2)

https://doi.org/10.28924/ada/ma.5.10


Eur. J. Math. Anal. 10.28924/ada/ma.5.10 7

is uniquely solvable and the solution is given by

ω(z) =
1

2πi

∫
∂M

⋂
C1

γ(ζ)

(
ζ − ai
ζ − z +

ζ̄ + ai

ζ̄ − z̄
− 1

+
z(ζ − ai)
ζz − a2

+
z̄(ζ̄ + ai)

ζ̄z̄ − a2
− 1

)
dζ

ζ − ai

+
1

2πi

∫
∂M

⋂
C2

γ(ζ)

(
ζ + ai

ζ − z +
ζ̄ − ai
ζ̄ − z̄

− 1

+
z(ζ + ai)

ζz − a2
+
z̄(ζ̄ − ai)
ζ̄z̄ − a2

− 1

)
dζ

ζ + ai

−
1

π

∫
M

f (ζ)G1(z, ζ)dξdη. (2.3)
where ζ = ξ + iη.

Proof. By the properties of the Green function and the harmonicity of the boundary integrals ωis seen to be a solution to the Poisson equation (see [7]). So, it remain remains to check theboundary relation. The study of integral boundary behavior requires calculations in different partsof the boundary.
Since, for z ∈ ∂M⋂C1,

Case1: ζ ∈ C1,

z̄(ζ̄ + ai)

ζ̄z̄ − a2
=
aiz−a2

ζz − a2
.

Case2: ζ ∈ C2,

ζ̄ − ai
ζ̄ − z̄

=
−aiz − a2

ζz − a2
,

z̄(ζ̄ − ai)
ζ̄z̄ − a2

=
−z − ai
ζ − z .

Thus, on ∂M⋂C1,

lim
z→ζ

ω(z) = lim
z→ζ

1

2πi

∫
∂M

⋂
C1

γ(ζ)

[
ζ − ai
ζ − z +

ζ̄ + ai

ζ̄ − z̄
− 1

]
dζ

ζ − ai

= lim
z→ζ

1

2πi

∫
C1

Υ(ζ)

[
ζ − ai
ζ − z +

ζ̄ + ai

ζ̄ − z̄
− 1

]
dζ

ζ − ai ,where
Υ(ζ) =

{
γ(ζ) ζ ∈ ∂M

⋂
C1,

0 ζ ∈ C1\(∂M).So based on the properties of the Poisson kernel for C1, [6]
lim
z→ζ

ω(z) = γ(ζ),

https://doi.org/10.28924/ada/ma.5.10


Eur. J. Math. Anal. 10.28924/ada/ma.5.10 8follows for ζ ∈ ∂M⋂C1 up to the corner points ±a of the domain M, because Υ fails to be con-tinuous there if γ not accidentally vanishes at these points.By the same way, for z ∈ ∂M⋂C2,

Case 1: ζ ∈C1,

ζ̄ + ai

ζ̄ − z̄
=
aiz − a2

ζz − a2
,

z̄(ζ̄ + ai)

ζ̄z̄ − a2
=
−z + ai

ζ − z .

Case 2: ζ ∈ C2,

z̄(ζ̄ + a)

ζ̄z̄ − a2
=
−aiz − a2

ζz − a2
.

Thus, on ∂M⋂C2,

lim
z→ζ

ω(z) = lim
z→ζ

1

2πi

∫
∂M

⋂
C2

γ(ζ)

[
ζ + ai

ζ − z +
ζ̄ − ai
ζ̄ − z̄

− 1

]
dζ

ζ + ai

= lim
z→ζ

1

2πi

∫
C2

Υ(ζ)

[
ζ + ai

ζ − z +
ζ̄ − ai
ζ̄ − z̄

− 1

]
dζ

ζ + ai
,

where
Υ(ζ) =

{
γ(ζ) ζ ∈ ∂M

⋂
C2,

0 ζ ∈ C2\(∂M).So based on the properties of the Poisson kernel for C2,

lim
z→ζ

ω(z) = γ(ζ),

follows for ζ ∈ ∂M⋂C2 up to the corner points ±a of the domain M, because Υ fails to be con-tinuous there if γ not accidentally vanishes at these points.
Now, we consider the boundary behaviors at the tips ±a. We represent the constant function1 as

1 =
1

2πi

∫
∂M

⋂
C1

[
ζ − ai
ζ − z +

ζ̄ + ai

ζ̄ − z̄
− 1 +

z(ζ − ai)
ζz − a2

+
z̄(ζ̄ + ai)

ζ̄z̄ − a2
− 1

]
dζ

ζ − ai

+
1

2πi

∫
∂M

⋂
C2

[
ζ + ai

ζ − z +
ζ̄ − ai
ζ̄ − z̄

− 1 +
z(ζ + ai)

ζz − a2
+
z̄(ζ̄ − ai)
ζ̄z̄ − a2

− 1

]
dζ

ζ + ai
.

Multiplying this relation with γ(±a) and subtracting the resulting equation from ω(z) shows for
z ∈ ∂M

⋂
C1,
lim
z→ζ

(ω(z)− γ(±a)) = lim
z→ζ

1

2πi

∫
∂M

⋂
C1

γ̃(ζ)

[
ζ − ai
ζ − z +

ζ̄ + ai

ζ̄ − z̄
− 1

]
dζ

ζ − ai ,

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Eur. J. Math. Anal. 10.28924/ada/ma.5.10 9

where γ̃(ζ) = γ(ζ)− γ(±a) and γ̃(±a) = 0,

lim
z→±a

ω(z) = γ(±a).

Similarly, for z ∈ ∂M⋂C2,
lim
z→ζ

(ω(z)− γ(±a)) = lim
z→ζ

1

2πi

∫
∂M

⋂
C2

γ̂(ζ)

[
ζ + ai

ζ − z +
ζ̄ − ai
ζ̄ − z̄

− 1

]
dζ

ζ + ai
,

where γ̂(ζ) = γ(ζ)− γ(±a) and γ̂(±a) = 0,therefore,
lim
z→±a

ω(z) = γ(±a).

Therefore, the proof is finished. �

References
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https://doi.org/10.1080/17476933.2013.799152.[11] İ. Gençtürk, K. Koca, Neumann Boundary Value Problem for Bitsadze Equation in a Ring Domain, J. Anal. 28 (2020),799–815. https://doi.org/10.1007/s41478-019-00202-3.[12] P.N. Ivanshin, E.A. Shirokova, The Solution of a Mixed Boundary Value Problem for the Laplace Equation in aMultiply Connected Domain, Iss. Anal. 26 (2019), 51–66. https://doi.org/10.15393/j3.art.2019.5570.[13] S.G. Pyatkov, Boundary Value and Inverse Problems for Some Classes of Nonclassical Operator-Differential Equa-tions, Sib. Math. J. 62 (2021), 489–502. https://doi.org/10.1134/S0037446621030125.[14] K.B. Sabitov, Dirichlet Problems for Mixed-Type Equations with Fractional Derivatives, Russ. Math. 66 (2022),71–81. https://doi.org/10.3103/S1066369X22090080.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.10 10

[15] K. Ravikumar, K. Ramkumar, D. Chalishajar, Existence and Stability Results for Second-Order Neutral StochasticDifferential Equations With Random Impulses and Poisson Jumps, Eur. J. Math. Anal. 1 (2021), 1–18. https:
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https://doi.org/10.28924/ada/ma.1.1
https://doi.org/10.28924/ada/ma.1.1

	1. Introduction
	2. The Dirichlet Problem for M
	References

