On the Construction of Beltrami Fields and Associated Boundary Value Problems

Abstract

In this paper, we present two simple methods for constructing Beltrami fields. The first one consists of a composition of operators, including a quaternionic transmutation operator as well as the computation of formal powers for the function $f(x)=e^{\textbf{i}\lambda x}$. For the second method, we generate Beltrami fields from harmonic functions, and using the intrinsic relation between the normal and tangential derivative, we solve an associated Neumann-type boundary value problem.

1 Introduction

We study the first-order partial differential equation

$$\begin{aligned} {\text{ curl } }{} {\textbf {w}}+\lambda {\textbf {w}}=0,\quad \text{ in } \Omega , \end{aligned}$$

(1.1)

where $\Omega$ is a domain in ${\mathbb {R}}^{3}$ and $\lambda \ne 0$ is an arbitrary complex number. We can easily observe that solutions of (1.1) describe vector fields that are proportional to their curl. Such fields are of particular interest and are known as Beltrami fields. In [2], boundary value problems for Beltrami fields (also called magnetohydrodynamics force-free magnetic fields) in exterior domains and in the framework of weighted Sobolev spaces were studied.

We will say that ${\textbf {v}}$ is a divergence-free solution of the Helmholtz equation if

$$\begin{aligned} \Delta {\textbf {v}}+\lambda ^{2}{} {\textbf {v}}=0,\quad {\text{ div } }{} {\textbf {v}}=0\quad \text{ in } \Omega . \end{aligned}$$

(1.2)

We have the following equivalence, ${\textbf {v}}$ is a divergence-free solution of the vector Helmholtz equation (1.2) if and only if ${\textbf {w}}=({\text {curl}}-\lambda )\,{\textbf {v}}$ is a Beltrami field in $\Omega$ [28]. This is a very common way to find Beltrami fields in practice.

Another classical result is due to Chandrasekhar and Kendall [5], they provided the most general solution of (1.1) as the sum of a poloidal and a toroidal vector field. More precisely, ${\textbf {w}}={\textbf {T}}+{\textbf {S}}$, where ${\textbf {T}}={\text{ curl } }({\textbf {a}}\,\varphi )$ and ${\textbf {S}}=-\dfrac{1}{\lambda } {\text{ curl } } {\textbf {T}}$, where $\textbf{a}$ is a fixed unit vector and $\varphi$ is an arbitrary scalar solution of the Helmholtz equation $\Delta \varphi +\lambda ^2\varphi =0$.

In the literature, there are several works focused on the construction of Beltrami fields because of their importance in hydrodynamics, plasma-physics, and astrophysics. In the works [24, 25], it is given an integral equation method to analyze a Neumann boundary value problem for the equation ${\text{ curl } } {\textbf {v}}+\lambda {\textbf {v}}={\textbf {g}}$, see also the references therein. More recently, in the works [6, 8] it was analyzed the same equation to provide a general solution of the time-harmonic Maxwell’s equations and to solve the Neumann BVP. The techniques employed in [6, 8] are based on quaternionic analysis, we first immersed the three-dimensional system into a quaternionic setting and we solve it (through an equivalent system), and after we project the quaternionic solution into a vector one.

In [23] explicitly, Beltrami fields for variable $\lambda$ were obtained by combining quaternionic analysis with transmutation operator theory. We apply the method provided in [23], for $\lambda$ a complex constant, taking advantage of the explicit expressions of the formal powers $\varphi _n$, and we give several explicit examples. Another recent construction of Beltrami fields was given in [7, Sec. 5]. The method is based on a uniformly convergent Neumann series in terms of a bounded right inverse of the curl operator; some integral operators of quaternionic analysis played an interesting role in the explicit expression of that inverse of curl.

Now, we will use the transmutation operator method to construct a family of Beltrami fields. The whole article is immersed in a biquaternionic setting. We will use some relations and properties of the biquaternionic projectors ${\mathcal {P}}^{\pm }=M^{(1\pm \textbf{i}\textbf{e}_3)/2}$, where $M^{\alpha } w=w\alpha$ is the right-hand side multiplication operator. These projectors appear in [16] and allow us to transform the original problem into finding solutions to $(D+M^{i\lambda \textbf{e}_3})w=0$, which we solve with the aid of transmutation operators and the explicit form of the formal powers.

The cornerstone of the construction is the fact of explicitly knowing the expression of the formal powers associated with the function $f(x)=e^{\textbf{i}\lambda x}$ (see [31]). Indeed

$$\begin{aligned} \varphi _n(x) = \left\{ \begin{array}{ll} (2k-1)!!\left( \frac{ x}{\lambda }\right) ^k\lambda x\big ( j_{k-1}(\lambda x)+\textbf{i} j_{k}(\lambda x)\big ) , &{}n=2k, \;\; \\ (2k+1)!!\left( \frac{x}{\lambda }\right) ^k x j_k(\lambda x), &{} n=2k+1, \end{array} \right. \end{aligned}$$

(1.3)

where $j_k$ denotes the spherical Bessel’s function of order k, see Subsection 3.1.

The outline of the paper is as follows. Section 2 defines the operators we will use throughout this work. In Sect. 3, we used the transmutation operator to obtain Beltrami fields in an explicit way since it is sufficient to know how the transmutation operator acts on certain elementary functions. For example, in the present work using the formal powers (1.3), we can find the images of basic polynomials under the action of the transmutation operator, which allows us to construct them.

Section 4 presents different explicit examples of Beltrami fields. The method presented here has no constraints concerning the geometry of the domain; even more, $\Omega$ can be a bounded or unbounded three-dimensional domain. Without further ado,

$$\begin{aligned} {\textbf {w}} e^{\textbf{i}\lambda x_3}\nabla u\, (1+\textbf{i}\textbf{e}_3) \end{aligned}$$

(1.4)

is a Beltrami field if and only if $u=u(x_1,x_2)$ is a harmonic function, see Proposition 4.1. In Subsection 4.1, we will analyze the Neumann-type boundary value problem for Beltrami fields

$$\begin{aligned} ({\text {curl}} +\lambda ){\textbf {w}}&=0, \quad \hspace{0.4 cm}\text { in }\Omega ,\nonumber \\ {\textbf {w}}|_{\partial \Omega }\cdot \eta&=\alpha , \qquad \text { on }\partial \Omega . \end{aligned}$$

(1.5)

Meanwhile, in Sect. 4.2, we study a Neumann boundary value problem (BVP) for the equation $(D+M^{\frac{f^{\prime }}{f} \textbf{e}_3}){\textbf {w}}=0$ and we will see that it is equivalent to an associate BVP for an appropriate conductivity equation. Later in Sect. 5, we close the work with a Hodge decomposition for the class of $L^2$ Beltrami fields.

2 Background

Let $\Omega \subseteq {\mathbb {R}}^{3}$ be a domain with a $C^{2}$-boundary. We are interested in functions $w=w_{0}+{\textbf {w}}:\Omega \rightarrow {\mathbb {B}}$ with $w_{0}={\text{ Sc } }w$, ${\textbf {w}}={\text{ Vec } }w$, where ${\mathbb {B}}$ denotes the algebra of biquaternions or complex quaternions. It is worth mentioning that the quaternionic imaginary units $\textbf{e}_{k}$, $k=1,2,3$ commutes with the complex imaginary unit $\textbf{i}$.

Unlike the quaternions algebra we know that ${\mathbb {B}}$ has zero divisors (see, e.g., [17]). Indeed, $\dfrac{1+\textbf{i}\textbf{e}_3}{2}$ and $\dfrac{1-\textbf{i}\textbf{e}_3}{2}$ are idempotent elements, giving rise to projectors

$$\begin{aligned} {\mathcal {P}}^{\pm }=M^{(1\pm \textbf{i}\textbf{e}_3)/2}, \end{aligned}$$

(2.1)

where $M^{\alpha } w=w\alpha$ is the right-hand side multiplication operator. Moreover, the modulus of the elements in ${\mathbb {B}}$ is defined as follows

$$\begin{aligned} |a|_{c}^{2}=|\text {Re }a|^{2}+|\text {{Im} }a|^{2}=\text {Sc }(a{\overline{a}}^{*})=\text {Sc }({\overline{a}}^{*}a), \end{aligned}$$

where $^{*}$ stands for the usual complex conjugation and $\bar{\text { } }$ for the quaternionic conjugation. Equivalently,

$$\begin{aligned} |a|_{c}^{2}=|a_{0}|^{2}+|a_{1}|^{2}+|a_{2}|^{2}+|a_{3}|^{2}, \end{aligned}$$

where $|a_{k}|^{2}=a_{k}a_{k}^{*}$, $k=0,1,2,3$.

