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

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

1SEPI-ESIME-ZAC-Instituto Politécnico Nacional, Ciudad México, Mexico2Facultad de Matemática, Universidad Autónoma de Guerrero, Chilpancingo, Guerrero, Mexico3Facultad de Ingeniería, Universidad del Desarrollo, Santiago de Chile, Chile

Correspondence should be addressed to Juan Bory Reyes; juanboryreyes@yahoo.com

Received 26 March 2019; Revised 31 August 2019; Accepted 10 September 2019; Published 31 January 2020

Academic Editor: Ping Li

Copyright © 2020 Yudier Peña Pérez et al. �is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 R2. �e study deals with a novel approach of ℎ-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 recti-fiable 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].

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

�e quaternionic analysis gives us a tool of wider applica-bility for the study of electromagnetic boundary value prob-lems. 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.

�ese 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-har-monic electromagnetic fields on the more challenging case of domains with fractal boundaries are discussed. Results con-cerning 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].

�e main goal of this work is the study of an electromag-netic Dirichlet type problem for a domain with fractal bound-ary in R2. 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.

�e outline of the paper is as follows: In Section 2 we pro-vide an outlook to the basics of quaternionic analysis and elements of fractals geometry; a new hyperholomorphic Cauchy type integral for a domain with ℎ−summable boundary in R2 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 ℎ-summable boundaries of domains in R2, 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

HindawiAdvances in Mathematical PhysicsVolume 2020, Article ID 4735357, 8 pageshttps://doi.org/10.1155/2020/4735357

Advances in Mathematical Physics2

quaternionic time-harmonic Maxwell system on a domain with ℎ-summable boundary in R2.

2. Preliminaries

�e noncommutative and associative algebra with zero divi-sors of the complex quaternions is denoted by H(C). For each complex quaternion �, one has �푎 = ∑3

�푘=0 �푎�푘�푒�푘 where {�푎�} ⊂ C, �0 is the multiplicative unit and {�푒�|�푘 = 1, 2, 3 } are standard quaternionic imaginary units. By definition, the complex imaginary unit � satisfies

Let �푎 = �푎0 + �㨀→�푎 = ∑3�푘=0 �푎�푘�푒�푘, where �푎0 =: �푆�푐(�푎) is called scalar

part and �㨀→�푎 =: �푉�푒�푐(�푎) is called vector part of the quaternion �. �e module of � coincides with its Euclidean norm: |�푎| = ‖�푎‖

R8

and the quaternionic conjugate is defined by −�푎 = �푎0 − �㨀→�푎 . If

�푆�푐(�푎) = 0 then �푎 = �㨀→�푎 is called a purely vector quaternion.�e multiplication of two quaternions �푎, �푏 can be rewritten

in vector terms:

where �㨀→�푎 ⋅ �㨀→�푏 and �㨀→� × �㨀→� are the scalar and the usual cross product in R2 respectively. We shall frequently write �푧 := (�푥, �푦) for a typical point of R2.

Let a domain Ω ⊂ R2, we will consider H(C)-valued

functions:

Properties of continuity, differentiability and integrability of � have to be understood component-wise. �e set of � times continuously differentiable functions is denoted by �퐶�(Ω;H(C)), �푘 ∈ N ∪ {0}.

Given �휆 ∈ C\{0}, let �훼 ∈ H(C) such that �2 = �. �is � generates the (le� and right) 2D Helmholtz operator, which acting on �퐶2(Ω;H(C)) are given by �Δ := Δ

R2 + ��푀 and

Δ � := ΔR

2 +�푀� respectively. Here and subsequently, ΔR

2 := �휕21 + �휕22, �휕� := �휕\�휕�푥� and for �휆 ∈ H(C), �푀�[�푓] := �푓�휆 and ��푀[�푓] := �휆�푓.

Additionally, the following partial differential operators will be considered

It follows easily that,

Set ��휕 := �휕�� + ��푀; �휕� := ���휕 +�푀�. �erefore, the Helmholtz operator can be factorized as follows:

Definition 1. A function �푓 ∈ �퐶1(Ω,H(C)) is called hyperholomorphic if it satisfies �휕�[�푓] ≡ 0 on Ω.

(1)�푖�푒� = �푒��푖, �푘 = 0, 1, 2, 3.

(2)�푎�푏 = �푎0�푏0 − �㨀→�푎 ⋅ �㨀→�푏 + �푎0�㨀→�푏 + �푏0�㨀→�푎 + �㨀→�푎 × �㨀→�푏 ,

(3)�푓 : Ω → H(C).

(4)�푠�푡�휕 := �푒1�휕1 + �푒2�휕2;�푠�푡 �휕 := �푒1 �휕1 + �푒2 �휕2;

(5)�휕�푠�푡 := �휕1 ∘ �푀�푒1 + �휕2 ∘ �푀�푒2 ;�휕�푠�푡 := �휕1 ∘ �푀�푒1 + �휕2 ∘ �푀�푒2 .

(6)�푠�푡�휕2 = �휕2�푠�푡 = −ΔR

2 .

(7)Δ �휆 = −�휕�훼 ∘ �휕−�훼 = −�휕−�훼 ∘ �휕�훼.

