A right inverse operator for $\operatorname{curl}+\lambda$ and applications

TL;DR

This paper derives a general solution for a complex curl equation with a parameter in three-dimensional domains using quaternionic analysis, and applies it to boundary value problems and Maxwell's equations in chiral media.

Contribution

It introduces a new right inverse operator for the curl+ operator and applies it to solve boundary value problems and Maxwell equations in complex media.

Findings

Explicit solution for curl+ equation in bounded domains.

Application to Neumann boundary value problems.

Application to time-harmonic Maxwell system in chiral media.

Abstract

A general solution of the equation $\operatorname{curl}\vec{w}+\lambda\vec {w}=\overrightarrow{g},\,\lambda\in\mathbb{C},\,\lambda\neq0$ is obtained for an arbitrary bounded domain $\Omega\subset\mathbb{R}^{3}$ with a Liapunov boundary and $\overrightarrow{g}\in W^{p,\operatorname{div}}\left( \Omega\right) =\left\{ \overrightarrow{u}\in L^{p}\left( \Omega\right) :\,\operatorname{div}\overrightarrow{u}\in L^{p}\left( \Omega\right) ,\,1<p<\infty\right\} $. The result is based on the use of classical integral operators of quaternionic analysis. Applications of the main result are considered to a Neumann boundary value problem for the equation $\operatorname{curl}\vec{w}+\lambda\vec {w}=\overrightarrow{g}$ as well as to the nonhomogeneous time-harmonic Maxwell system for achiral and chiral media.

Equations

$$\mathrm{curl}\vec{w}+\lambda \vec{w}=\overrightarrow{g},\lambda \in \mathbb{C}$$

$$(D+\lambda )\vec{w}=g$$

$$\mathrm{curl}\vec{w}+\lambda \vec{w}=\overrightarrow{g}$$

$$D=e_1{\partial }_1+e_2{\partial }_2+e_3{\partial }_3,$$

$$Dw=-\mathrm{div}\vec{w}+\mathrm{grad}{w}_0+\mathrm{curl}\vec{w},$$

$$(D+\lambda)w=0\iff\left\{\begin{array}[c]{rcl}\operatorname{div}\vec{w}\!\!&=&\!\!\lambda w_{0},\\ \operatorname{curl}\vec{w}+\lambda\vec{w}\!\!&=&\!\!-\operatorname{grad}w_{0}.\end{array}\right.$$

$$(\Delta +{\lambda }^2)w=0.$$

$$\mathrm{curl}\vec{u}+\lambda \vec{u}=0$$

$$(D+\lambda )\vec{w}=g$$

$$-\mathrm{div}\vec{w}=g_0,$$

$$\mathrm{curl}\vec{w}+\lambda \vec{w}=\vec{g}.$$

$${\mathrm{Sol}}_{\lambda }^p(\Omega ):=\{g=g_0+\vec{g}\in L^p(\Omega ,\mathbb{B}):\mathrm{div}\vec{g}+\lambda g_0=0\}$$

$${\int }_{\Omega }(\mathrm{grad}{v}_0)\vec{w}d\vec{y}={\int }_{\Omega }(\lambda \vec{w}-g)v_{0}d\vec{y},\forall v_0\in W_0^{1,q}(\Omega ),1/p+1/q=1.$$

$${\int }_{\Omega }(\mathrm{grad}{v}_0)\cdot \vec{g}d\vec{y}=\lambda {\int }_{\Omega }{g}_0{v}_{0}d\vec{y},\forall v_0\in W_0^{1,q}(\Omega ).$$

$$E_{\pm \lambda }(\vec{x})=\pm \lambda \theta (\vec{x})-\mathrm{grad}\theta (\vec{x})=\theta (\vec{x})\left(\pm \lambda +\frac{\vec{x}}{|\vec{x}{|}^2}-i\lambda \frac{\vec{x}}{|\vec{x}|}\right),\vec{x}\in {\mathbb{R}}^3\setminus \{0\}.$$

$$T_{\lambda }[w](\vec{x})$$

$$(D+\lambda )T_{\lambda }[w](\vec{x})=w(\vec{x}),\vec{x}\in \Omega .$$

$$\Delta +{\lambda }^2=-(D-\lambda )(D+\lambda )$$

$$(\Delta +{\lambda }^2)T_{\lambda }[w]=-(D-\lambda )[w]=\mathrm{div}\vec{w}+\lambda w_0-\mathrm{curl}\vec{w}-\nabla w_0+\lambda \vec{w}$$