The three-dimensional Moisil-Teodorescu differential operator D is defined by

$$\begin{aligned} D=\textbf{e}_{1}\partial _{1}+\textbf{e}_{2}\partial _{2}+\textbf{e}_{3}\partial _{3}, \end{aligned}$$

where $\partial _{k}=\partial /\partial x_{k}$, $k=1,2,3$. It is well known that the action of the operator D to any differentiable function $w=w_{0}+{\textbf {w}}$ can be written in terms of the classical differential operators of vector calculus in the form

$$\begin{aligned} Dw=-{\text{ div } }{} {\textbf {w}}+{\text{ grad } }w_{0}+{\text{ curl } } {\textbf {w}}, \end{aligned}$$

(2.2)

where the scalar and vector parts are ${\text{ Sc }}\left( Dw\right) =-{\text{ div } }{} {\textbf {w}}$ and ${\text{ Vec }}\left( Dw\right) ={\text{ grad } }w_{0}+{\text{ curl } }{} {\textbf {w}}$, respectively.

Let $w\in C^{1}(\Omega ,{\mathbb {B}})$ and $\lambda \in {\mathbb {C}}$, $\lambda$ will be considered a complex number throughout the manuscript unless otherwise specified. We will say that w is $\lambda$-monogenic in $\Omega$ if $w\in {\text {Ker}} (D+\lambda )$ in $\Omega$. Or equivalently,

$$\begin{aligned} (D+\lambda )w=0\iff \left\{ \begin{array}{rcl} {\text{ div } }{} {\textbf {w}}\!\! &{}{} = &{}{} \!\!\lambda w_{0},\\ {\text{ curl } }{} {\textbf {w}}+\lambda {\textbf {w}}\!\! &{}{} = &{}{} \!\!-{\text{ grad } } w_{0}. \end{array} \right. \end{aligned}$$

(2.3)

When $w_0=0$, a $\lambda$-monogenic is a Beltrami field. In [6], it was defined the vector-field valued operator on ${\mathbb {B}}$-valued functions as follows

$$\begin{aligned} {{\mathcal {V}}} :{\text{ Ker }}(D+\lambda ){} & {} \rightarrow {\text{ Ker }}(D+\lambda )\cap C(\Omega ,{\text{ Vec } }{\mathbb {B}}), \nonumber \\ {{\mathcal {V}}}[u]{} & {} = {\text{ Vec } } u + \frac{1}{\lambda }D {\text{ Sc } } u. \end{aligned}$$

(2.4)

This operator ${\mathcal {V}}$ maps $\lambda$-monogenic functions into Beltrami fields in $\Omega$.

Later, we will use that ${\text {Ker}} (D+\lambda )$ is a biquaternionic right module. Then means that for $u,v \in {\text {Ker}} (D+\lambda )$ and $\alpha \in {\mathbb {B}}$ constant, then $(u\alpha +v) \in {\text {Ker}}(D+\lambda )$.

We will say that w satisfies the Helmholtz equation if

$$\begin{aligned} (\Delta +\lambda ^{2})w=0,\quad \text { in }\Omega . \end{aligned}$$

(2.5)

Solutions of (2.5) are sometimes known as metaharmonic functions with parameter $\lambda$ [33]. It is well-known that the perturbed Dirac operator $D+\lambda$ factorizes the Helmholtz operator as follows

$$\begin{aligned} \Delta +\lambda ^{2}=-(D+\lambda )(D-\lambda )=-(D-\lambda )(D+\lambda ). \end{aligned}$$

(2.6)

Consequently, if $(D+\lambda )w=0$ in $\Omega$, by the linearity of the Helmholtz operator, then w satisfies (2.5) component-wise.

3 Beltrami Fields

In this Section, we will present a construction method for Beltrami fields in three-dimensional bounded domains. It is based on the Transmutation Operator Method, some classical references are [1, 4, 10, 11, 26, 27], and some more recent ones are [15, 18, 20, 21, 32].

3.1 Transmutation Operators Revisited

Here we summarize a construction from [3, 23, 31]; for our purposes, we only work with an exponential function, but in general, we can consider more general functions f, to construct the following systems (see [3, 23]). Let us consider $f\in C^2[-a,a]$ a complex-valued function such that $f(x)\ne 0$ for all $x\in [-a,a]$ and $f(0)=1$. Define the recursive integrals associated to f as follows

$$\begin{aligned} X^{(0)}(x)\equiv & {} {\widetilde{X}}^{(0)}(x)\equiv 1, \nonumber \\ X^{(n)}(x)= & {} n\int _0^x X^{(n-1)}(s)\left( f^2(s)\right) ^{(-1)^{n}}ds, \nonumber \\ {\widetilde{X}}^{(n)}(x)= & {} n\int _0^x {\widetilde{X}}^{(n-1)}(s) \left( f^2(s)\right) ^{(-1)^{n-1}}ds. \end{aligned}$$

(3.1)

With the aid of these integrals, the formal powers $\varphi _n$ $n=0,1,\dots$, are given by

$$\begin{aligned} \varphi _n(x) = \left\{ \begin{array}{ll} f(x)X^{(n)}(x), &{}{} n \mathrm {\ odd,} \\ f(x){\widetilde{X}}^{(n)}(x), &{}{}n \mathrm {\ even,} \end{array} \right. \end{aligned}$$

(3.2)

Moreover, there exists a Volterra operator [3]

$$\begin{aligned} T_f[u](x)=u(x)+\int _{-x}^{x} K_f(x,t;h)\, u(t)\, dt \end{aligned}$$

where $K_f$ is a continuous kernel such that

$$\begin{aligned} K_f(x,t;h)=\dfrac{h}{2}+K(x,t)+\dfrac{h}{2}\int _{-t}^x(K(x,s)-K(x,-s))\, ds, \end{aligned}$$

and K satisfies a Goursat problem (see [22, Th. 6]). Moreover, for $k=0,1,\dots$,

$$\begin{aligned} T_f(x^k) = \varphi _k(x). \end{aligned}$$

(3.3)

In particular, if $f(x)=e^{\textbf{i}\lambda x}$, then we have the following relation for the recursive integrals defined in (3.1).

Proposition 3.1

Let $f(x)=e^{\textbf{i}\lambda x}$, then the following relation holds

$$\begin{aligned} {\widetilde{X}}^{(n)}(x)=(X^{(n)}(x))^*, \qquad \forall n \in {\mathbb {N}}. \end{aligned}$$

Proof

The proof is by induction. For $n=0$, we have ${\widetilde{X}}^{(0)}(x)=1=X^{(0)}(x)= X^{(0)}(x)^*$ by definition. Suppose that ${\widetilde{X}}^{(n-1)}(x)= (X^{(n-1)}(x))^*$. By the definition of ${\widetilde{X}}^{(n)}$, we have

$$\begin{aligned} \quad \quad {\widetilde{X}}^{(n)}(x)=n\int _0^x {\widetilde{X}}^{(n-1)}(s)\,e^{(-1)^{n-1}2\textbf{i}\lambda s} \,ds = (X^{(n)}(x))^*. \end{aligned}$$

$\square$

If $f(x)=e^{\textbf{i}\lambda x}$, it is well-known that transmutation operator $T_f$ takes the form (see [3, Ex. 12])

$$\begin{aligned} T_f[u](x)= & {} u(x)- \dfrac{\lambda }{2}\displaystyle \int _{-x}^x\frac{\sqrt{x^2-y^2}J_1\Big (\lambda \sqrt{x^2-y^2}\Big )u(y)}{x-y} \nonumber \\{} & {} - {\textbf {i }}u(y) J_0\Big (\lambda \sqrt{x^2-y^2}\Big ) \; dy, \end{aligned}$$

(3.4)

where $J_0$ and $J_1$ are the Bessel functions of the first kind. Moreover, $T_f$ is a continuous operator on the space of continuous functions with respect to the maximum norm, $||u||=\max |u|$. The following relation it was established on $C^1[-a,a]$ (see [19, 22])

$$\begin{aligned} \partial _x \frac{1}{f}T_f=\frac{1}{f}T_{1/f}\partial _x. \end{aligned}$$

(3.5)

A formula for the computation of the formal powers was founded in [31], when we consider the function $f(x)=e^{\textbf{i}\lambda x}$. Nevertheless, for a general function f, finding an explicit expression for the formal powers is difficult. Thus, the formal powers $\varphi _n(x)$ associated to $f(x)=e^{\textbf{i}\lambda x}$ defined in (3.2) has the following form:

$$\begin{aligned} \varphi _n(x)&= \left\{ \begin{array}{ll} (2k-1)!!\left( \frac{ x}{\lambda }\right) ^k\lambda x\big ( j_{k-1}(\lambda x)+\textbf{i} j_{k}(\lambda x)\big ) , &{}n=2k, \;\; \\ (2k+1)!!\left( \frac{x}{\lambda }\right) ^k x j_k(\lambda x) &{} n=2k+1 \end{array} \right. \end{aligned}$$

(3.6)

where $j_k$ denotes the spherical Bessel’s function of order k.