If we write �푓 = ∑3�푘=0 �푓�푘�푒�푘 = �푓0 + �㨀→�푓 , then we obtain by

straightforward calculation

From [17], if � = �2 ∈ C, a fundamental solution �� of Δ � is given by

where

and �(�푠)�푛 is the Hankel function of the kind � and of order

�푛 ∈ {0, 1, 2} (see [18]).If �훼 ̸= 0, Im(�훼) = 0 the functions −(�푖/4)�퐻(1)

0 (�훼|�푧|) and (�푖/4)�퐻(2)

0 (�훼|�푧|) are fundamental solutions of the Helmholtz operator Δ �2.

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 �훼 = �훼0 ∈ C.

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 [0,∞) such that �휙(0) = 0 and �휙(�푡)/�푡 is nonincreasing. One such function � said to be a majorant. Note that �휙(�푡) = �푡�, 0 < �휈 < 1, are majorants.

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

Suppose E ⊂ R2 be a bounded set. �e generalized Hölder

space, denoted by �퐶0,�휙(E,H(C)), is defined to be the family of all H(C)-valued functions � on E such that

where � is a given majorant. For �휙(�푡) = �푡�, 0 < �휈 < 1, we write �퐶0,�휈(E,H(C)) instead of �퐶0,�휙(E,H(C)).

We recall that a nonnegative and almost increasing (or almost decreasing) function � means that there exists �푐 ≥ 1 such that �휙(�푥) ≤ �푐�휙(�푦) for all � ≤ � (� ≤ �), respectively.

Following [21, Definition 1.1], we say that a majorant � has order �� if there exists a �� (0 < �휈� < 1) and a positive real number �0 such that

(8)�푠�푡�휕[�푓] = −div�㨀→�푓 + grad�푓0 + rot�㨀→�푓 .

(9)�휃�훼(�푧) :={{{

(−1)�푠 �푖4�퐻(�푠)0 (�훼|�푧|), if �훼 ̸= 0,

12�휋 log |�푧|, if �훼 = 0,

(10)�푠 := { 1, if Im (�훼) > 0 or�훼 > 0,2, if Im (�훼) < 0 or�훼 < 0,

(11)K�푠�푡,�훼(�푧) := −�휕−�훼�휃�훼(�푧), �푧 ∈ R2\{0}.

(12)

K�푠�푡,�훼(�푧) ={{{{{

(−1)�푆 �푖�훼4 (�퐻(�푠)1 (�훼|�푧|) �푧|�푧| + �퐻(�푠)

0 (�훼|�푧|)), if �훼 ̸= 0,

− �푧2�휋|�푧|2

, if �훼 = 0.

(13)�儨�儨�儨�儨�푓(�푥) − �푓(�푦)�儨�儨�儨�儨 ≤ �푐�휙(�儨�儨�儨�儨�푥 − �푦�儨�儨�儨�儨), �푥, �푦 ∈ E,

(14)�휈�휙 = sup{�휈 : �휙(�푡)�푡�휈 is almost increasing, 0 < �푡 < �푡0}

= inf{�휈 : �휙(�푡)�푡�휈 is almost decreasing, 0 < �푡 < �푡0}.

3Advances in Mathematical Physics

�is guarantees that �푐−1�푡�휈� ≤ �휙(�푡) ≤ �푐�푡�휈� , 0 < �푡 < �푡0.If a majorant � has order ��, then we let � = �� and we will

use the symbol �퐶0,�휙�(E,H(C)) instead �퐶0,�휙(E,H(C)).�e Whitney extension theorem (see [22]) in the quater-

nionic analysis context is stated as follows.

Theorem 1. Suppose E ⊂ R2 be a compact set and let

�푓 ∈ �퐶0,�휑�(E,H(C)). �ere exists a compactly supported function �̃such that

(i) �푓�儨�儨�儨�儨�儨E = �푓;(ii) �푓 ∈ �퐶∞(R2\E);(iii)

�e function �̃ is called a Whitney type extension of �. With the notation �푑�푖�푠�푡(�퐴, �퐵) we stand for the distance between the subsets � and � of R2.

2.1. Elements of Fractal Geometry. Let ℎ : (0,∞) → (0,∞) be a gauge function, i.e., a continuous and nondecreasing interval function with lim�푡→0+ℎ(�푡) = 0.

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. �e set E is called ℎ-summable if there exists δ > 0 such that

where �푁E(�푡) stands for the least number of open balls of radius

� needed to cover E. When ℎ(�푡) = �푡� with �푑 ∈ (1, 2), we recover the d-summability of E.

Definition 2 is unchanged if �푁E(�푡) with 2−�푘 ≤ �푡 < 2−�푘+1 is

replaced by the number of �-squares intersecting E. By a �-square we mean one of the form

where �푘, �푙1, �푙2 are integers.We follow [22] considering the Whitney decomposition

of Ω

�e squares � in W have disjoint interiors and satisfy

Here and subsequently, |E| stands for the diameter of a bounded set E ⊂ R

2.