$$T_{0,\pm \lambda }[w](\vec{x})$$

$$={\int }_{\Omega }\left[\pm \lambda \theta (\vec{x}-\vec{y})w_0(\vec{y})+{\mathrm{grad}}_{\vec{x}}\theta (\vec{x}-\vec{y})\cdot \vec{w}(\vec{y})\right]d\vec{y},$$

$${\overrightarrow{T}}_{1,\pm \lambda }[w_0](\vec{x})$$

$$=-{\int }_{\Omega }{\mathrm{grad}}_{\vec{x}}\theta (\vec{x}-\vec{y})w_0(\vec{y})d\vec{y}$$

$${\overrightarrow{T}}_{2,\pm \lambda }[\vec{w}](\vec{x})$$

$$={\int }_{\Omega }\left[\pm \lambda \theta (\vec{x}-\vec{y})\vec{w}(\vec{y})-{\mathrm{grad}}_{\vec{x}}\theta (\vec{x}-\vec{y})\times \vec{w}(\vec{y})\right]d\vec{y}.$$

$$T_{\pm \lambda }[w]=T_{0,\pm \lambda }[w]+{\overrightarrow{T}}_{1,\pm \lambda }[w_0]+{\overrightarrow{T}}_{2,\pm \lambda }[\vec{w}].$$

$$L_{\lambda }[w](\vec{x})={\int }_{\Omega }\theta (\vec{x}-\vec{y})w(\vec{y})d\vec{y}$$

$$T_{0,\pm \lambda }[w](\vec{x})=\mathrm{div}{L}_{\lambda }[\vec{w}](\vec{x})\pm \lambda L_{\lambda }[w_0](\vec{x}),$$

$${\overrightarrow{T}}_{1,\pm \lambda }[w_0](\vec{x})=-\mathrm{grad}{L}_{\lambda }[w_0](\vec{x})$$

$${\overrightarrow{T}}_{2,\pm \lambda }[\vec{w}](\vec{x})=-\mathrm{curl}{L}_{\lambda }[\vec{w}](\vec{x})\pm \lambda L_{\lambda }[\vec{w}](\vec{x}).$$

$$\vec{w}\in \mathrm{Ker}(\Delta +{\lambda }^2)\left(\Omega \right).$$

$$\mathrm{curl}(\mathrm{curl}+\lambda )\vec{w}=0,$$

Full text

Applications of the main result are considered to a Neumann boundary value problem for the equation $\operatorname{curl}\vec{w}+\lambda\vec{w}=\overrightarrow{g}$ as well as to the nonhomogeneous time-harmonic Maxwell system for achiral and chiral media.

Keywords: Div-curl system, monogenic functions, Helmholtz equation, metaharmonic

conjugate function, Neumann boundary value problem, Maxwell’s equations.

Classification: 30G20, 30G35, 35Q60.

1 Introduction

We study the nonhomogeneous equation

[TABLE]

in a bounded domain $\Omega\subset\mathbb{R}^{3}$ and propose an integral representation for its general solution or in other words we construct a right-inverse operator for the operator $\operatorname{curl}+\lambda$ in an appropriate functional space. In the situation when the boundary values of $\vec{w}$ are given a representation of $\vec{w}$ is known (e.g., in the context of quaternionic analysis it is obtained directly from the Borel-Pompeiu formula [13, pp. 59, 60]). However, such representation not always is convenient. It is clear, e.g., that the Newton potential $L[u](\vec{x}):=-\frac{1}{4\pi}\int_{\Omega}\frac{{u(\vec{y})}}{\left|{\vec{x}-\vec{y}}\right|}{\,d\vec{y}}$ (the right inverse operator for the Laplacian) is often used with no relation to concrete boundary values of a solution to a Poisson equation.