Next table shows us the computations of the first five formal powers associated to $f(x)=e^{\textbf{i}\lambda x}$.

Table 1 Explicit computations of the first formal powers

If we interchange the function $f=e^{\textbf{i}\lambda x}$, by $1/f=e^{-\textbf{i}\lambda x}$, then the functions $Y^{(n)},{\widetilde{Y}}^{(n)}$ associated to 1/f defined in (3.1) coincides with the functions ${\widetilde{X}}^{(n)},X^{(n)}$ respectively associated to f. Using Proposition 3.1, we can verify that the formal powers $\psi _n$ associated with 1/f are complex conjugates of the formal powers $\varphi _n$ associated with f. Indeed,

$$\begin{aligned} \psi _n(x)&= \left\{ \begin{array}{ll} e^{-{\textbf {i}}\lambda x}n\int _0^x Y^{(n-1)}(s)e^{2{\textbf {i}}\lambda s(-1)^{n+1}} ds, &{}{} n \mathrm {\ odd,} \\ e^{-{\textbf {i}}\lambda x}n\int _0^x {\widetilde{Y}}^{(n-1)}(s)e^{2{\textbf {i}}\lambda s(-1)^{n}}ds, &{}{}n \mathrm {\ even,} \end{array} \right. \\ {}&= \left\{ \begin{array}{ll} (e^{{\textbf {i}}\lambda x})^*n\int _0^x {\widetilde{X}}^{(n-1)}(s)\left( e^{2{\textbf {i}}\lambda s(-1)^{n}}\right) ^* ds, &{}{} n \mathrm {\ odd,} \\ (e^{{\textbf {i}}\lambda x})^* n\int _0^x X^{(n-1)}(s)\left( e^{2{\textbf {i}}\lambda s(-1)^{n-1}}\right) ^*ds, &{}{}n \mathrm {\ even,} \end{array} \right. \\ {}&= \varphi _n(x)^* \end{aligned}$$

and due to continuity of the operator $T_f$ this implies that $T_{1/f} = (T_f)^*$.

3.2 Construction Through Transmutation Operators

In this Subsection, $\Omega$ is a bounded open subset of ${\mathbb {R}}^3$ satisfying the following symmetry property with respect to $x_3$:whenever $(x_1,x_2,x_3) \in \Omega$, the straight segment from $(x_1,x_2,-x_3)$ to $(x_1,x_2,x_3)$ also lies in $\Omega$ and we say that $\Omega$ is convex on $x_3$, similar condition can be defined on the variables $x_1$ and $x_2$. We require this so that the complex-valued operators $T_f$, $T_{1/f}$ can be applied to functions in $\Omega$ with respect to the variable $x_3$ for fixed $(x_1,x_2)$.

In [23], with the aid of $T_f$ and $T_{1/f}$ the following continuous quaternionic operator was defined.

Definition 3.2

The biquaternionic transmutation operator $\textbf{T}_f$ acts on $v\in C(\Omega ,{\mathbb {B}})$ in the variable $x_3$ as follows:

$$\begin{aligned} \textbf{T}_f[v](x)&= T_{1/f} [v_0](x) + T_{f}[v_1](x)\textbf{e}_{1} + T_{f}[v_2](x)\textbf{e}_{2} + T_{1/f} [v_3](x)\textbf{e}_3. \end{aligned}$$

(3.7)

Using that the operators $T_f$ and $T_{1/f}$ are complex conjugates, then (3.7) reduces to

$$\begin{aligned} \textbf{T}_f[v](x)&= T_{f}^* [v_0](x) + T_{f}[v_1](x)\textbf{e}_{1} + T_{f}[v_2](x)\textbf{e}_{2} + T_f^* [v_3](x)\textbf{e}_3. \end{aligned}$$

When we interchange f by 1/f we also consider the operator $\textbf{T}_{1/f}$. In [23] it was proved that any function w, $\lambda$-monogenic admits the decomposition:

$$\begin{aligned} w&={\mathcal {P}}^+\textbf{T}_f[u_1] +{\mathcal {P}}^-\textbf{T}_{1/f}[u_2], \end{aligned}$$

(3.8)

where the projectors were defined in (2.1) and with $u_1,\,u_2$ arbitrary monogenic functions. More generally, we have the following the Runge approximation theorem.

Proposition 3.3

[ [23]] Every ${\mathbb {B}}$-valued solution w of $(D+\lambda )w=0$ in $\Omega$ can be uniformly approximated on each compact $K \subset \Omega$ by right-linear combinations of the elements $w={\mathcal {P}}^+\textbf{T}_f[u_1] +{\mathcal {P}}^-\textbf{T}_{1/f}[u_2]$, where $u_1,u_2 \in {\text {Ker}} D$.

Now, we rewrite the decomposition (3.8) for $\lambda$-monogenic functions using the conjugation identities of the transmutation operator.

Lemma 3.4

The following formula holds for any functions $u_1, u_2 \in C(\Omega , {\mathbb {B}})$,

$$\begin{aligned} {\mathcal {P}}^+\textbf{T}_f[u_1] +{\mathcal {P}}^-\textbf{T}_{1/f}[u_2]={\mathcal {P}}^+\textbf{T}_f[u_1]+\left( {\mathcal {P}}^+\textbf{T}_f\right) ^*[u_2]. \end{aligned}$$

(3.9)

Moreover, if $u_1=u_2$, then

$$\begin{aligned} ({\mathcal {P}}^+ \textbf{T}_{f}+{\mathcal {P}}^- \textbf{T}_{1/f})u= {\text {Re}}\textbf{T}_f[u]- {\text {Im}}\textbf{T}_f [u] \textbf{e}_3. \end{aligned}$$

(3.10)

Proof

Let $u_i \in C(\Omega , {\mathbb {B}})$, $i=1,2$. Using the fact $\textbf{T}_{1/f}=\textbf{T}^*_f$ we straightforward obtain

$$\begin{aligned} {\mathcal {P}}^+\textbf{T}_f[u_1] +{\mathcal {P}}^-\textbf{T}_{1/f}[u_2]={\mathcal {P}}^+\textbf{T}_f[u_1] +{\mathcal {P}}^-\textbf{T}_{f}^*[u_2]={\mathcal {P}}^+\textbf{T}_f[u_1]+\left( {\mathcal {P}}^+\textbf{T}_f\right) ^*[u_2]. \end{aligned}$$

Besides,

$$\begin{aligned} ({\mathcal {P}}^+ \textbf{T}_{f}+{\mathcal {P}}^- \textbf{T}_{1/f})u&= \textbf{T}_{f}[u]\left( \frac{1+\textbf{ie}_3}{2}\right) + \textbf{T}_{1/f}[u]\left( \frac{1-\textbf{ie}_3}{2}\right) \\ {}&=\frac{1}{2}\left( \textbf{T}_{f}[u]+\textbf{T}_{1/f}[u]\right) +\frac{1}{2}\left( \textbf{T}_{f}[u]-\textbf{T}_{1/f}[u]\right) \textbf{ie}_3 \\ {}&=\frac{1}{2}\left( \textbf{T}_{f}[u]+\textbf{T}^*_{f}[u]\right) +\frac{1}{2}\left( \textbf{T}_{f}[u]-\textbf{T}^*_{f}[u]\right) \textbf{ie}_3 \\&={\text {Re}}\textbf{T}_f[u]- {\text {Im}}\textbf{T}_f [u] \textbf{e}_3, \end{aligned}$$

which proves (3.10). $\square$

Corollary 3.5

Let u be a $C(\Omega ,{\mathbb {B}})$ monogenic function in $\Omega$, then

$$\begin{aligned} w&= {\text {Re}}\textbf{T}_f[u]-{\text {Im}}\textbf{T}_f[u]\textbf{e}_3, \end{aligned}$$

(3.11)

satisfies $(D+\lambda )w=0$ in $\Omega$. Moreover,

$$\begin{aligned} {\textbf {w}}={\mathcal {V}}[{\text {Re}}\textbf{T}_f[u]- {\text {Im}}\textbf{T}_f [u] \textbf{e}_3], \end{aligned}$$

is a Beltrami field in $\Omega$, where ${\mathcal {V}}$ is the operator defined in (2.4).