3. Hyperholomorphic Cauchy Type Integral for Fractal Curves

�e Cauchy type integral associated to quaternionic analysis has been involved recently with fractional metric dimensions

�儨�儨�儨�儨�儨���휕[�푓](�푥, �푦)�儨�儨�儨�儨�儨 ≤ �푐(�휑�(�푑�푖�푠�푡(�푧,E))/(�푑�푖�푠�푡(�푧,E))), �푧 =

(�푥, �푦) ∈ R2\E.

(15)∫�훿

0�푁

E(�푡)ℎ(�푡)�푡 �푑�푡 < +∞,

(16)[�푙12−�푘, (�푙1 + 1)2−�푘] × [�푙22−�푘, (�푙2 + 1)2−�푘],

(17)Ω = +∞⋃�푘=1

W�푘 =: W = +∞⋃

�푘=1⋃

�푄∈W�

�푄 ≡ ⋃�푄∈W

�푄.

(18)�푑�푖�푠�푡(�푧, Γ) ≥ 1√2|�푄| = 2−�푘+1, �푧 ∈ �푄,�푄 ∈ W

�푘.

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 �퐿�(Ω,H(C)), �푝 > 1, the set of �-integrable functions, the Teodorescu transform ��0

[�] for �푓 ∈ �퐿�(Ω,H(C)), is given by

Looking at the kernel function K�푠�푡,�훼0(�푧) we can decompose it

in the following way

where K1�푠�푡,�훼0

(�푧) := (�훼0/(2�휋)) ln|�푧| and the continuous function K

2�푠�푡,0(�푧) = 0.

Remark 3.1. It is to be expected that ��0 shares many of the

properties of �0. For example, �푇�훼0[�푓] ∈ �퐶0,(�푝−2)/�푝(R2,H(C)),

if �푓 ∈ �퐿�(Ω,H(C)) for �푝 > 2, by analogy with [7, Subsection 8.1].

In what follows, given �� a majorant and �푑 ∈ (1, 2), we will take Ω ⊂ R

2 to be a Jordan domain with ℎ-summable bound-ary Γ, for ℎ(�푡) = �푡�푑−1�휑�휈(�푡) and �푡 ∈ (0, |Γ|].Definition 3. We define the hyperholomorphic Cauchy type integral of �푓 ∈ �퐶0,�휑�(Γ,H(C)) by the formula

where �Ω is the indicator function of Ω.

�e following proposition makes Definition 3 legitimate.

Proposition 1. �e integral (21) is independent of the choice of �̃푓.

Proof. By definition, �휕�0[�푓] = ���휕[�푓] +�푀�0[�푓]. As

�푓 ∈ �퐶0,�휑�(Ω ∪ Γ,H(C)) we have �푀�0[�푓] ∈ �퐿�(Ω,H(C)) for any �푝 > 0. �e proof is completed by showing that �푠�푡�휕[�푓] ∈ �퐿1(Ω,H(C)).

We have

which is a consequence of �eorem 1 (iii).According to (18) and taking account that �휑�(�푡)/�푡 does not

increase, we have

(19)�푇�훼0

[�푓](�푥, �푦) := ∫ΩK�푠�푡,�훼0

(�푥 − �푢, �푦 − v)�푓(�푢, v)�푑�푢 ∧ �푑v, (�푥, �푦) ∈ R2.

(20)K�푠�푡,�훼0(�푧) = K�푠�푡,0(�푧) +K

1�푠�푡,�훼0

(�푧) +K2�푠�푡,�훼0

(�푧),

(21)�퐾∗

�훼0[�푓](�푥, �푦) = �휒Ω�푓(�푥, �푦) − �푇�훼0

∘ �휕�훼0[�푓](�푥, �푦), (�푥, �푦) ∈ R

2\Γ,

(22)

∫Ω

�儨�儨�儨�儨�儨�푠�푡�휕[�푓](�푥, �푦)�儨�儨�儨�儨�儨�푑�푥 ∧ �푑�푦 = ∑�푄∈W

∫�푄

�儨�儨�儨�儨�儨�푠�푡�휕[�푓](�푥, �푦)�儨�儨�儨�儨�儨v�푑�푥 ∧ �푑�푦 ≤ �푐

⋅ ∑�푄∈W

∫�푄

�휑�휈(�푑�푖�푠�푡(�푧, Γ))�푑�푖�푠�푡(�푧, Γ) �푑�푥 ∧ �푑�푦,

(23)�휑�(�푑�푖�푠�푡(�푧, Γ))�푑�푖�푠�푡(�푧, Γ) ≤ �휑�(�푑�푖�푠�푡(�푄, Γ))

�푑�푖�푠�푡(�푄, Γ) ≤ �휑�(|�푄|)|�푄| , �푧 ∈ �푄.