The right-inverse operator for the operator $\operatorname{curl}+\lambda$ is obtained by using a quaternionic approach. Equation (1) is considered as a vector part of a quaternionic equation

[TABLE]

where $D$ is the Moisil-Teodorescu operator and $g_{0}=\operatorname*{Sc}g$ is related with $\overrightarrow{g}$ by the equality $\operatorname{div}\vec{g}+\lambda g_{0}=0$. A right inverse operator for $D+\lambda$ is well known (see, e.g., [8], [16], [13]). Sometimes it is called the Teodorescu transform and denoted by $T_{\lambda}$. However $T_{\lambda}\left[g\right]$ is in general a complete quaternion (whose scalar part is not necessarily zero), and, of course, simply resting a scalar part from $T_{\lambda}\left[g\right]$ does not lead to a solution of (2). Thus, the main problem for constructing a right inverse for $\operatorname{curl}+\lambda$ reduces to finding a right inverse for the operator $\left(D+\lambda\right)V$ where $V$ is a projection operator $Vw=\operatorname*{Vec}\left(w\right)=\vec{w}$. This problem we solve in three steps. First, we introduce certain component operators conforming $T_{\lambda}$ and study their properties. Second, we give a complete solution to the problem of constructing so-called metaharmonic conjugate functions, considering the $(D+\lambda)w=0$ (for a full quaternion $w$) construct $\overrightarrow{w}$ from a given $w_{0}=\operatorname*{Sc}w$ and vice versa, given $\overrightarrow{w}$ find $w_{0}$. Finally, with the aid of these results we find out what term should be rested from $T_{\lambda}\left[g\right]$ in order that the resulting function still be a solution of the equation $(D+\lambda)w=g$ at the same time being purely vectorial.

The outline of this paper is as follows. In Section 2 is given the notation and some basic results on equation (2). In Section 3 we introduce a decomposition of the $\lambda$-Teodorescu transform and some properties of the component operators. In Section 4 the procedure for constructing metaharmonic conjugate functions is presented. It is worth mentioning that in the case $\lambda\neq 0$ it resulted to be far more elementary and explicit than in the case $\lambda=0$ (for which we refer to [6, Prop. 2.3] and [7]). In Section 5 the main result of this work is presented which consists in a general solution of (1) and an explicit expression for the right inverse operator for the operator $\operatorname{curl}+\lambda$. In the rest of the paper we show some applications of this result. In Section 6 a Neumann problem for (1) is reduced to a boundary integral equation. In Section 7 a general weak solution of the nonhomogeneous time-harmonic Maxwell system is obtained. In Section 8 it is applied to a standard boundary value problem for the Maxwell system, well studied in the homogeneous case (e.g., in [3]) but not in the nonhomogeneous situation. Finally, in Section 9 a general weak solution of the nonhomogeneous Maxwell system for chiral media is presented.

2 Background for the $D+\lambda$ system

Together with the equation

[TABLE]

it is convenient to consider a quaternionic equation whose vectorial part coincides with (3). We begin by introducing the necessary notations.

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded domain. We are interested in functions $w=w_{0}+\vec{w}\colon\Omega\rightarrow\mathbb{B}$ with $w_{0}=\operatorname{Sc}w$, $\vec{w}=\operatorname{Vec}w$, where $\mathbb{B}$ denotes the algebra of biquaternions.

From now on $\vec{x}\in\mathbb{R}^{3}$. The Moisil-Teodorescu differential operator $D$ (also known as the generalized Cauchy-Riemann or occasionally the Dirac operator but in fact was introduced by R. W. Hamilton) is defined by

[TABLE]

where $\partial_{i}=\partial/\partial x_{i}$, $i=1,2,3$ and $e_{i}$ stand for basic quaternionic units. We remind that in terms of the classical differential operators of vector calculus the action of $D$ can be written as

[TABLE]

meaning that $\operatorname{Sc}\left(Dw\right)=-\operatorname{div}\vec{w}$ and $\operatorname{Vec}\left(Dw\right)=\operatorname{grad}w_{0}+\operatorname{curl}\vec{w}$.

Definition 1

Let $w\in C^{1}(\Omega,\mathbb{B})$ and $\lambda\in\mathbb{C}$. We will say that $w$ is $\lambda$-monogenic in $\Omega$ if $w$ belongs to the kernel of $D+\lambda$ in $\Omega$.

Or equivalently,

[TABLE]

When $\lambda=0$ the system (4) represents the Moisil-Teodorescu system that defines the quaternionic monogenic functions (see, e.g., [8, 9]).

If $w$ is $\lambda$-monogenic, it necessarily satisfies the Helmholtz equation

[TABLE]

Definition 2

The purely vectorial $\lambda$-monogenic functions (when $\lambda\neq 0$) are called force-free fields (or sometimes force-free magnetic fields).

They satisfy the equation

[TABLE]

which additionally implies that $\operatorname{div}\vec{u}=0$. Observe that equation (6) implies that $\operatorname{curl}\vec{u}\times\vec{u}=0$. Some references for the force-free fields are [17, 18, 19, 14] and references therein.

