Dirichlet Type Problem for 2D Quaternionic Time-Harmonic Maxwell System in Fractal Domains

Pérez, Yudier Peña; Blaya, Ricardo Abreu; Bosch, Paul; Reyes, Juan Bory

doi:https://doi.org/10.1155/2020/4735357

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2020 | Article ID 4735357 | https://doi.org/10.1155/2020/4735357

Dirichlet Type Problem for 2D Quaternionic Time-Harmonic Maxwell System in Fractal Domains

Yudier Peña Pérez,¹Ricardo Abreu Blaya,²Paul Bosch,³and Juan Bory Reyes¹

Academic Editor: Ping Li

Received26 Mar 2019

Revised31 Aug 2019

Accepted10 Sept 2019

Published31 Jan 2020

Abstract

We investigate an electromagnetic Dirichlet type problem for the 2D quaternionic time-harmonic Maxwell system over a great generality of fractal closed type curves, which bound Jordan domains in . The study deals with a novel approach of -summability condition for the curves, which would be extremely irregular and deserve to be considered fractals. Our technique of proofs is based on the intimate relations between solutions of time-harmonic Maxwell system and those of the Dirac equation through some nonlinear equations, when both cases are reformulated in quaternionic forms.

1. Introduction

A theory of hyperholomorphic functions of two real variables is the most natural and close generalization of complex analysis that preserves many of its important features. Some integral representation type formulas in closed Jordan rectifiable curves are proved in [1, 2] and [3, Appendix 4]. Applications in physical problems with elliptic geometries and potential theory can be found in [4–6].

The Maxwell equations govern the behavior of the electromagnetic field. Despite the fact that these equations are more than hundred years old, they still are subject to changes in content, notation, and frameworks.

The quaternionic analysis gives us a tool of wider applicability for the study of electromagnetic boundary value problems. In particular, a quaternionic hyperholomorphic approach to time-harmonic solutions of the Maxwell system is established in [3, 7–9] and the references given there.

These studies confine attention to Lipschitz domains in the worst case scenario. For pure and applied mathematical interest, in [10] some boundary value problems for time-harmonic electromagnetic fields on the more challenging case of domains with fractal boundaries are discussed. Results concerning boundary value problems for the time-harmonic Maxwell system along classical lines can be found in [11]. An overview of different methods that are useful in the analysis of the time-harmonic Maxwell equations was given in [12].

The main goal of this work is the study of an electromagnetic Dirichlet type problem for a domain with fractal boundary in . For a deeper discussion of some electromagnetic problems in two dimensions, we refer the reader to [13–16].

Our main motivation for the introduction of quaternionic analysis in electromagnetics is the difficulty in solving the Maxwell equations in fractal domains involving boundary condition on fractal boundaries, which requires the use of very advanced mathematical techniques.

The outline of the paper is as follows: In Section 2 we provide an outlook to the basics of quaternionic analysis and elements of fractals geometry; a new hyperholomorphic Cauchy type integral for a domain with summable boundary in is described in Section 3, where we also state theoretical results on integral representation formulas in domains bounded by such curves. Results on jump boundary value problems across an -summable boundaries of domains in , as well as certain Dirichlet type problems for hyperholomorphic solutions of two dimensional Helmholtz equation are presented and discussed in Section 4. Finally, Section 5 analyzes some Dirichlet type problems involving electromagnetism in the form of 2D quaternionic time-harmonic Maxwell system on a domain with -summable boundary in .

2. Preliminaries

The noncommutative and associative algebra with zero divisors of the complex quaternions is denoted by . For each complex quaternion , one has where , is the multiplicative unit and are standard quaternionic imaginary units. By definition, the complex imaginary unit satisfies

Let , where is called scalar part and is called vector part of the quaternion . The module of coincides with its Euclidean norm: and the quaternionic conjugate is defined by . If then is called a purely vector quaternion.

The multiplication of two quaternions can be rewritten in vector terms:

where and are the scalar and the usual cross product in respectively. We shall frequently write for a typical point of .

Let a domain , we will consider -valued functions:

Properties of continuity, differentiability and integrability of have to be understood component-wise. The set of times continuously differentiable functions is denoted by , .

Given , let such that . This generates the (left and right) 2D Helmholtz operator, which acting on are given by and respectively. Here and subsequently, , and for , and .

Additionally, the following partial differential operators will be considered

It follows easily that,

Set . Therefore, the Helmholtz operator can be factorized as follows:

Definition 1. A function is called hyperholomorphic if it satisfies on .

If we write , then we obtain by straightforward calculation

From [17], if , a fundamental solution of is given by

where

and is the Hankel function of the kind and of order (see [18]).

If , the functions and are fundamental solutions of the Helmholtz operator .

By (7), the fundamental solution of the operator , i.e., the quaternionic Cauchy kernel, is defined as

Hence

Remark 2.1. In what follows we suppose that .