Advances in Mathematical Physics4

Because each side of Γ� belongs to some �푄 ∈ W�, we have for

�푘 > �푘0,

�e conclusion of [23, Lemma 1], implies that

Combining (28) with (29) yields (26) for w ∈ Ω. �e case w ∈ R

2\{Ω ∪ Γ} can be handled in the same way; the difference is in the fact that now �푑�푖�푠�푡(w, Γ�) ≥ �푑�푖�푠�푡(w, Γ). ☐

In the rest of this paper we assume Ω+ := Ω and Ω− := R

2\{Ω+ ∪ Γ}.3.1. Integral Representation Formulae. �e 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 �푓 ∈ �퐶1(Ω+ ∪ Γ,H(C)), then

(i)

(ii) �휕�훼0∘ �푇�훼0

[�푓](�푥, �푦) = �푓(�푥, �푦), (�푥, �푦) ∈ Ω+

hold.

Let us mention two important consequences of �eorem 2.

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

Theorem 4 (Cauchy formula). Let �푓 ∈ �퐶1(Ω+ ∪ Γ,H(C)) such that be hyperholomorphic in Ω+, then

Let us now establish and prove the following auxiliary lemma.

Lemma 1. Let �푓 ∈ �퐶0,�휑�(Γ,H(C)), then �휕�훼0[�푓] ∈ �퐿�푝(Ω+, H(C))

for all

Proof. �e proof will be divided into two steps. First �푀�훼0[�푓] ∈ �퐿�푝(Ω+,H(�퐶)) for any �푝 > 0, which follows from the fact that �푓 ∈ �퐶0,�휑�(Ω+ ∪ Γ, H(�퐶)) with Ω+ bounded. �e next step is to prove that �푠�푡�휕[�푓] ∈ �퐿�푝(Ω+,H(C)) for �푝 < (3 − �휈� − �푑)/(1 − �휈�). Indeed, application of �eorem 1 (iii) enables us to write

(32)

�儨�儨�儨�儨�儨�儨�儨�儨�儨∫Γ�푘K�푠�푡,�훼0

(�푥 − �푢, �푦 − v)�푛�푘(�푢, v)̃�푙(�푢, v) �푑Γ(�푢,v)�儨�儨�儨�儨�儨�儨�儨�儨�儨

≤ �푐�儨�儨�儨�儨�푄0�儨�儨�儨�儨 ∑�푄∈�푊�푘

|�푄|�휑�휈(|�푄|) ≤ �푐�儨�儨�儨�儨�푄0�儨�儨�儨�儨 ∑�푄∈�푊�푘

|�푄|�푑−1�휑�휈(|�푄|).

(33)lim�푘→∞

∫Γ�푘K�푠�푡,�훼0

(�푥 − �푢, �푦 − v)�푛�푘(�푢, v)̃�푙(�푢, v)�푑Γ(�푢,v) = 0.

�퐾∗�훼0[�푓](�푥, �푦) + �푇�훼0

∘ �휕�훼0[�푓](�푥, �푦) = {�푓(�푥, �푦), (�푥, �푦) ∈ Ω+0, (�푥, �푦) ∈ Ω−

(34)�퐾∗

�훼0[�푓](�푥, �푦) + �푇�훼0

∘ �휕�훼0[�푓](�푥, �푦) + �휕�훼0

∘ �푇�훼0[�푓](�푥, �푦)

= { 2�푓(�푥, �푦), (�푥, �푦) ∈ Ω+0, (�푥, �푦) ∈ Ω−.

(35)�퐾∗�훼0[�푓](�푥, �푦) = {�푓(�푥, �푦), (�푥, �푦) ∈ Ω+0, (�푥, �푦) ∈ Ω−.

(36)�푝 < 3 − �휈� − �푑1 − �휈� .

�e inequality (23) and the fact that �푑 − 1 < 1, gives

Consequently

where the last sum is finite due to [23, Lemma 1].Now suppose that �̃ and g̃ are two different Whitney type

extensions of �. �en �̃푙 := �푓 − g̃, is a Whitney type extension of the null function and hence ̃�푙|Γ = 0. It remains to prove that

To this end, let us consider the following connected domains

�e boundary of Ω�, denoted by Γ�, consists of sides of some squares �푄 ∈ W

�.�us, we have

Now, take �푧 ∈ Ω and choose �0 sufficiently large such that �푧 ∈ Ω�0 and for � > �0, �푑�푖�푠�푡(�푧, Γ�푘) ≥ �儨�儨�儨�儨�푄0

�儨�儨�儨�儨/2√2, where �0 is a square of W�0. �e quaternionic Borel–Pompeiu formula, see [4, �eorem 4.1, �eorem 4.4], applied to Ω�, yields

where �푛�(�푢, v) is the unit normal vector on Γ� and �푑Γ(�푢,v) denotes the surface measure. Next, let w := (�푢 + �푖v) ∈ Γ�, � ∈ W

� a square containing w, and �푧 ∈ Γ such that |w − �푧| = �푑�푖�푠�푡(w, Γ). Since �̃푙|Γ = 0, it follows that

If Σ is a side of Γ� and �푄 ∈ W� is the �-square containing Σ, we

have for �푘 > �푘0

(24)

∑�푄∈W

∫�푄

�휑�휈(�푑�푖�푠�푡(�푧, Γ))�푑�푖�푠�푡(�푧, Γ) �푑�푥 ∧ �푑�푦 ≤ ∑

�푄∈W∫

�푄

�휑�휈(|�푄|)|�푄| �푑�푥 ∧ �푑�푦

= ∑�푄∈W

|�푄|�휑�휈(|�푄|)≤ ∑

�푄∈W|�푄|�푑−1�휑�휈(|�푄|).