We are especially interested in purely vectorial solutions of the nonhomogeneous equation

[TABLE]

where $g=g_{0}+\vec{g}\in L^{p}(\Omega,\mathbb{B})$, $1<p<\infty$. A purely vectorial function $\vec{w}$ is a solution of (7) iff it solves the system of equations

[TABLE]

[TABLE]

The second equation of the system coincides with (3) meanwhile the first one is not independent. Indeed, application of $\operatorname{div}$ to (9) leads to the compatibility condition $\operatorname{div}\vec{g}+\lambda g_{0}=0$. Defining the subspace of functions in $L^{p}(\Omega,\mathbb{B})$, $1<p<\infty$, where the system (8), (9) is well-posed,

[TABLE]

we obtain that (3) is equivalent to (7) for $g\in\operatorname{Sol}_{\lambda}^{p}(\Omega)$.

Remark 3

In the special case $\lambda=0$ the system (8), (9) reduces to the classical div-curl system. In [6] a general solution for this first order partial differential system for star-shaped domains was presented and in [7] for Lipschitz domains in $\mathbb{R}^{3}$ with connected complement.

Using the well-known generalized Green’s formulas [5] a weak characterization of the solutions of (8), (9) is

[TABLE]

In particular, by (10) the elements of $\operatorname{Sol}_{\lambda}^{p}(\Omega)$ satisfy the equality

[TABLE]

3 Some integral operators

In this section we study the component operators of the $\lambda$-Teodorescu transform.

Using the fact that $\theta(\vec{x})=-e^{i\lambda|\vec{x}|}/(4\pi|\vec{x}|)$ is a fundamental solution of the Helmholtz operator $\Delta+\lambda^{2}$, corresponding fundamental solutions of the operators $D\pm\lambda$ are given by [16, Th. 3.16], [13]

[TABLE]

The $\lambda$-Teodorescu transform is defined as follows (see, e.g., [13])

[TABLE]

Moreover, for every $w\in L^{p}(\partial\Omega,\mathbb{B})$, $1<p<\infty$, $F_{\lambda}[w]$ is $\lambda$-monogenic and $T_{\lambda}$ is a right inverse of $D+\lambda$.

Proposition 4

([16, Th. 4.14]) Let $\Omega$ be a bounded domain with a Liapunov boundary, $w\in L^{p}(\Omega,\mathbb{B})$, $p>1$. Then in the generalized sense

[TABLE]

Proposition 5

Let $\Omega$ be a bounded domain with a Liapunov boundary, $w\in L^{p}(\Omega,\mathbb{B})$, $1<p<\infty$. Then

1. (i)

$\operatorname{Sc}T_{\lambda}[w]$* is a solution of the Helmholtz equation (5) if and only if $w\in\operatorname{Sol}_{\lambda}^{p}(\Omega)$.* 2. (ii)

$\operatorname{Vec}T_{\lambda}[w]$* is a solution of the Helmholtz equation (5) if and only if $\operatorname{curl}\vec{w}+\nabla w_{0}=\lambda\vec{w}$.*

Proof. Due to the factorization of the Helmholtz operator

[TABLE]

and by Proposition 4, we have that

[TABLE]

from where after separating the scalar and the vector parts the assertion follows.

Following the decomposition of the Teodorescu transform for $\lambda=0$ [6] and using the relations $\operatorname{Sc}E_{\pm\lambda}=\pm\lambda\theta$ and $\operatorname{Vec}E_{\pm\lambda}=-\operatorname{grad}\theta$, let us define

[TABLE]

[TABLE]

and

[TABLE]

Thus,

[TABLE]

Notice that ${\overrightarrow{T}}_{\!\!1,\lambda}={\overrightarrow{T}}_{\!\!1,-\lambda}$, meanwhile the difference for the other pair of operators consists in a change of one sign. Under the hypothesis of Proposition 5, we have that $T_{0,\lambda}[w]$ and ${\overrightarrow{T}}_{\!\!1,\lambda}[w_{0}]+\overrightarrow{T}_{\!\!2,\lambda}[\vec{w}]$ are scalar and vector solutions of the Helmholtz equation (5), respectively.

Let us consider the Newton potential $L_{\lambda}\colon L^{p}(\Omega)\rightarrow W^{2,p}(\Omega)$ defined by

[TABLE]

representing a right inverse for the Helmholtz operator $\Delta+\lambda^{2}$ (see, e.g., [2, p. 155]). Using the fact that $E_{\lambda}=-(D-\lambda)\theta$, we obtain the following relations.