Let us now take a quick look at the notion of majorant, with the purpose of considering the generalized Hölder spaces, see [19, 20]. Let be a continuous increasing function on such that and is nonincreasing. One such function said to be a majorant. Note that , , are majorants.

In what follows will denote a positive constant, not necessarily the same at different occurrences.

Suppose be a bounded set. The generalized Hölder space, denoted by , is defined to be the family of all -valued functions on such that

where is a given majorant. For , , we write instead of .

We recall that a nonnegative and almost increasing (or almost decreasing) function means that there exists such that for all (), respectively.

Following [21, Definition 1.1], we say that a majorant has order if there exists a and a positive real number such that

This guarantees that .

If a majorant has order , then we let and we will use the symbol instead .

The Whitney extension theorem (see [22]) in the quaternionic analysis context is stated as follows.

Theorem 1. Suppose be a compact set and let There exists a compactly supported function such that
(i);(ii);(iii).The function is called a Whitney type extension of . With the notation we stand for the distance between the subsets and of .

2.1. Elements of Fractal Geometry

Let be a gauge function, i.e., a continuous and nondecreasing interval function with .

In [23] a variation of the geometric concept of -summability, which is due to Harrison and Norton in [24] is introduced.

Definition 2 [23, Definition 1]. Let ℎ be a gauge function. The set is called -summable if there exists such that
where stands for the least number of open balls of radius needed to cover . When with , we recover the d-summability of .

Definition 2 is unchanged if with is replaced by the number of -squares intersecting . By a -square we mean one of the form

where are integers.

We follow [22] considering the Whitney decomposition of

The squares in have disjoint interiors and satisfy

Here and subsequently, stands for the diameter of a bounded set .

3. Hyperholomorphic Cauchy Type Integral for Fractal Curves

The Cauchy type integral associated to quaternionic analysis has been involved recently with fractional metric dimensions and fractals, see [9, 10, 25]. In this section, we define and characterize the hyperholomorphic Cauchy type integral on fractal type curves. Before giving the definition, we will state some preliminary results.

Let , , the set of -integrable functions, the Teodorescu transform for , is given by

Looking at the kernel function we can decompose it in the following way

where and the continuous function .

Remark 3.1. It is to be expected that shares many of the properties of . For example, , if for , by analogy with [7, Subsection 8.1].

In what follows, given a majorant and , we will take to be a Jordan domain with -summable boundary , for and .

Definition 3. We define the hyperholomorphic Cauchy type integral of by the formula
where is the indicator function of .

The following proposition makes Definition 3 legitimate.

Proposition 1. The integral (21) is independent of the choice of .

Proof. By definition, . As we have for any . The proof is completed by showing that .
We havewhich is a consequence of Theorem 1 (iii).
According to (18) and taking account that does not increase, we haveThe inequality (23) and the fact that , givesConsequentlywhere the last sum is finite due to [23, Lemma 1].
Now suppose that and are two different Whitney type extensions of . Then , is a Whitney type extension of the null function and hence . It remains to prove thatTo this end, let us consider the following connected domainsThe boundary of , denoted by , consists of sides of some squares .
Thus, we haveNow, take and choose sufficiently large such that and for , , where is a square of . The quaternionic Borel–Pompeiu formula, see [4, Theorem 4.1, Theorem 4.4], applied to , yieldswhere is the unit normal vector on and denotes the surface measure. Next, let , a square containing , and such that . Since , it follows thatIf is a side of and is the -square containing , we have for Because each side of belongs to some , we have for ,The conclusion of [23, Lemma 1], implies thatCombining (28) with (29) yields (26) for . The case can be handled in the same way; the difference is in the fact that now .
In the rest of this paper we assume and .

3.1. Integral Representation Formulae

The following formulas represent extensions to those given in [25] for the case of a Jordan domain with a d-summable boundary.

Theorem 2 (Borel–Pompeiu formula). Suppose , then
(i)(ii)hold.

Let us mention two important consequences of Theorem 2.

Theorem 3 (Koppelman formula). Let satisfy the hypotheses of above theorem, then the following equality holds

Theorem 4 (Cauchy formula). Let such that be hyperholomorphic in , then

Let us now establish and prove the following auxiliary lemma.

Lemma 1. Let , then for all

Proof. The proof will be divided into two steps. First for any , which follows from the fact that with bounded. The next step is to prove that for . Indeed, application of Theorem 1 (iii) enables us to write
According to (18) and taking account that does not increase, we haveConsequentlywhich is due to the fact that is a majorant of order .
The inequality implies that , which providesThereforeCombining the inequalities (37), (39), and (41) we can conclude thatIt remains to use that .

Remark 3.2. Obviously, is a hyperholomorphic function in , which is clear from Theorem 2 (ii). The question of whether admits continuous extensions (to be denoted by ) to will be answered affirmatively in the next theorem. We see at once that .