Example

Consider the monogenic function $u=x_1x_2-x_1x_3\textbf{e}_1+x_2x_3\textbf{e}_2$.

$$\begin{aligned} \nonumber {\text{ Re }}{} {\textbf {T}}_f[u]&=x_1x_2{\text{ Re } }\varphi _0(x_3)-x_1{\text{ Re } }\varphi _1(x_3){\textbf {e}}_1+x_2{\text{ Re } }\varphi _1(x_3){\textbf {e}}_2 \\ \nonumber&= x_1x_2\cos (\lambda x_3)-\frac{x_1\sin (\lambda x_3)}{\lambda }{} {\textbf {e}}_1+\frac{x_2\sin (\lambda x_3)}{\lambda }{} {\textbf {e}}_2 \\ \nonumber {\text{ Im } }{} {\textbf {T}}_f[u]&=x_1x_2{\text{ Im } }\varphi _0(x_3)-x_1{\text{ Im } }\varphi _1(x_3){\textbf {e}}_1+x_2{\text{ Im } }\varphi _1(x_3){\textbf {e}}_2 \\ {}&= -x_1x_2\sin (\lambda x_3) \end{aligned}$$

By Corollary 3.5, we obtain that

$$\begin{aligned} {\textbf {w}}={\mathcal {V}}[w]= \frac{1}{\lambda }( x_2\cos (\lambda x_3)-x_1\sin (\lambda x_3))\textbf{e}_1 +\frac{1}{\lambda }(x_1\cos (\lambda x_3)+x_2\sin (\lambda x_3))\textbf{e}_2, \end{aligned}$$

is a Beltrami field in $\Omega$.

Example

Now, let us consider as initial data a purely vectorial monogenic function. That is, if ${\textbf {u}}=\nabla h$, where h is a real-valued harmonic function in $\Omega$. Using (3.5) and Corollary 3.5, it is easy to verify that

$$\begin{aligned} \begin{aligned} {\textbf {w}}&=\left( \dfrac{1}{\lambda }{\text{ Im } }T_f[\partial _1\partial _3 h]+ {\text{ Re } }T_f[\partial _1h]-{\text{ Im } }T_f[\partial _2 h]\right) {\textbf {e}}_1\\&+\left( \dfrac{1}{\lambda }{\text{ Im } }T_f[\partial _2\partial _3 h]+{\text{ Re } }T_f[\partial _2h]+{\text{ Im } }T_f[\partial _1 h]\right) {\textbf {e}}_2\\&+\left( \dfrac{1}{\lambda }{\text{ Im } }T_f[\partial _3^2 h]+2 {\text{ Re } }T_{f}[\partial _3 h]\right) {\textbf {e}}_3, \end{aligned} \end{aligned}$$

(3.12)

is a Beltrami field in $\Omega$. In particular, if ${\textbf {u}}=\nabla h$ has vanishing $\textbf{e}_3$ part, then the above Beltrami field (3.12) also has vanishing $\textbf{e}_3$ part. More precisely, if $\partial _3 h=0$, then

$$\begin{aligned} \begin{aligned} {\textbf {w}}&=({\text {Re}}T_f[\partial _1h]-{\text {Im}}T_f[\partial _2 h])\textbf{e}_1+({\text {Re}}T_f[\partial _2h]+{\text {Im}}T_f[\partial _1 h])\textbf{e}_2 \\&= ( \cos (\lambda x_3) \partial _1 h- \sin (\lambda x_3)\partial _2 h)\textbf{e}_1 + (\cos (\lambda x_3)\partial _2 h + \sin (\lambda x_3)\partial _1 h)\textbf{e}_2 \end{aligned}\nonumber \\ \end{aligned}$$

(3.13)

is a Beltrami field in $\Omega$ with vanishing $\mathbf {e_3}$ part.

Theorem 3.6

Let us consider as initial data a purely vectorial monogenic function, ${\textbf {u}}=\nabla h$, where h is a real-valued harmonic function in $\Omega$. Then the following statements hold

• 1. The function

  $$\begin{aligned} w=2{\mathcal {P}}^+\left( \nabla -\lambda \right) T_f[h], \end{aligned}$$

  is $\lambda$-monogenic function in $\Omega$.

• 2. The function

  $$\begin{aligned} {\textbf {w}}=\nabla _{1,2} T_f[h] \, (1+\textbf{i}\textbf{e}_3)+\dfrac{\textbf{i}}{\lambda } \textbf{T}_f[\nabla \partial _3 h], \end{aligned}$$

  is a Beltrami field in $\Omega$.

• 3. If $\partial _3 h=0$, then the function

  $$\begin{aligned} {\textbf {w}}=f\nabla h\, (1+\textbf{i}\textbf{e}_3), \end{aligned}$$

  is a Beltrami field with vanishing $\textbf{e}_3$ component.

Proof

Now, let us consider as initial data a purely vectorial monogenic function. That is, if ${\textbf {u}}=\nabla h$, where h is a scalar harmonic function in $\Omega$. Using (3.5) with $f=e^{\textbf{i}\lambda x_3}$, we obtain that

$$\begin{aligned} \begin{aligned} \textbf{T}_{f}[\nabla h]&= T_f[\partial _1 h]\textbf{e}_1+T_f[\partial _2 h]\textbf{e}_2+T_{1/f}[\partial _3 h]\textbf{e}_3, \\&= \partial _1 T_f[h]\textbf{e}_1+\partial _2T_f[h]\textbf{e}_2+ \partial _3 T_f[h]\textbf{e}_3-\textbf{i}\lambda T_f[h]\textbf{e}_3, \\&= \nabla T_f[h]-\textbf{i}\lambda T_f[h]\textbf{e}_3. \end{aligned} \end{aligned}$$

(3.14)

Taking $u_2=0$ and $2u_1=\nabla h$ in (3.9) we have that

$$\begin{aligned} w=\textbf{T}_f[\nabla h]+\textbf{T}_f[\nabla h]\textbf{ie}_3=2{\mathcal {P}}^+\textbf{T}_f[\nabla h]\in {\text {Ker}}(D+\lambda ). \end{aligned}$$

(3.15)

Now, if we substitute (3.14) into (3.15) we have the following function w is $\lambda$-monogenic

$$\begin{aligned} \begin{aligned} w&= \nabla T_f[h]+\textbf{i}\nabla T_f[h]\textbf{e}_3-\textbf{i}\lambda T_f[h]\textbf{e}_3-\lambda T_f[h]\\&=\nabla T_f[h] (1+\textbf{ie}_3)-\lambda T_f[h](1+\textbf{ie}_3)\\&=(\nabla T_f[h]- \lambda T_f[h])(1+\textbf{ie}_3)\\&= 2{\mathcal {P}}^+\left( \nabla -\lambda \right) T_f[h] \end{aligned} \end{aligned}$$

(3.16)

For the second point, we will apply the operator ${\mathcal {V}}$ to (3.16) to get an explicit expression for the Beltrami field. Thus,

$$\begin{aligned} {\textbf {w}}&={\mathcal {V}}[2{\mathcal {P}}^+\left( \nabla -\lambda \right) T_f[h]]\\&=\left( \partial _1 T_f[h]+\textbf{i}\partial _2 T_f[h]\right) \mathbf {e_1}+\left( \partial _2 T_f[h]-\textbf{i}\partial _1 T_f[h]\right) \mathbf {e_2}\\&+\left( \partial _3 T_f[h]-\textbf{i}\lambda T_f[h]\right) \mathbf {e_3}-\dfrac{1}{\lambda }\nabla (\lambda T_f[h]+\textbf{i}\partial _3T_f[h])\\&=\textbf{i}\left( \partial _2 T_f[h]-\dfrac{1}{\lambda }\partial _1\partial _3 T_f[h]\right) \mathbf {e_1}+\textbf{i}\left( -\partial _1 T_f[h]-\dfrac{1}{\lambda }\partial _2\partial _3 T_f[h]\right) \mathbf {e_2}\\&+\textbf{i}\left( -\lambda T_f[h]-\dfrac{1}{\lambda } \partial _3^2 T_f[h]\right) \mathbf {e_3} \end{aligned}$$