(25)∫Ω

�儨�儨�儨�儨�儨�푠�푡�휕[�푓](�푥, �푦)�儨�儨�儨�儨�儨�푑�푥 ∧ �푑�푦 ≤ �푐 ∑�푄∈W

|�푄|�푑−1�휑�휈(|�푄|),

(26)�휒Ω �̃푙(�푥, �푦) − �푇�훼0∘ �휕�훼0

[�̃푙](�푥, �푦) = 0, (�푥, �푦) ∈ R2\Γ.

(27)Ω� := {�푧 ∈ �푄 |�푄 ∈ W�, for some �푗 ≤ �푘}.

(28)∫

ΩK�푠�푡,�훼0

(�푥 − �푢, �푦 − v)�휕�훼0[�̃푙](�푢, v)�푑�푢 ∧ �푑v

= lim�푘→∞

∫Ω�푘

K�푠�푡,�훼0(�푥 − �푢, �푦 − v)�휕�훼0

[�̃푙](�푢, v) �푑�푢 ∧ �푑v.

(29)

�̃푙(�푥, �푦) + ∫Ω�푘

K�푠�푡,�훼0(�푥 − �푢, �푦 − v)�휕�훼0

[�̃푙](�푢, v)�푑�푢 ∧ �푑v= ∫

Γ�푘K�푠�푡,�훼0

(�푥 − �푢, �푦 − v)�푛�푘(�푢, v)̃�푙(�푢, v) �푑Γ(�푢,v),

(30)�儨�儨�儨�儨�儨�푙(w)�儨�儨�儨�儨�儨 = �儨�儨�儨�儨�儨�푙(w) − �̃푙(�푧)�儨�儨�儨�儨�儨 ≤ �푐�휑�(|w − �푧|) ≤ �푐�휑�(|�푄|).

(31)

�儨�儨�儨�儨�儨�儨�儨∫ΣK�푠�푡,�훼0

(�푥 − �푢, �푦 − v)�푛�푘(�푢, v)̃�푙(�푢, v) �푑Γ(�푢,v)�儨�儨�儨�儨�儨�儨�儨

≤ �푐�儨�儨�儨�儨�푄0�儨�儨�儨�儨∫Σ

�儨�儨�儨�儨�儨�푙(�푢, v)�儨�儨�儨�儨�儨�푑Γ(�푢,v) ≤ �푐�儨�儨�儨�儨�푄0�儨�儨�儨�儨 |�푄|�휑�휈(|�푄|).

5Advances in Mathematical Physics

Proof. We have 2 < (3 − �휈� − �푑)/(1 − �휈�), because �푑 − 1 < �휈�. �us we are at liberty to choose � such that

For any such �, we conclude that �휕�훼0[�푓] ∈ �퐿�푝(Ω+,H(�퐶)) (see

Lemma 1). From Remark 3.1 it follows that the integral term in (21), represents a continuous function in R2. By the above, �∗

�훼0[�] admits a continuous extension to Ω+ ∪ Γ. �e inequal-

ity (45) implies that �� and (�푝 − 2)/�푝 are both greater than �� and consequently [�∗

�훼0]+[�] belongs to �퐶0,�휓�(Γ,H(C)). �e

rest of the proof runs as before. ☐

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

Corollary 1. Let �푑 − 1 < �휈 and consider �푓 ∈ �퐶0,�휈(Γ,H(C)), then �∗

�훼0[�] admits continuous extensions to Ω± ∪ Γ such that

its boundary values

belong to �퐶0,�훽(Γ,H(C)), whenever �훽 < (1 + �휈 − �푑)/(3 − �휈 − �푑).

4. Boundary Value Problems

We deal with three boundary value problems for hyperholo-morphic solutions of the two dimensional Helmholtz equation in a fractal domain of R2.

Theorem 6. Let �푓 ∈ �퐶0,�휑�(Γ,H(C)) with �푑 − 1 < �휈�, then the jump problem

has a solution explicitly given by

where the hyperholomorphic components �푓± ∈ �퐶0,�휓�(Ω±,H(C)) whenever �� is a majorant with order �훽� < (1 + �휈� − �푑)/(3 − �휈� − �푑), and moreover �푓−(∞) = 0.

Proof. It is sufficient to use �eorem 5. ☐

Theorem 7. Suppose that �퐺 ∈ �퐶0,�휑�(Γ,H(C))with �푑 − 1 < �휈�. If there exists �푓 ∈ �퐶0,�휑�(Ω+ ∪ Γ,H(C))such that

(46)2 < �푝 < 3 − �휈� − �푑1 − �휈� .

(47)[�퐾∗

�훼0]+[�푓](�푥, �푦) = �푓(�푥, �푦) − �푇�훼0

∘ �휕�훼0[�푓](�푥, �푦), (�푥, �푦) ∈ Γ,

(48)[�퐾∗�훼0]−[�푓](�푥, �푦) = −�푇�훼0

∘ �휕�훼0[�푓](�푥, �푦), (�푥, �푦) ∈ Γ

(49)�푓 = �푓+ − �푓−,

(50)�푓± = [�퐾∗�훼0]±[�푓].