Proposition 6

Let $\Omega$ be a bounded domain with a Liapunov boundary and $w\in L^{p}(\Omega,\mathbb{B})$, $1<p<\infty$. Then

[TABLE]

[TABLE]

and

[TABLE]

Moreover, ${\overrightarrow{T}}_{1,\lambda}[w_{0}]$ is an irrotational vector field and $-T_{0,\lambda}[\vec{w}]+\overrightarrow{T}_{\!\!2,\lambda}[\vec{w}]\in\operatorname{Sol}_{\lambda}^{p}(\Omega)$.

The proof is straightforward.

The result of Proposition 6 can be summarized in the following expression for the $\lambda$-Teodorescu operator $T_{\lambda}[w]=-(D-\lambda)L_{\lambda}[w]$.

4 Construction of metaharmonic conjugate functions

Definition 7

We will say that a scalar function $w_{0}$ and a vector function $\vec{w}$ are metaharmonic conjugate functions to each other in $\Omega$ if the biquaternion valued function $w:=w_{0}+\vec{w}$ is $\lambda$-monogenic in $\Omega$, that is $(D+\lambda)(w_{0}+\vec{w})=0$.

In this section we propose a procedure for constructing $\vec{w}$ when $w_{0}$ is known and vice versa.

Remark 8

The construction of the conjugate functions in the case $\lambda=0$ in general is not an elementary operation. For star-shaped domains it can be performed explicitly based on a radial integral operator [6, Prop. 2.3] (see also [22] for star-shaped domains in $\mathbb{H}$), meanwhile for a more general domain the procedure is far less explicit and can be defined in terms of the Cauchy integral operator requiring the inversion of a boundary integral operator (see [7, Appendix] where this procedure was introduced for bounded Lipschitz domains with a connected complement). Both methods were fundamental in the solution of the div-curl system provided in their respective references.

It is worth mentioning that the construction of the metaharmonic conjugate functions (for $\lambda\neq 0$) is far more simple and can be performed explicitly in general.

Theorem 9

(i) Let $w_{0}$ be a solution of $(\Delta+\lambda^{2})w_{0}=0$ in a domain $\Omega$. Then the function $\vec{w}:=-\frac{1}{\lambda}\nabla w_{0}$ is its metaharmonic conjugate. It is defined up to an arbitrary purely vectorial solution of $(D+\lambda)\overrightarrow{v}=0$.

(ii) Let

[TABLE]

Then a necessary and sufficient condition for the existence of a $\lambda$-monogenic function $w$ in $\Omega$ such that $\operatorname{Vec}w=\vec{w}$ is the equality

[TABLE]

and if it is satisfied then the unique metaharmonic conjugate to $\vec{w}$ has the form $w_{0}=\frac{1}{\lambda}\operatorname{div}\overrightarrow{w}$.

Proof. As a corollary of (12) we have that if $(\Delta+\lambda^{2})w_{0}=0$, the function $w:=-\frac{1}{\lambda}\left(D-\lambda\right)w_{0}$ is a solution of (4) and additionally $\operatorname*{Sc}w=w_{0}$. Thus, (i) is proved.

Let us prove (ii). The necessity of (16) follows directly from (4). For the sufficiency, consider $w_{0}=\frac{1}{\lambda}\operatorname{div}\overrightarrow{w}$. Hence the scalar equation in (4) is satisfied. Application of the gradient leads then to the equalities

[TABLE]

Now, using (15) and (16) we obtain

[TABLE]

which is the vector equation in (4).

5 Solution of system (1)

Theorem 10

Let $\Omega$ be a bounded domain with a Liapunov boundary, $\lambda\in\mathbb{C}$, $\lambda\neq 0$ and $\overrightarrow{g}\in W^{p,\operatorname{div}}\left(\Omega\right)$, $1<p<\infty$. Then a weak general solution of the system (3) is given by

[TABLE]

where $\vec{u}$ is an arbitrary solution of (6).