Applying (3.5) in the derivatives with respect to the variable $x_3$, the above expression reduces to

$$\begin{aligned} {\textbf {w}}=\nabla _{1,2} T_f[h] \, (1+\textbf{i}\textbf{e}_3)+\dfrac{\textbf{i}}{\lambda } \textbf{T}_f[\nabla \partial _3 h], \end{aligned}$$

(3.17)

where $\nabla _{1,2}$ means that the gradient is taken only with respect to the variables $x_1$ and $x_2$. For the last point, we will suppose that $\partial _3 h=0$, using that $T_f$ only acts on the variable $x_3$, this easily implies that $T_f[h(x_1,x_2)]=h(x_1,x_2)T_f[1]=h(x_1,x_2)f=h(x_1,x_2)e^{\textbf{i}\lambda x_3}$. Thus, the Beltrami field (3.17) turns into the desired expression

$$\begin{aligned} {\textbf {w}}=f\,\nabla _{1,2}h\, (1+\textbf{i}\textbf{e}_3). \end{aligned}$$

$\square$

Remark 3.7

Consider $B_r=\{x\in {\mathbb {R}}^3:|x|<r\}$ the ball centered at 0 of radius r and let h be a harmonic function written in the form of its Taylor series expansion as follows

$$\begin{aligned} h(x)=\sum _{k_1,k_2,k_3\ge 0}\frac{\partial _1^{k_1}\partial _2^{k_2}\partial _3^{k_3} h(0)}{k_1!k_2!k_3!}x_1^{k_1}x_2^{k_2}x_3^{k_3}. \end{aligned}$$

(3.18)

Applying the continuous operator $T_f$ to the above Taylor series, we get

$$\begin{aligned} T_f[h](x)&=T_f\left[ \sum _{k_1,k_2,k_3\ge 0}\frac{\partial _1^{k_1}\partial _2^{k_2}\partial _3^{k_3} h(0)}{k_1!k_2!k_3!}x_1^{k_1}x_2^{k_2}x_3^{k_3}\right] \\&=\sum _{k_1,k_2,k_3\ge 0}\frac{\partial _1^{k_1}\partial _2^{k_2}\partial _3^{k_3} h(0)}{k_1!k_2!k_3!}T_f[x_1^{k_1}x_2^{k_2}x_3^{k_3}] \\&=\sum _{k_1,k_2,k_3\ge 0}\frac{\partial _1^{k_1}\partial _2^{k_2}\partial _3^{k_3} h(0)}{k_1!k_2!k_3!}x_1^{k_1}x_2^{k_2}\varphi _{k_3}(x_3). \end{aligned}$$

Analogously,

$$\begin{aligned} \textbf{T}_f[\nabla \partial _3 h]&=\left( \sum _{k_1,k_2,k_3\ge 0}\frac{\partial _1^{k_1}\partial _2^{k_2}\partial _3^{k_3} h(0)}{(k_1-1)!k_2!(k_3-1)!}x_1^{k_1-1}x_2^{k_2}\varphi _{k_3-1}(x_3)\right) \textbf{e}_1\\&\quad +\left( \sum _{k_1,k_2,k_3\ge 0}\frac{\partial _1^{k_1}\partial _2^{k_2}\partial _3^{k_3} h(0)}{k_1!(k_2-1)!(k_3-1)!}x_1^{k_1}x_2^{k_2-1}\varphi _{k_3-1}(x_3)\right) \textbf{e}_2\\&\quad +\left( \sum _{k_1,k_2,k_3\ge 0}\frac{\partial _1^{k_1}\partial _2^{k_2}\partial _3^{k_3} h(0)}{k_1!k_2!(k_3-2)!}x_1^{k_1}x_2^{k_2}\varphi _{k_3-2}^{*}(x_3)\right) \textbf{e}_3. \end{aligned}$$

Substituting the above expressions into (3.17) and using the explicit expression of the formal powers (3.6), we will be able to explicitly know the form of a family of Beltrami fields generated by the action of transmutation operators on harmonic functions.

The following table shows examples of Beltrami fields constructed through the transmutation method developed in this Subsection 3.2. We compute them using Mathematica program with the package QuaternionAnalysis [12].

Table 2 Transmutations of monogenic polynomials

4 Construction through harmonic functions

In this Section, we will leave aside the transmutation operator method, and using classical properties of vector calculus, we will prove that it is possible to generate a large family of Beltrami fields from harmonic functions. Following [30], we embed ${\mathbb {C}}^3$ into the space ${\mathbb {B}}$ by considering biquaternions with vanishing $\textbf{e}_3$ component, we called them reduced biquaternions. We will use the well-know property of reduced biquaternions $\textbf{e}_3\, w \,\textbf{e}_3=-{\overline{w}}$, where w is a reduced biquaternion. In particular, if w is a purely vectorial reduced biquaternion $w=w_1\textbf{e}_1+w_2\textbf{e}_2$, then $\textbf{e}_3\, w \,\textbf{e}_3=w$. Likewise, the symmetry of the domain required by the transmutation operators will not be necessary in the upcoming construction, and we can even consider unbounded domains. This class of Beltrami fields provided in Proposition 4.1 will be the key to solving a Neumann-type boundary value problem analyzed in Subsection 4.1.

Proposition 4.1

Let $\Omega$ be a bounded or unbounded domain and let $f=f(x_3)=e^{\textbf{i}\lambda x_3}$. Then

$$\begin{aligned} {\textbf {w}}=f\,\nabla u+\textbf{i}f\,\nabla u \,\textbf{e}_3=f\nabla u\, (1+\textbf{i}\textbf{e}_3) \end{aligned}$$

(4.1)

is a Beltrami field in $\Omega$ with vanishing $\textbf{e}_3$ part if and only if $u=u(x_1,x_2)$ is harmonic in $\Omega$.

Proof

Using the following property of the reduced quaternions $\textbf{e}_3\nabla _{1,2} u\textbf{e}_3=\nabla _{1,2} u$, let us verify that ${\textbf {w}}$ defined in (4.1) is a Beltrami field in $\Omega$

$$\begin{aligned} {\text{ curl } } {\textbf {w}}&={\text{ curl } } \left( f\,\nabla u+{\textbf {i}}f\,\nabla u \,{\textbf {e}}_3\right) \\ {}&=\nabla f \times \nabla u+{\textbf {i}} \nabla f\times \nabla u {\textbf {e}}_3+{\textbf {i}} f\, {\text{ curl } } (\nabla u {\textbf {e}}_3)\\ {}&={\textbf {i}}f\lambda \, {\textbf {e}}_3\times \nabla u-f\lambda \,{\textbf {e}}_3\times \nabla u\, {\textbf {e}}_3+{\textbf {i}}f \left( ({\textbf {e}}_3\cdot \nabla )\nabla u-(\Delta u){\textbf {e}}_3\right) \\ {}&=-\lambda {\textbf {w}}-{\textbf {i}}f(\Delta u)\,{\textbf {e}}_3. \end{aligned}$$

More precisely, ${\textbf {w}}$ is a Beltrami field if and only if $\Delta u=0$ in $\Omega$ as we desired. $\square$

Notice that we can arrive at this simplified expression of Beltrami fields (4.1) using some properties of the transmutation operators as illustrated in the proof of Theorem 3.6, part 3. However, the proof was shown here for completeness and using only some vector calculus identities.

Remark 4.2

The above construction of Beltrami fields with vanishing $\textbf{e}_3$ part can be replied to obtain Beltrami fields with vanishing $\textbf{e}_2$ part by taking the exponential function depending only on the variable $x_2$, namely $g=g(x_2)=e^{\textbf{i}\lambda x_2}$. In that way, the sum of both Beltrami fields will be a Beltrami field without vanishing components.

Corollary 4.3

Let $f(x_3)=e^{\textbf{i}\lambda x_3}$ and $g(x_2)=e^{\textbf{i}\lambda x_2}$. If $u_1=u_1(x_1,x_2)$ and $u_2=u_2(x_1,x_3)$ are harmonic functions in $\Omega$, then an explicit Beltrami field in $\Omega$ is given by

$$\begin{aligned} {\textbf {w}} =\left( f\partial _1u_1(1-\textbf{i})+g\partial _1u_2(1+\textbf{i})\right) \textbf{e}_1-f\partial _2 u_1(1+\textbf{i})\textbf{e}_2+g\partial _3u_2(-1+\textbf{i})\textbf{e}_3. \end{aligned}$$

Proof

By Proposition 4.1, ${\textbf {w}}_1=f\nabla u_1+\textbf{i}f\nabla u_1\textbf{e}_3$ is a Beltrami field in $\Omega$. The fact that ${\textbf {w}}_2=g\nabla u_2+\textbf{i}g\nabla u_2\textbf{e}_2$ is also a Beltrami field is analogous, now using the identity $\textbf{e}_2\, w \,\textbf{e}_2=w$ for any $w=w_1\textbf{e}_1+w_3\textbf{e}_3$.