(51)�휕�훼0

[�푓] = 0, inΩ+�푓 = �퐺, on Γ

According to (18) and taking account that �휑�(�푡)/�푡 does not increase, we have

Consequently

which is due to the fact that �� is a majorant of order ��.�e inequality �푝 < (3 − �휈� − �푑)/(1 − �휈�) implies that

�푑 < �휈�(�푝 − 1) + 3 − �푝, which provides

�erefore

Combining the inequalities (37), (39), and (41) we can con-clude that

It remains to use that �휕�0[�푓] = ���휕[�푓] +�푀�0[�푓]. ☐

Remark 3.2. Obviously, �∗�훼0[�] is a hyperholomorphic

function in R2\Γ, which is clear from �eorem 2 (ii). �e question of whether �∗

�훼0[�] admits continuous extensions

(to be denoted by [�∗�훼0]±[�]) to Ω± ∪ Γ will be answered

affirmatively in the next theorem. We see at once that [�퐾∗

�훼0]−[�푓](∞) = 0.

Theorem 5. Let �푑 − 1 < �휈� and consider �푓 ∈ �퐶0,�휑�(Γ,H(C)), then �∗

�훼0[�] admits continuous extensions to Ω± ∪ Γ such that

belong to �퐶0,�휓�(Γ,H(C)), whenever

(37)

∫Ω

�儨�儨�儨�儨�儨�푠�푡�휕[�푓](�푥, �푦)�儨�儨�儨�儨�儨�푝�푑�푥 ∧ �푑�푦 = ∑�푄∈W

∫�푄

�儨�儨�儨�儨�儨�푠�푡�휕[�푓](�푥, �푦)�儨�儨�儨�儨�儨�푝�푑�푥 ∧ �푑�푦

≤ �푐 ∑�푄∈W

∫�푄[�휑�휈(�푑�푖�푠�푡(�푧, Γ))

�푑�푖�푠�푡(�푧, Γ) ]�푝�푑�푥 ∧ �푑�푦.

(38)�휑�(�푑�푖�푠�푡(�푧, Γ))�푑�푖�푠�푡(�푧, Γ) ≤ �휑�(�푑�푖�푠�푡(�푄, Γ))

�푑�푖�푠�푡(�푄, Γ) ≤ �휑�(|�푄|)|�푄| , �푧 ∈ �푄.

(39)

∑�푄∈W

∫�푄[�휑�휈(�푑�푖�푠�푡(�푧, Γ))

�푑�푖�푠�푡(�푧, Γ) ]�푝�푑�푥 ∧ �푑�푦 ≤ ∑

�푄∈W∫

�푄[�휑�휈(|�푄|)

|�푄| ]�푝�푑�푥 ∧ �푑�푦

= ∑�푄∈W

�휑�푝�휈 (|�푄|)|�푄|2−�푝 ≤ �푐 ∑

�푄∈W|�푄|�푝�휈� |�푄|2−�푝,

(40)∑�푄∈W

|�푄|�휈�(�푝−1)+3−�푝 ≤ ∑�푄∈W

|�푄|�푑.

(41)∑�푄∈W

|�푄|�푝�휈�+2−�푝 ≤ ∑�푄∈W

|�푄|�푑−1|�푄|�휈� ≤ �푐 ∑�푄∈W

|�푄|�푑−1�휑�휈(|�푄|).

(42)∫Ω

������푠�푡�휕[�푓](�푥, �푦)������푝�푑�푥 ∧ �푑�푦 ≤ �푐 ∑�푄∈W

|�푄|�푑−1�휑�휈(|�푄|) < +∞.

(43)[�퐾∗

�훼0]+[�푓](�푥, �푦) = �푓(�푥, �푦) − �푇�훼0

∘ �휕�훼0[�푓](�푥, �푦), (�푥, �푦) ∈ Γ,

(44)[�퐾∗�훼0]−[�푓](�푥, �푦) = −�푇�훼0

∘ �휕�훼0[�푓](�푥, �푦), (�푥, �푦) ∈ Γ

(45)�훽� < 1 + �휈� − �푑3 − �휈� − �푑.

Advances in Mathematical Physics6

5. Main Results

Between 1861 and 1862, J. C. Maxwell published the funda-mental papers “A treatise on electricity and magnetism” and “A Dynamical �eory of the Electromagnetic Field”, where the behavior of electromagnetic fields was described and com-pletely formulated the system of partial differential equations, named a�er him, which form the foundation of classical elec-tromagnetism, 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. �is has fasci-nated 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 electromag-netic fields, where all fields vary sinusoidally in time with a single frequency of oscillations �, i.e., with the dependence on the time as �−�푖�휔�푡.

Here and subsequently, �㨀→�퐸 and �㨀→�퐻 denote the complex ampli-tudes of the electric respectively magnetic field and � and � are, respectively, the complex-valued absolute permittivity and permeability of the medium. �e current density and the charge density are related by the equality div

�㨀→�푗 = �푖�휔�휌.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.