Theorem 5. Let and consider , then admits continuous extensions to such that
belong to , whenever

Proof. We have , because . Thus we are at liberty to choose such thatFor any such , we conclude that (see Lemma 1). From Remark 3.1 it follows that the integral term in (21), represents a continuous function in . By the above, admits a continuous extension to . The inequality (45) implies that and are both greater than and consequently belongs to . The rest of the proof runs as before.

The following direct corollary is a refinement of [25, Theorem 6] (also see [26]). We check at once that requirement on and has been weakened.

Corollary 1. Let and consider , then admits continuous extensions to such that its boundary values
belong to , whenever .

4. Boundary Value Problems

We deal with three boundary value problems for hyperholomorphic solutions of the two dimensional Helmholtz equation in a fractal domain of .

Theorem 6. Let with , then the jump problem has a solution explicitly given bywhere the hyperholomorphic components whenever is a majorant with order , and moreover .

Proof. It is sufficient to use Theorem 5.

Theorem 7. Suppose that with . If there exists such that then On the contrary, if (52) holds, there exists a solution of (51), whenever is a majorant with order .

Proof. Assume (51) holds, which signifies that is a Whitney type extension of . Application of Theorem 4 gives , but as in .
Now (52) follows after passage to the limit from inside . On the other hand, if (52) holds, our claim follows directly by taking . Analysis similar to that in the proof of Theorem 5 shows that .

Theorem 8. Let with and (). If there exists a solution of
thenConversely, if (54) holds, there exists a solution of (53), whenever is a majorant with order

Proof. Take and let be a solution of (53). By Theorem 2 (ii), we have that is a solution of (51) with replaced by . The equality.
which is clear from Theorem 7 applied to this case, implies (54). Taking , the second assertion follows directly.

Under the assumptions of Theorem 8 we have that the function does not depend on the . Consequently, the following equality holds

Remark 4.1. For a vector-valued function in Theorem 8, (57) clearly forces and to satisfy

5. Main Results

Between 1861 and 1862, J. C. Maxwell published the fundamental papers “A treatise on electricity and magnetism” and “A Dynamical Theory of the Electromagnetic Field”, where the behavior of electromagnetic fields was described and completely formulated the system of partial differential equations, named after him, which form the foundation of classical electromagnetism, radio-electronics, wave propagation theory, and many other branches of physics and engineering. Since this time, the fact that solutions of the Maxwell system (for time-harmonic fields) can be related to solutions of the Dirac equation, through some nonlinear equations. This has fascinated several generations of physicists and mathematicians in various branches of science, because of their general, even philosophical significance, see for instance [3, 5, 8–10]. To the best of the author’s knowledge, this deep relation was properly formulated and documented originally in [27].

We shall focus attention on time-harmonic electromagnetic fields, where all fields vary sinusoidally in time with a single frequency of oscillations , i.e., with the dependence on the time as .

Here and subsequently, and denote the complex amplitudes of the electric respectively magnetic field and and are, respectively, the complex-valued absolute permittivity and permeability of the medium. The current density and the charge density are related by the equality .

In recent decades, interest in the time-harmonic Maxwell system has never dropped. Works noted in [8, 28–35] are some examples of theses achievements in literature.

The current research is oriented towards a quaternionic reformulation of the Maxwell system (59), which is adapted from [8], and given a more simply algebraical structure in the form

where , are purely vectorial -valued functions and the wave number is chosen such that .

This equivalence is the key to obtaining in this section our main results concerning the solvability of an inhomogeneous Dirichlet type problem for 2D quaternionic time-harmonic Maxwell system.

Theorem 9. Let and belong to , with . Let and be complex vector-valued functions in . If there exists and , both in , satisfying in the system (59) such that on .
then we haveandConversely, if (62)–(65) are valid, thenandare solutions of the system (59) and the boundary conditions (61) are satisfied. Moreover, and belong to if is a majorant with order

Proof. As a first step, we put and . Moreover, the conditions (61) now read.

Next, Theorem 8 applied to and , together with the equality

implies the conditions (62) and (64). Additionally, the vectorial nature of the complex amplitudes and Remark 4.1 yield (63) and (65). If we use the identities

a trivial verification completes the proof.

6. Conclusions

This paper established a new hyperholomorphic Cauchy type integral for a domain with fractal boundary in , which plays a remarkable role in the theoretical results on integral representation formulas. Three boundary value problems for hyperholomorphic solutions of a two dimensional Helmholtz equation in a fractal domain of are studied. They have proven successful in the solution of an inhomogeneous Dirichlet type problem for a 2D quaternionic time-harmonic Maxwell system in a domain with fractal boundary in .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgments

Yudier Peña-Pérez gratefully acknowledges the financial support of the Postgraduate Study Fellowship of the Consejo Nacional de Ciencia y Tecnología (CONACYT) (Grant number 744134). Juan Bory-Reyes was partially supported by Instituto Politécnico Nacional in the framework of SIP programs (SIP20195662).

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