Then the sum is also a Beltrami field, developing we have the desired expression

$$\begin{aligned} {\textbf {w}}&=f\nabla u_1+\textbf{i}f\nabla u_1\textbf{e}_3+g\nabla u_2+\textbf{i}g\nabla u_2\textbf{e}_2\\&=\left( f\partial _1u_1(1-\textbf{i})+g\partial _1u_2(1+\textbf{i})\right) \textbf{e}_1-f\partial _2 u_1(1+\textbf{i})\textbf{e}_2+g\partial _3u_2(-1+\textbf{i})\textbf{e}_3. \end{aligned}$$

$\square$

Using the previous Corollary 4.3 it is possible to provide many examples of Beltrami fields, not only generated by polynomial functions. Let us construct two explicit examples:

Example

Taking $u_1=x_1-x_2$ and $u_2=x_1^2-x_3^2$ in Corollary 4.3, then

$$\begin{aligned} {\textbf {w}}_1&=e^{{\textbf {i}}\lambda x_3}\left( {\textbf {e}}_1(1-{\textbf {i}})-{\textbf {e}}_2(1+{\textbf {i}})\right) ,\\ {\textbf {w}}_2&=2e^{{\textbf {i}}\lambda x_2}\left( x_1{\textbf {e}}_1(1+{\textbf {i}})+x_3{\textbf {e}}_3(-1+{\textbf {i}})\right) , \end{aligned}$$

are Beltrami fields in ${\mathbb {R}}^3$ with vanishing $\textbf{e}_3$ and $\textbf{e}_2$ components, respectively. Consequently,

$$\begin{aligned} {\textbf {w}}=\left( e^{\textbf{i}\lambda x_3}(1-\textbf{i})+2e^{\textbf{i}\lambda x_2} x_1 (1+\textbf{i})\right) \textbf{e}_1-e^{\textbf{i}\lambda x_3}(1+\textbf{i})\textbf{e}_2+2x_3e^{\textbf{i}\lambda x_2}(-1+\textbf{i})\textbf{e}_3, \end{aligned}$$

is a Beltrami field in ${\mathbb {R}}^3$.

Example

Taking $u_1=\ln (x_1^2+x_2^2)$ and $u_2=x_1^2-x_3^2$ in Corollary 4.3, then

$$\begin{aligned} \nabla u_1=\tfrac{2x_1}{x_1^2+x_2^2}\textbf{e}_1+\tfrac{2x_2}{x_1^2+x_2^2}\textbf{e}_2, \qquad \nabla u_2=2(x_1\textbf{e}_1-x_3\textbf{e}_3). \end{aligned}$$

Furthermore,

$$\begin{aligned} {\textbf {w}}_1&=\frac{2e^{\textbf{i}\lambda x_3}}{x_1^2+x_2^2}\left( (x_1-\textbf{i}x_2)\textbf{e}_1+(x_2-\textbf{i}x_1)\textbf{e}_2\right) ,\\ {\textbf {w}}_2&=2e^{\textbf{i}\lambda x_2}\left( x_1\textbf{e}_1(1+\textbf{i})+x_3\textbf{e}_3(-1+\textbf{i})\right) , \end{aligned}$$

are Beltrami fields in $\Omega ={\mathbb {R}}^3{\setminus } \{(x_1,x_2,x_3)\in {\mathbb {R}}^3:x_1=x_2=0\}$ with vanishing $\textbf{e}_3$ and $\textbf{e}_2$ components, respectively. Thus,

$$\begin{aligned} {\textbf {w}}&=\left( \frac{2e^{\textbf{i}\lambda x_3}}{x_1^2+x_2^2}(x_1-\textbf{i}x_2)+2e^{\textbf{i}\lambda x_2} x_1 (1+\textbf{i})\right) \textbf{e}_1+\frac{2e^{\textbf{i}\lambda x_3}}{x_1^2+x_2^2}(x_2-\textbf{i}x_1)\textbf{e}_2\\&\qquad +2x_3e^{\textbf{i}\lambda x_2}(-1+\textbf{i})\textbf{e}_3, \end{aligned}$$

is a Beltrami field in $\Omega$.

4.1 Boundary Value Problems for Beltrami Fields

Let us consider the following boundary value problem of Neumann-type for Beltrami fields

$$\begin{aligned} ({\text {curl}} +\lambda ){\textbf {w}}&=0,\, \hspace{.6 cm} \quad \text { in }\Omega ,\nonumber \\ {\textbf {w}}|_{\partial \Omega }\cdot \eta&=\alpha , \qquad \text { on }\partial \Omega . \end{aligned}$$

(4.2)

We will construct solutions of (4.2) using the explicit expression of Beltrami fields given in Proposition 4.1 as well as some interesting facts concerning the normal and tangential derivative of conjugates harmonic functions.

In the following Theorem 4.4, the harmonic functions u and v will be considered in the complex variable $z=x_1+\textbf{i}x_2$. Thus, $\nabla u=\partial _1 u\,\textbf{e}_1+\partial _2 u\, \textbf{e}_2$ and $\Delta u=\partial _1^2 u+\partial _2 ^2 u$. Recall that the tangential derivative is usually defined by

$$\begin{aligned} \partial _T u=\partial _2 u\,\eta _1-\partial _1 u\,\eta _2, \end{aligned}$$

where $\eta _1$ and $\eta _2$ are the first two components of the outward normal vector $\eta$ to $\partial \Omega$, and the normal derivative is given by

$$\begin{aligned} \partial _{\eta } u=\partial _1 u\,\eta _1+\partial _2 u\,\eta _2. \end{aligned}$$

Moreover, the intrinsic relationship between the tangential and normal derivative of harmonic conjugates functions is well known. Indeed, if $w=u+\textbf{i}v$ is holomorphic in $\Omega$, then its real and imaginary parts satisfy

$$\begin{aligned} \dfrac{\partial u}{\partial \eta }=\partial _T v,\qquad \dfrac{\partial v}{\partial \eta }=-\partial _T u. \end{aligned}$$

(4.3)

For the following result, it will be sufficient to work with three-dimensional domains $\Omega$ with sufficient regularity for the normal vector to be well defined, and also, we will assume that $\partial \Omega$ is a two-dimensional surface, a classic example would be the sphere. The latter will guarantee that we can express the trace of the scalar and vector fields in terms only of the variables $x_1$ and $x_2$.

Theorem 4.4

Let $f=f(x_3)=e^{\textbf{i}\lambda x_3}$ and $\alpha =\alpha (x_1,x_2)\in C^1(\partial \Omega )$. Then

$$\begin{aligned} {\textbf {w}}={\textbf {w}}_1+f\,\nabla u \,(1+\textbf{i}\textbf{e}_3) +f^{-1}\nabla v\, (\textbf{i}+\textbf{e}_3) \end{aligned}$$

(4.4)

satisfies the BVP (4.2), where ${\textbf {w}}_1$ is an arbitrary $C^1({\overline{\Omega }})$ Beltrami field and $u=u(x_1,x_2)$ is a solution of de BVP

$$\begin{aligned} \Delta u&=0, \qquad \qquad \qquad \hspace{.9 cm}\text { in }\Omega ,\nonumber \\ \dfrac{\partial u}{\partial \eta }&=\dfrac{1}{2f}\left( \alpha -{\textbf {w}}_1|_{\partial \Omega }\cdot \eta \right) \quad \text { on }\partial \Omega , \end{aligned}$$

(4.5)

and $vf^{-1}$ is a harmonic conjugate of fu.

Proof

Let u be the unique solution of (4.5) up to constants and let $f^{-1}v$ be a harmonic conjugate of fu, considering the complex variable $z=x_1+\textbf{i}x_2$. By Proposition 4.1, ${\textbf {v}}:=f\nabla v(1+\textbf{i}\textbf{e}_3)$ is a Beltrami field, then ${\textbf {v}}^*\in {\text {Ker}} (D+\lambda )$. Using that ${\text {Ker}} (D+\lambda )$ is a quaternionic right module, then ${\textrm{v}}^*\textbf{e}_3=f^{-1}\nabla v (\textbf{i}+\textbf{e}_3)$ is also a Beltrami field.