�e 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 �㨀→�휗 = −�푖�휔�휀�㨀→�퐸 + �훼0

�㨀→�퐻, �㨀→�휂 = �푖�휔�휀�㨀→�퐸 + �훼0�㨀→�퐻 are purely vecto-

rial H(C)-valued functions and the wave number �훼0 = �휔√�휀�휇 is chosen such that Im�훼0 ≥ 0.

�is 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 �㨀→�푗 belong to �퐿�푝(Ω+,H(C)), with �푝 > 2.

Let �㨀→�푒 and �㨀→ℎ be complex vector-valued functions in �퐶0,�휑�(Γ,H(C))

. If there exists �㨀→�퐸 and �㨀→�퐻, both in �퐶0,�휑�(Ω+ ∪ Γ,H(C)), satisfying in Ω+ the system (59) such that on Γ.

(59)

rot�㨀→�퐻 = −�푖�휔�휀�㨀→�퐸 + �㨀→�푗 ,

rot�㨀→�퐸 = �푖�휔�휇�㨀→�퐻,

div�㨀→�퐸 = �휌

�휀 ,div

�㨀→�퐻 = 0.

(60)�휕−�훼0

[�㨀→�휗 ] = div�㨀→�푗 + �훼0

�㨀→�푗

�휕�훼0[�㨀→�휂 ] = −div�㨀→�푗 + �훼0

�㨀→�푗 ,

then

On the contrary, if (52) holds, there exists a solution �푓 ∈ �퐶0,�휓�(Ω+ ∪ Γ,H(C)) of (51), whenever �� is a majorant with order �훽� < (1 + �휈� − �푑)/(3 − �휈� − �푑).

Proof. Assume (51) holds, which signifies that � is a Whitney type extension of �. Application of �eorem 4 gives �푓 = �퐾∗

�훼0[�푓], but �퐾∗

�훼0[�푓] = �퐾∗

�훼0[�퐺] as �푓 = �퐺 in Γ.

Now (52) follows a�er passage to the limit from inside Ω+. On the other hand, if (52) holds, our claim follows directly by taking �푓 = �퐾∗

�훼0[�퐺]. Analysis similar to that in the proof of

�eorem 5 shows that �푓 ∈ �퐶0,�휓�(Ω+ ∪ Γ,H(C)). ☐

Theorem 8. Let �퐺 ∈ �퐶0,�휑�(Γ,H(C)) with �푑 − 1 < �휈� and �퐹 ∈ �퐿�푝(Ω+,H(C))(�푝 > 2). If there exists �푓 ∈ �퐶0,�휑�(Ω+ ∪ Γ,H(C)) a solution of

then

Conversely, if (54) holds, there exists a solution �푓 ∈ �퐶0,�휓�(Ω+ ∪ Γ,H(C)) of (53), whenever �� is a majorant with order

Proof. Take g = �푓 − �푇�0[�퐹] and let � be a solution of (53).

By �eorem 2 (ii), we have that g is a solution of (51) with � replaced by �퐺 − [�푇�훼0

[�퐹]]|Γ ∈ �퐶0,�휓�훽(Γ,H(C)). �e equality.

which is clear from �eorem 7 applied to this case, implies (54). Taking �푓 = �퐾∗

�훼0[�퐺] + �푇�훼0

[�퐹], the second assertion follows directly.

Under the assumptions of �eorem 8 we have that the function �푓 = �퐾∗

�훼0[�퐺] + �푇�훼0

[�퐹] does not depend on the �̃ . Consequently, the following equality holds

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

(52)[�퐾∗�훼0]−[�퐺](�푥, �푦) = 0, (�푥, �푦) ∈ Γ.

(53)�휕�훼0[�푓] = �퐹, �푖�푛Ω+

�푓 = �퐺, �표�푛Γ

(54)[�퐾∗�훼0]−[�퐺](�푥, �푦) = −�푇�훼0

[�퐹](�푥, �푦), (�푥, �푦) ∈ Γ.

(55)�훽� < min{1 + �휈� − �푑3 − �휈� − �푑,

�푝 − 2�푝 }.

(56)[�퐾∗�훼0]−[�퐺 − �푇�훼0

[�퐹]](�푥, �푦) = 0, (�푥, �푦) ∈ Γ,

(57)�푓(�푥, �푦) = �퐺(�푥, �푦) − �푇�훼0

∘ �휕�훼0[�퐺](�푥, �푦)

+ �푇�훼0[�퐹](�푥, �푦), (�푥, �푦) ∈ Ω+.

(58)�푆�푐(�푇�0∘ �휕�0

[�퐺](�푥, �푦)) = �푆�푐(�푇�0[�퐹](�푥, �푦)).

7Advances in Mathematical Physics

implies the conditions (62) and (64). Additionally, the vecto-rial 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

�is paper established a new hyperholomorphic Cauchy type integral for a domain with fractal boundary in R2, which plays a remarkable role in the theoretical results on integral rep-resentation formulas. �ree boundary value problems for hyperholomorphic solutions of a two dimensional Helmholtz equation in a fractal domain of R2 are studied. �ey 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 R2.