Proof. Let $g_{0}:=-\frac{1}{\lambda}\operatorname{div}\vec{g}$. Then $g:=g_{0}+\vec{g}\in\operatorname{Sol}_{\lambda}^{p}(\Omega)$. Using Proposition 5 (i), we have that $T_{0,\lambda}[g]$ is a solution of the Helmholtz equation (5). Thus, by Theorem 9 (i),

[TABLE]

Consider

[TABLE]

Since $T_{\lambda}$ is a right inverse of $D+\lambda$ (Proposition 4), we obtain

[TABLE]

Due to the decomposition (13) of the Teodorescu transform, the vector part of the last equality can be written as follows

[TABLE]

Thus,

[TABLE]

or, equivalently,

[TABLE]

Hence (18) takes the form

[TABLE]

By Proposition 6, $\operatorname{curl}{\overrightarrow{T}}_{\!\!1,\lambda}[g_{0}]=0$, thus (17) is obtained.

Corollary 11

Under the hypothesis of Theorem 10 the operator

[TABLE]

is a right inverse of the operator $\operatorname{curl}+\lambda I$ on $W^{p,\operatorname{div}}\left(\Omega\right)$, $1<p<\infty$.

Remark 12

Theorem 10 allows one to solve also slightly more general systems of the form

[TABLE]

where $\varphi$ is an arbitrary continuously differentiable scalar function. Indeed, $\overrightarrow{v}$ is a solution of (19) iff $\overrightarrow{w}:=e^{\varphi}\overrightarrow{v}$ satisfies (3) with $\overrightarrow{g}=e^{\varphi}\vec{h}$.

6 A Neumann boundary value problem

Following [19] we consider the Neumann problem for the equation $\operatorname{curl}\vec{w}+\lambda\vec{w}=\vec{g}$ in a domain with a connected boundary belonging to the class $C^{2}$. The problem consists in finding $\vec{w}\in C^{1}(\Omega)\cap C(\overline{\Omega})$ such that

[TABLE]

[TABLE]

where $\vec{g}\in C^{1,\gamma}(\Omega)$, $\varphi_{0}\in C^{0,\gamma}(\partial\Omega)$ and $0<\gamma<1$. A necessary condition for the existence of a solution is

[TABLE]

The next definition was introduced in [19] in order to study the Neumann problem (20), (21) using the relation with the Helmholtz equation.

Definition 13

We will say that $\lambda$ is regular with respect to the Neumann problem (20), (21) if for all solutions $\vec{w}\in C^{1}(\Omega)\cap C(\overline{\Omega})$, $a\in C^{2}(\Omega)\cap C(\overline{\Omega})$ of the system of differential equations

[TABLE]

satisfying the boundary conditions

[TABLE]

there follows $a\equiv 0$ in $\Omega$.

Using the operator $R_{\lambda}$ we transform the boundary value problem (20), (21) into a Neumann problem for force-free fields.

Theorem 14

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded domain with a $C^{2}$-boundary. Let $\lambda$ be regular with respect to (20), (21). Then a solution of the Neumann problem (20), (21) is given by

[TABLE]

where

[TABLE]

and $\vec{\psi}=\vec{u}|_{\partial\Omega}\times\overrightarrow{n}$ is a continuous solution of the boundary integral equation

[TABLE]

Proof. By Theorem 10 we have that $\vec{w}=R_{\lambda}[\vec{g}]+\vec{u}$ satisfies (20), where $\vec{u}$ is an arbitrary force-free field. Therefore solution of the Neumann problem (20), (21) reduces to solution of the Neumann problem for a force-free field with a corresponding boundary condition

[TABLE]

[TABLE]

Since (22) is assumed to be fulfilled we have that $\psi_{0}$ also satisfies (22),

[TABLE]

(due to the divergence theorem). The rest of the proof consists in applying the solution given in [19, Th. 3.3] to the Neumann problem for force-free fields (25), (26).

7 Time-harmonic Maxwell’s equations

Let us consider the time-harmonic Maxwell equations

[TABLE]

[TABLE]

for a homogeneous isotropic medium. The quantities $\epsilon$ and $\mu$ are complex numbers. The wave number $\lambda=\omega\sqrt{\epsilon\mu}$ is chosen such that $\operatorname{Im}\lambda\geq 0$. The charge density and the current density are related by the equality $\rho=\frac{1}{i\omega}\operatorname{div}\vec{j}$. Some references to the theory of time-harmonic Maxwell’s equations are [3, 4, 21].

Following [12] (see also [16] and [13]) the Maxwell system can be diagonalized with the aid of a pair of purely vectorial biquaternion valued functions $\vec{\varphi}:=-i\omega\epsilon\vec{E}+\lambda\vec{H}$ and $\vec{\psi}:=i\omega\epsilon\vec{E}+\lambda\vec{H}$, obtaining

[TABLE]

Notice that the pair of equations (29) is equivalent to the system (27), (28).

