Abstract
In this study, we have determined Cauchy–Pompeiu representation formula in octant unit ring by modulation the main Cauchy–Pompeiu representation; this modulation was obtained by applying the parqueting-reflection method. Moreover, the boundedness of the modulated Cauchy–Pompeiu’s operator in is proved by applying Schmitz’s inequality.
1. Introduction
Many studies have investigated the modulation of Cauchy–Pompeiu representation formula in different particular domains, e.g., the ring [1], half-disk and half-ring [2], quarter-ring [3], half-hexagon [4], and lens and lune [5], to obtain an explicit solution to the Schwartz problem [6]:
The Cauchy–Pompeiu representation formula is the fundamental tool for solving Schwartz problem which just has to be properly modified.
The main Cauchy–Pompeiu representation formula is given as follows.
Theorem 1 (Cauchy–Pompeiu representation [1]). Any function for a regular complex domain can be represented as follows:
The area integral appearing in the complex Cauchy–Pompeiu representation defines a weakly singular integral operator ; we call it as “the Cauchy–Pompeiu’s operator” [1]:
Its properties have been studied by Vekua [7]. The operator helps in solving Schwartz problem because it holdsin a weak sense (the derivative in Sobolev space [6]).
When Cauchy–Pompeiu representation formula is modulated, the operator is also modulated.
In this study, we modulate the Cauchy–Pompeiu representation formula in octant unit ring by using the parqueting-reflection method; then, we define the modulated operator in , and we prove the boundedness of this operator.
2. The Octant Unit Ring
We define the octant unit ring as follows (Figure 1):
First, we will find the reflections of the domain at its boundaries.
Let be the segment , be the circular arc , where , be the segment , and be the circular arc , where it is .
Let ; then, (1) The point is reflected at the segment onto . (2) The points are reflected at y-axis onto , respectively. (3) The points are reflected at x-axis onto , respectively.
We obtain the following eight sets:
We can easily notice that (see Figure 2)
The points are reflected at onto the points:
The points are reflected at onto the points:where
Point (7) is reflected at onto the points:
Point (10) is reflected at onto
Point (11) is reflected at onto
When continuing that way, we obtainwhere .
Point (9) is reflected at onto
Points (14) are reflected at onto
When continuing that way, we obtainwhere
Each point of sets (13) and (16) belong to a different domain ; this domains parquets the complex plane:
3. Main Result
In this section, we have determined the Cauchy–Pompeiu representation formula in octant unit ring by applying Theorem 1 at points (13) and (16).
Theorem 2. Any function can be represented as follows:where , and .
Proof. Using Theorem 1 for the points , for every , we findWhen adding (19)–(22) and fixing the terms, we obtainSimilarly, when using Theorem 1 for the points, , for , and fixing the terms, we findWhen using Theorem 1 for the points, , for , and fixing the terms we also findFinally, when using Theorem 1 for the points, we obtainFor , we findSince , we can rewrite (23)–(31) as follows:Adding (32) and (33) givesAdding (36) and (37) givesAdding (34) and (35) givesAdding (38) and (39) givesWhen taking the complex conjugate of (41) and (43), we can obtainWhen adding (40)–(42) and then subtracting (44) and (45) from the result, we can obtainWe noticeBy putting on and fixing the terms, we end up the proof.
The Cauchy–Pompeiu’s operator in is the area integral in the modulated Cauchy–Pompeiu representation in ; it is given by the following formula:
4. The Operator
The Cauchy–Pompeiu’s operator in includes an infinite series; therefore, we have to prove its convergence; this will be done through the following two lemmas.
Lemma 1. For every , with , we have the following:where .
Proof. Let and ; then,where ,Since , for every , we haveBy the symmetry properties, we haveThis leads tofor every .
On the contrary, we haveBy the symmetry properties, we haveThis leads tofor every .
We haveTherefore,Sincewe can obtainBy symmetry properties, we havefor every .
This means
Lemma 2. For every , with , , the series is as follows:Is it uniformly converged.
Proof. First, we haveFrom Lemma 1, we haveSecond, we haveFrom Lemma 1, we haveThird, we haveFrom Lemma 1, we haveFourth, we haveFrom Lemma 1, we haveWhen using (66), (68), (70), and (72), we can obtainAnd, the series is uniformly converged.
5. The Boundedness of Cauchy–Pompeiu’s Operator in
The boundedness of the operator is proved by Tutschke and Mshimba in [8] by applying the Schmitz inequality. In this section, we will use the same technique to prove the boundedness of the modulated Cauchy–Pompeiu’s operator in .
Theorem 3. (Schmitz’s inequality [8]). For any regular domain , we havewhere , , is a real number consist , and is the area of the domain .
Theorem 4. Let be a bounded domain; then, for , , T is a completely linear operator from into with .
Theorem 5. The Cauchy–Pompeiu’s operator in the octant ring iswhich is bounded, where and .
Proof. First, we haveFrom Lemmas 1 and 2, we can writewhere , with ,whereBy Schmitz’s inequality, we haveOn the contrary, we havewhere .
Therefore, we haveAnd, the operator is bounded.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.