Data Availability

No data were used to support this study.

Conflicts of Interest

�e 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 sup-port 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).

References

[1] O. F. Gerus and M. Shapiro, “On a cauchy-type integral related to the Helmholtz operator in the plane,” Boletín de la Sociedad Matemática Mexicana, vol. 10, no. 1, pp. 63–82, 2004.

[2] O. F. Gerus, V. N. Kutrunov, and M. Shapiro, “On the spectra of some integral operators related to the potential theory in the plane,” Mathematical Methods in the Applied Science, vol. 33, no. 14, pp. 1685–1691, 2010.

[3] V. V. Kravchenko and M. Shapiro, Integral Representations for Spatial Models of Mathematical Physics, vol. 351 of Research Notes in Mathematics, Pitman Advanced Publishing Program, London, 1st edition, 1996.

(71)

�㨀→�퐸 = 12�푖�휔�휀(

�㨀→�휂 − �㨀→�휗 ),�㨀→�퐻 = 1

2�훼0(�㨀→�휂 + �㨀→�휗 ),

then we have

and

Conversely, if (62)–(65) are valid, then

and

are solutions of the system (59) and the boundary conditions (61) are satisfied. Moreover, �㨀→�퐸 and �㨀→�퐻 belong to �퐶0,�휓�(Ω+ ∪ Γ,H(C)) if �� is a majorant with order

Proof. As a first step, we put �㨀→�휗 = −�푖�휔�휀�㨀→�퐸 + �훼0

�㨀→�퐻 and �㨀→�휂 = �푖�휔�휀�㨀→�퐸 + �훼0�㨀→�퐻. Moreover, the conditions (61) now read.

Next, �eorem 8 applied to �㨀→�휗 and �㨀→�휂 , together with the

equality

(61)�㨀→�퐸 �儨�儨�儨�儨�儨�儨Γ = �㨀→�푒 ,�㨀→�퐻�儨�儨�儨�儨�儨�儨Γ =

�㨀→ℎ ,

(62)�푇−�훼0∘ �휕−�훼0

[−�푖�휔�휀�㨀→�푒 + �훼0�㨀→ℎ ]��������Γ = �푇−�훼0

[div �㨀→�푗 + �훼0�㨀→�푗 ]������Γ,

(63)�푆�푐(�푇−�훼0

∘ �휕−�훼0[−�푖�휔�휀�㨀→�푒 + �훼0

�㨀→ℎ ]) = �푆�푐(�푇−�훼0[div �㨀→�푗 + �훼0

�㨀→�푗 ]) �푖�푛 Ω+

(64)�푇�훼0∘ �휕�훼0

[�푖�휔�휀�㨀→�푒 + �훼0�㨀→ℎ ]��������Γ = �푇�훼0

[−div �㨀→�푗 + �훼0�㨀→�푗 ]������Γ,

(65)

�푆�푐(�푇�훼0∘ �휕�훼0

[�푖�휔�휀�㨀→�푒 + �훼0�㨀→ℎ ]) = �푆�푐(�푇�훼0

[−div �㨀→�푗 + �훼0�㨀→�푗 ]) �푖�푛 Ω+.

(66)

�㨀→�퐸 =∼�㨀→�푒 − 1

2�푖�휔�휀{�푖�휔�휀(�푇�훼0∘ �휕�훼0

+ �푇−�훼0∘ �휕−�훼0

)

⋅ [∼�㨀→�푒 ] + �훼0(�푇−�훼0

∘ �휕−�훼0− �푇�훼0

∘ �휕�훼0)

⋅ [∼�㨀→ℎ ] − (�푇�훼0

+ �푇−�훼0)[div �㨀→�푗 ] + �훼0(�푇�훼0

− �푇−�훼0)[�㨀→�푗 ]}

(67)

�㨀→�퐻 =∼�㨀→ℎ − 1

2�훼0{�푖�휔�휀(�푇�훼0

∘ �휕�훼0− �푇−�훼0

∘ �휕−�훼0)

⋅ [∼�㨀→�푒 ] − �훼0(�푇−�훼0

∘ �휕−�훼0+ �푇�훼0

∘ �휕�훼0)[

∼�㨀→ℎ ]

− (�푇�훼0− �푇−�훼0

)[div �㨀→�푗 ] + �훼0(�푇�훼0+ �푇−�훼0

)[�㨀→�푗 ]},

(68)�훽� < min{1 + �휈� − �푑3 − �휈� − �푑,

�푝 − 2�푝 }.

(69)

�㨀→�휗 �儨�儨�儨�儨�儨�儨�儨Γ = −�푖�휔�휀�㨀→�푒 + �훼0�㨀→ℎ ,

�儨�儨�儨�儨�儨�㨀→�휂 �儨�儨�儨�儨�儨Γ = �푖�휔�휀�㨀→�푒 + �훼0�㨀→ℎ .

(70)[�퐾∗�훼0]−[�푓](�푥, �푦) = −�푇�훼0

∘ �휕�훼0[�푓](�푥, �푦),

Advances in Mathematical Physics8

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