Theorem 15

Let $\Omega$ be a bounded domain with a Liapunov boundary and $\vec{j}\in W^{p,\operatorname{div}}(\Omega)$, $1<p<\infty$. Then a general weak solution of the time-harmonic Maxwell system (27), (28) is given by

[TABLE]

where $\vec{u}$ and $\vec{v}$ are arbitrary force-free fields associated to the wave numbers $\lambda$ and $-\lambda$, respectively. That is, $(D+\lambda)\vec{u}=0$ and $(D-\lambda)\vec{v}=0$.

Proof. Notice that if $\vec{j}\in W^{p,\operatorname{div}}(\Omega)$, then for the right hand side of (29) we have $-\operatorname{div}\vec{j}+\lambda\vec{j}\in\operatorname{Sol}_{\lambda}^{p}(\Omega)$ and $\operatorname{div}\vec{j}+\lambda\vec{j}\in\operatorname{Sol}_{-\lambda}^{p}(\Omega)$. Thus, by Theorem 10 we have that

[TABLE]

are weak solutions of (29), where $\vec{u}$ and $\vec{v}$ are arbitrary force-free fields associated to the wave numbers $\lambda$ and $-\lambda$, respectively.

Now (15) is obtained by noting that the pairs of vector fields $(\vec{E},\vec{H})$ and $(\vec{\varphi},\vec{\psi})$ are related by the equations $2i\omega\epsilon\vec{E}=\vec{\psi}-\vec{\varphi}$ and $2\lambda\vec{H}=\vec{\psi}+\vec{\varphi}$.

Remark 16

Using Proposition 6 to compute $\overrightarrow{T}_{\!\!2,\lambda}+\overrightarrow{T}_{\!\!2,-\lambda}$ and $\overrightarrow{T}_{\!\!2,\lambda}-\overrightarrow{T}_{\!\!2,-\lambda}$ one can write (15) in the form

[TABLE]

where $L_{\lambda}$ is the right inverse of the operator $\Delta+\lambda^{2}$ defined in (14).

By (13), we have

[TABLE]

Therefore the weak solution (15) of the time-harmonic Maxwell’s system can be rewritten as follows

[TABLE]

Compare with the solution given in [13, p. 62] where the boundary values of $\vec{E}$ and $\vec{H}$ are assumed to be known.

8 Boundary value problems for the time-harmonic Maxwell

equations

With the aid of Theorem 15 the method of integral equations developed for boundary value problems for homogeneous Maxwell equations (see, e.g., [3, Ch. 4]) can be extended onto the nonhomogeneous equations. As an example, we study the following boundary value problem. Find a solution of the Maxwell system (27), (28) provided with the boundary condition

[TABLE]

The system is considered in a bounded domain $\Omega$ with a $C^{2}$-boundary, $\vec{j}\in C^{0,\gamma}(\overline{\Omega})$, $0<\gamma<1$ and $\vec{\varphi}\in\mathcal{F}_{\text{Div}}(\partial\Omega):=\{\vec{\phi}\in C^{0,\gamma}(\partial\Omega)\colon\operatorname{Div}\vec{\phi}\in C^{0,\gamma}(\partial\Omega)\}$. That is the surface divergence of $\vec{\varphi}$ (see [3, Def. 2.28]) exists and belongs to the Hölder space $C^{0,\gamma}(\partial\Omega)$.

Analogously to the procedure used in Section 6 this boundary value problem is transformed into a boundary value problem for a homogeneous Maxwell system. Denote

[TABLE]

where $\vec{u}$ and $\vec{v}$ are arbitrary force-free fields from (15). Since $\operatorname{curl}(\vec{u}\pm\vec{v})=-\lambda(\vec{u}\mp\vec{v})$, it is immediate that $(\vec{E}^{\ast},\vec{H}^{\ast})$ satisfy the homogeneous time-harmonic Maxwell equations. By Theorem 15 and (16), the boundary value problem (27), (28), (33) is equivalent to finding a pair of vector fields $(\vec{E}^{\ast},\vec{H}^{\ast})\in C^{1}(\Omega)\cap C(\overline{\Omega})$ satisfying

[TABLE]

[TABLE]

Theorem 17

Let $\Omega$ be a bounded domain with a $C^{2}$-boundary. Let $\vec{\varphi}=\vec{E}|_{\partial\Omega}\times\overrightarrow{n}\in\mathcal{F}_{\text{Div}}(\partial\Omega)$ and $\vec{j}|_{\partial\Omega}\in\mathcal{F}_{\text{Div}}(\partial\Omega)$. If $\operatorname{Im}\lambda>0$, then there exists at most one solution of the Maxwell boundary value problem (27), (28), (33).