Now, in order to verify that the boundary condition ${\textbf {w}} |_{\partial \Omega }\cdot \eta =\alpha$ is fulfilled, we use that u is the unique solution of the BVP (4.5), then

$$\begin{aligned} {\textbf {w}}|_{\partial \Omega }\cdot \eta&={\textbf {w}}_1|_{\partial \Omega }\cdot \eta +f\dfrac{\partial u}{\partial \eta }+{\textbf {i}} f\, \partial _T(u)+\dfrac{{\textbf {i}}}{f}\, \dfrac{\partial v}{\partial \eta }+\dfrac{1}{f}\partial _T(v)\\ {}&={\textbf {w}}_1|_{\partial \Omega }\cdot \eta +\dfrac{\partial (f\,u)}{\partial \eta }+{\textbf {i}}\, \partial _T(f\, u)+{\textbf {i}}\,\dfrac{\partial (f^{-1}\, v)}{\partial \eta }+\partial _T(f^{-1}\,v) \end{aligned}$$

using the fact that f only depends on the variable $x_3$ and $u=u(x_1,x_2)$ and $v=v(v_1,v_2)$ only depend on the variables $x_1$ and $x_2$. Now, by the interrelation between the normal and tangential derivative between harmonic conjugates functions (4.3), we obtain

$$\begin{aligned} {\textbf {w}}|_{\partial \Omega }\cdot \eta =w_1|_{\partial \Omega }\cdot \eta +2 \dfrac{\partial (f\,u)}{\partial \eta }=\alpha , \end{aligned}$$

which is exactly our Neumann boundary condition. $\square$

Observe that we can replace the hypothesis $\alpha =\alpha (x_1,x_2)\in C^1(\partial \Omega )$ by $\alpha =\alpha (x_1,x_2,x_3)\in C^1({\overline{\Omega }})$, because by the condition impose on the domain the trace $\alpha |_{\partial \Omega }$ will depend only on the variables $x_1$ and $x_2$.

4.2 Associated BVP’s

Let us consider ${\varvec{\alpha }}=\nabla f/f$, it is well-known that

$$\begin{aligned} (D+M^{{\varvec{\alpha }}})\,{\textbf {w}}=0 \end{aligned}$$

if and only if

$$\begin{aligned} {\text {div}} \left( f {\textbf {w}}\right) =0, \qquad {\text {curl}} \left( \dfrac{{\textbf {w}}}{f}\right) =0. \end{aligned}$$

(4.6)

In [7, Ch. 6] it was analyzed the previous system for the in-homogeneous case, using a divergence-free invariant right inverse of the curl operator. In the same work, the equivalence of the system (4.6) was used to analyze the time-independent Maxwell’s system in inhomogeneous media.

Following [23] we will say that $f=f(x_3)$ is a non-vanishing coefficient in the direction of $x_3$ if $f\in C^2[-a,a]$ such that $f(x)\not =0$ for all $x \in [-a,a]$ and $f(0)=1$. If we take this function into (3.2), it is known that there exists a Volterra operator $T_f$ which satisfies (3.3) (see [18]). But in general, obtaining a closed form for the formal powers related to any non-vanishing coefficient is very difficult.

Previously in [23, Th. 4.3], it was constructed a transmutation operator which sends monogenic functions into solutions of $\left( D+M^{\frac{f^{\prime }}{f}\textbf{e}_3}\right) u=0$. More precisely, it was established the following transmutation relation

$$\begin{aligned} \left( D+M^{\frac{f^{\prime }}{f}{} {\textbf {e}}_3}\right) {\textbf {T}}_f[u]={\textbf {T}}_{1/f}\,[D\, u], \qquad \forall u\in C^1(\Omega ), \end{aligned}$$

(4.7)

where the transmutation operator $\textbf{T}_f$ was previously defined in (3.7).

In this subsection, we will analyze the following boundary value problem

$$\begin{aligned} \left( D+M^{\frac{f^{\prime }}{f}\textbf{e}_3}\right) {\textbf {w}}&=0,\, \qquad \text { in }\Omega ,\nonumber \\ {\textbf {w}}|_{\partial \Omega }\cdot \eta&=\beta , \qquad \text { on }\partial \Omega , \end{aligned}$$

(4.8)

where $f=f(x_3)$ be a non-vanishing coefficient for $\Omega$ in the direction of $x_3$.

There exists a close relationship between the above BVP (4.8) and an associated BVP for the conductivity equation. Indeed,

Proposition 4.5

Let $f=f(x_3)$ be a non-vanishing coefficient for $\Omega$ in the direction of $x_3$, ${\textbf {v}}$ an arbitrary $C^1({\overline{\Omega }}, {\mathbb {B}})$ purely vectorial monogenic function in $\Omega$ and $\beta \in C^1(\partial \Omega )$. Then

$$\begin{aligned} {\textbf {w}}=\textbf{T}_f[{\textbf {v}}]+f\,\nabla u \end{aligned}$$

satisfies the BVP (4.8) if and only if u is solution of the following BVP for the conductivity equation

$$\begin{aligned} \begin{aligned} {\text{ div } }(f^2\nabla u)= & {} 0,\, \qquad \qquad \qquad \qquad \,\,\,\quad \text{ in } \Omega ,\\ f^2\, \nabla u|_{\partial \Omega }\cdot \eta= & {} f\,\beta -f\,{\textbf {T}}_f[{\textbf {v}}]|_{\partial \Omega }\cdot \eta , \qquad \text{ on } \partial \Omega . \end{aligned} \end{aligned}$$

(4.9)

Proof

By (4.7), we have that $\textbf{T}_f[{\textbf {v}}]\in {\text {Ker}} \left( D+M^{\frac{f^{\prime }}{f}\textbf{e}_3}\right)$, for any purely vectorial monogenic function ${\textbf {v}}\in C^1({\overline{\Omega }})$. On the other hand, by (4.6) we obtain that $f\,\nabla u\in {\text {Ker}} \left( D+M^{\frac{f^{\prime }}{f}\textbf{e}_3}\right)$ if and only if u is a solution of the conductivity equation ${\text{ div } }(f^2 \nabla u)=0$ in $\Omega$. The equivalence of the boundary condition is straightforward. $\square$

It is well-known that the BVP (4.9) is well-defined, and the existence of a unique solution is usually analyzed through variational methods. More precisely, the solution u of the BVP (4.9) is given by the minimum of the following functional (see [29, Ch. 4])

$$\begin{aligned} L[v]=\int _{\Omega } f^2\, \nabla v \cdot \nabla v-\int _{\partial \Omega } \gamma \,v, \end{aligned}$$

for all $v\in H^1(\Omega )$, where $\gamma$ stands for the Neumann boundary data $\gamma =f\,\beta -f\,\textbf{T}_f[{\textbf {v}}]|_{\partial \Omega }\cdot \eta$.

5 Orthogonal Decomposition

Let $\lambda$ be a fixed complex number. We are interested in studying the space of $L^p$ Beltrami fields defined in $\Omega$ and taking values in ${\mathbb {C}}^3$, so we shall define

$$\begin{aligned} {\mathcal {B}}^p(\Omega )=\left\{ {\textbf {u}}\in L^p(\Omega ) :({\text{ curl }}+\lambda )\,{\textbf {u}}=0 \text{ in } \Omega \right\} , \end{aligned}$$

in the sense of distributions. It is straightforward that ${\mathcal {B}}^p(\Omega )\subset {\text {Sol}}^p(\Omega )$, where ${\text {Sol}}^p(\Omega )$ means the space of $L^p$ divergence-free complex-valued vector fields. In the following, we will provide an orthogonal decomposition of $L^2(\Omega ,{\mathbb {C}}^3)$, under the scalar product

$$\begin{aligned} \left\langle {\textbf {f}},{\textbf {g}}\right\rangle = \int _{\Omega } {\textbf {f}}^*\cdot {\textbf {g}}\, dx. \end{aligned}$$

(5.1)

Recall the Teodorescu transform

$$\begin{aligned} T_{\Omega }[w](x)=\int _{\Omega }\dfrac{y-x}{4\pi |y-x|^3}\,w(y)\,dy,\quad x\in {\mathbb {R}}^{3}, \end{aligned}$$

which is a right inverse of the Moisil-Teodorescu operator. That is, $DT_{\Omega }=I$ in $\Omega$. Let us denote by $A^p(\Omega )=A^p(\Omega ,{\mathbb {B}})$ the subspace of monogenic functions in $L^p(\Omega ,{\mathbb {B}})$, $1<p<\infty$, which is also a right ${\mathbb {B}}$-module, with the usual norm given by

$$\begin{aligned} \Vert w\Vert _{A^p(\Omega )}^p=\Vert w\Vert _{L^p(\Omega )}^p=\int _{\Omega }{|w|_c^p \, dx}. \end{aligned}$$

It is well-known that $A^p(\Omega ,{\mathbb {B}})$ is a closed subspace in $L^p(\Omega ,{\mathbb {B}})$ $1<p<\infty$ [13, Prop. 3.73].