Proof. Due to the reduction of the problem (27), (28), (33) to the homogeneous problem (34), (35) we just need to verify that $\vec{E}^{\ast}|_{\partial\Omega}\times\overrightarrow{n}$ belongs to $\mathcal{F}_{\text{Div}}(\partial\Omega)$ and then application of [3, Sec. 4.3] gives us the result. Let $\{S_{n}\}$ be a sequence of surfaces contained in $\partial\Omega$ with boundaries $\partial S_{n}$ of class $C^{2}$ that converges to the point $\vec{x}\in\partial\Omega$ in the sense of [3, Def. 2.28] and $\overrightarrow{n}_{n}$ the outward unit normal vector to $\partial S_{n}$. Then

[TABLE]

Hence by the regularity of $\vec{\varphi}$ and $\vec{j}$ we obtain that $\vec{E}^{\ast}|_{\partial\Omega}\times\overrightarrow{n}\in\mathcal{F}_{\text{Div}}(\partial\Omega)$. See [3, Th. 4.16] for the uniqueness of the solution.

9 Maxwell’s equations in chiral media

In this section we apply the general solution of the system (8) provided by Theorem 10 to Maxwell’s equations in chiral media. The concept of chirality has played an important role in chemistry, optics, among others fields (see, e.g., [10, 20]). Consider the corresponding Maxwell equations

[TABLE]

where $\beta$ is the chirality measure of the medium. Following the notation and results of [11, 13], we denote

[TABLE]

and consider the following purely vectorial biquaternion valued functions

[TABLE]

Then the system (36) can be written in a biquaternionic form as follows

[TABLE]

where $\lambda=\omega\sqrt{\epsilon\mu}$. Or equivalently,

[TABLE]

where the new wave numbers $\alpha_{1}=\lambda/(1+\lambda\beta)$ and $\alpha_{2}=\lambda/(1-\lambda\beta)$ physically correspond to the propagation of waves of opposing circular polarizations. This reduction to of (36) to a couple of equations for sometimes called Beltrami fields is often used in relation with electromagnetics in chiral media (see, e.g., [1]).

Theorem 18

Let $\Omega$ be a bounded domain with a Liapunov boundary and $\vec{j}\in W^{p,\operatorname{div}}(\Omega)$, $1<p<\infty$. Then a general weak solution of the Maxwell system (36) is given by

[TABLE]

where $\vec{u}_{\alpha_{1}}$ and $\vec{u}_{-\alpha_{2}}$ are arbitrary force-free fields associated to the wave numbers $\alpha_{1}$ and $-\alpha_{2}$, respectively.

Proof. Notice that if $\vec{j}\in W^{p,\operatorname{div}}(\Omega)$, then the $L^{p}$ functions on the right hand side of (40) belong to $\operatorname{Sol}_{\alpha_{1}}^{p}(\Omega)$ and $\operatorname{Sol}_{-\alpha_{2}}^{p}(\Omega)$, respectively. Thus, by Theorem 10 we have that

[TABLE]

are weak solutions of (40), where $\vec{u}_{\alpha_{1}}$ and $\vec{u}_{-\alpha_{2}}$ are arbitrary force-free fields associated to the wave numbers $\alpha_{1}$ and $-\alpha_{2}$, respectively.

Since $\vec{E}=\frac{1}{2}\left(\vec{\varphi}+\vec{\psi}\right)$ and $\vec{H}=\frac{1}{2i}(\vec{\varphi}-\vec{\psi})$ we have

[TABLE]

Due to Proposition 6 and (37) we obtain (18).

10 Conclusions

A right inverse operator for the operator $\operatorname{curl}+\lambda$ is constructed for any bounded domain with a Liapunov boundary and as a corollary a convenient representation for a general weak solution of equation (1) is presented. Several applications of this result are developed which include the nonhomogeneous time-harmonic Maxwell system for achiral and chiral media. No doubt that the restriction on the smoothness of the boundary can be weakened. Not less interesting would be obtaining an analogous result for unbounded domains. The main result of the present work admits a natural extension onto the $n$-dimensional situation with the aid of analogous Clifford analysis’ tools.