Following the ideas in [9], we can use the continuity of the Teodorescu operator and the closedness of the classical Bergman spaces $A^p(\Omega )$ to prove that ${\mathcal {B}}^p(\Omega )$ is a closed subspace of $L^p(\Omega )$.

Proposition 5.1

${\mathcal {B}}^p(\Omega )$ is a closed subspace of $L^p(\Omega )$, for $1<p<\infty$.

Proof

Let $\left\{ {\textbf {u}}_n\right\} \subseteq {\mathcal {B}}^p(\Omega )$ be a sequence such that ${\textbf {u}}_n\rightarrow {\textbf {u}}$ in $L^p(\Omega )$. Then,

$$\begin{aligned} ({\text{ curl } }+\lambda )\,{\textbf {u}}_n=0, \qquad {\text{ div } } {\textbf {u}}_n=0, \quad \text{ in } \Omega . \end{aligned}$$

In other words, $(D+\lambda ){\textbf {u}}_n=0$ in $\Omega$ for all n. Using that $DT_{\Omega }=I$ in $\Omega$, we equivalently obtain that $\left\{ {\textbf {u}}_n+T_{\Omega }[\lambda {\textbf {u}}_n]\right\} \subseteq A^p(\Omega )$. Further, it converges to ${\textbf {u}}+T_{\Omega }[\lambda {\textbf {u}}]$ in $L^p(\Omega )$. Indeed, by continuity of the Teodorescu transform $T_{\Omega }$, we have

$$\begin{aligned} \Vert {\textbf {u}}_n+T_{\Omega }[\lambda {\textbf {u}}_n]-({\textbf {u}}+T_{\Omega }[\lambda {\textbf {u}}])\Vert _{L^p(\Omega )} \le (1+\Vert T_{\Omega }\Vert \lambda )\Vert {\textbf {u}}_n-{\textbf {u}}\Vert _{L^p(\Omega )}, \end{aligned}$$

where $\Vert T_{\Omega }\Vert$ is the operator norm from $L^p(\Omega )$ to itself. Consequently, by the closedness of the classical Bergman spaces $A^p(\Omega )$ in $L^p(\Omega )$, we get that ${\textbf {u}}+T_{\Omega }[\lambda {\textbf {u}}]\in A^p(\Omega )$. Therefore, ${\textbf {u}}\in {\mathcal {B}}^p(\Omega )$, which implies the result. $\square$

Let us consider the subspace of vector fields with vanishing tangential trace. Namely, $W_{\eta }^{1,2}(\Omega , {\mathbb {C}}^3)=\{{\textbf {u}}\in W^{1,2}(\Omega , {\mathbb {C}}^3):{\textbf {u}}|_{\partial \Omega }\times \eta =0\}$.

Theorem 5.2

Let $\Omega$ be a Lipschitz domain. The following orthogonal decomposition holds

$$\begin{aligned} L^2(\Omega ,{\mathbb {C}}^3)={\mathcal {B}}^2(\Omega ) \oplus ({\text {curl}}+\lambda ^*)W_{\eta }^{1,2}(\Omega ,{\mathbb {C}}^3), \end{aligned}$$

(5.2)

under the scalar product (5.1).

Proof

Let ${\textbf {h}}\in W_{\eta }^{1,2}(\Omega )$. Using the Green’s formula

$$\begin{aligned} \int _{\Omega } ({\text{ curl } } {\textbf {w}}^*\cdot {\textbf {h}}-{\textbf {w}}^*\cdot {\text{ curl } } {\textbf {h}})\, dx=\int _{\partial \Omega }({\textbf {w}}^*|_{\partial \Omega }\times {\textbf {h}}|_{\partial \Omega })\cdot \eta \, ds_x. \end{aligned}$$

(5.3)

Then, the right-hand side (5.3) vanishes. Consequently,

$$\begin{aligned} \int _{\Omega } ({\text {curl}}+\lambda ^*) {\textbf {w}}^*\cdot {\textbf {h}}\, dx=\int _{\Omega }{} {\textbf {w}}^*\cdot ({\text {curl}}+\lambda ^*) {\textbf {h}}\, dx. \end{aligned}$$

In other words,

$$\begin{aligned} \langle ({\text {curl}}+\lambda ) {\textbf {w}}, {\textbf {h}}\rangle =\langle {\textbf {w}}, ({\text {curl}}+\lambda ^*) {\textbf {h}}\rangle , \qquad \forall {\textbf {h}}\in W_{\eta }^{1,2}(\Omega ). \end{aligned}$$

(5.4)

If ${\textbf {w}}\in {\mathcal {B}}^2(\Omega )$, then we get $({\text {curl}}+\lambda ^*)W_{\eta }^{1,2}(\Omega )\subset {\mathcal {B}}(\Omega )^{2\,\bot }$. Now, we will prove that $({\text {curl}}+\lambda ^*)C_c^{\infty }(\Omega )^{\bot }\subset {\mathcal {B}}^2(\Omega )$. Let ${\textbf {u}}\in ({\text {curl}}+\lambda ^*)C_c^{\infty }(\Omega )\,^{\bot }$, then by (5.4) we get

$$\begin{aligned} 0=\langle {\textbf {u}}, ({\text {curl}}+\lambda ^*) {\textbf {h}}\rangle =\langle ({\text {curl}}+\lambda ) {\textbf {u}}, \textbf{h}\rangle ,\qquad \forall {\textbf {h}}\in C_c^{\infty }(\Omega ). \end{aligned}$$

Consequently, ${\textbf {u}}\in {\mathcal {B}}^2(\Omega )$, as we desired. Using that $C_c^{\infty }(\Omega )$ is dense in $W_0^{1,2}(\Omega )$, we obtain

$$\begin{aligned} {\mathcal {B}}^2(\Omega )^{\bot }\subset \overline{({\text {curl}}+\lambda ^*)C_c^{\infty }(\Omega )}=({\text {curl}}+\lambda ^*)W^{1,2}_{0}(\Omega )\subset ({\text {curl}}+\lambda ^*)W^{1,2}_{\eta }(\Omega ). \end{aligned}$$

$\square$

In particular, if we consider $\lambda \in {\mathbb {R}}$, then the above decomposition changes as follows

$$\begin{aligned} L^2(\Omega ,{\mathbb {R}}^3)={\mathcal {B}}^2(\Omega ) \oplus ({\text {curl}}+\lambda )W_{\eta }^{1,2}(\Omega ), \end{aligned}$$

(5.5)

under the inner product

$$\begin{aligned} \langle {\textbf {f}},{\textbf {g}}\rangle = \int _{\Omega } {\textbf {f}}\cdot {\textbf {g}}\, dx, \qquad \forall {\textbf {f}}, {\textbf {g}}\in L^2(\Omega , {\mathbb {R}}^3). \end{aligned}$$

Compare the orthogonal decompositions (5.2), (5.5) with the Hodge decomposition obtained in [9, Th. 11] for the kernel of the Vekua operator $D-\alpha C-\beta$, where $\alpha$ and $\beta$ are bounded functions, and C means Clifford conjugation, by taking $\alpha =0$ and $\beta =-\lambda$.

6 Conclusions

This study presents a closed-form solution for Beltrami fields using transmutation theory and quaternionic analysis. The rich theory of harmonic functions and Taylor series, combined with well-established transmutation techniques, allows the development of numerous examples.

Furthermore, these techniques can be adapted to solve other differential equations in mathematical physics. For instance, solutions to partial differential operators like $D +\lambda$ and $D-\lambda$ or $\Delta +\lambda ^2$ can be constructed using the same ideas.

The motivation for this work lies in its potential applications, by providing a complete system of solutions, this method paves the way for implementing computational methods to solve boundary value problems.