A right inverse of curl operator which is divergence free invariant and some applications to generalized Vekua type problems

TL;DR

This paper develops a divergence-free, curl-invariant right inverse operator for vector fields in bounded domains, with applications to Beltrami fields, Vekua problems, and Maxwell's equations, advancing mathematical tools for vector calculus and PDEs.

Contribution

It introduces a novel divergence-free, curl-invariant right inverse of the curl operator and applies it to solve various PDE problems in physics and mathematics.

Findings

Constructed a bounded right inverse of curl operator.

Extended the inverse to divergence-free fields with boundary conditions.

Applied the inverse to problems in electromagnetism and fluid dynamics.

Abstract

In this work, we investigate the system formed by the equations $\text{div } \vec w=g_0$ and $\text{curl } \vec w=\vec g$ in bounded star-shaped domains of $\mathbb{R}^3$. A Helmholtz-type decomposition theorem is established based on a general solution of the above-mentioned div-curl system which was previously derived in the literature. When $g_0\equiv 0$, we readily obtain a bounded right inverse of $\text{curl}$ which is a divergence-free invariant. The restriction of this operator to the subspace of divergence-free vector fields with vanishing normal trace is the well-known Biot--Savart operator. In turn, this right inverse of $\text{curl}$ will be modified to guarantee its compactness and satisfy suitable boundary-value problems. Applications to Beltrami fields, Vekua-type problems as well as Maxwell's equations in inhomogeneous media are included.

Equations

$$\vec{w}=-\mathrm{grad}{v}_0+\mathrm{curl}\vec{v}.$$

$$v_0(\vec{x})$$

$$\vec{v}(\vec{x})$$

$$L_{{\mathbb{R}}^3}[u](\vec{x})=-\frac{1}{4\pi }{\int }_{{\mathbb{R}}^3}\frac{u(\vec{y})}{\left|\vec{x}-\vec{y}\right|}d\vec{y},$$

$$\mathrm{curl}\vec{v}(\vec{x})=\mathrm{curl}\frac{1}{4\pi }{\int }_{{\mathbb{R}}^3}\frac{\mathrm{curl}\vec{w}(\vec{y})}{\left|\vec{x}-\vec{y}\right|}d\vec{y}=-{\int }_{{\mathbb{R}}^3}\frac{(\vec{x}-\vec{y})\times \mathrm{curl}\vec{w}(\vec{y})}{4\pi |\vec{x}-\vec{y}{|}^3}d\vec{y},$$

$$\vec{w}=\mathrm{grad}{v}_0^{\ast }-\mathrm{curl}{\vec{v}}^{\ast }.$$

$$\{\begin{array}{rl}{v}_0^{\ast }(x)& ={\int }_{\Omega }\frac{\mathrm{div}\vec{w}(\vec{y})}{4\pi |\vec{x}-\vec{y}|}dy,\\ {\vec{v}}^{\ast }(\vec{x})& ={\int }_{\Omega }\frac{\mathrm{curl}\vec{w}(\vec{y})}{4\pi |\vec{x}-\vec{y}|}dy+{\int }_0^1\frac{t|\vec{x}{|}^2}{2}\mathrm{grad}{\psi }_0(t\vec{x})dt.\end{array}$$

$${\psi }_0(\vec{x})=-{\mathrm{div}}_{\vec{x}}{\int }_{\Omega }\frac{\mathrm{curl}\vec{w}(\vec{y})}{4\pi |\vec{x}-\vec{y}|}d\vec{y}.$$

$$T_{\Omega }[-\mathrm{div}\vec{w}+\mathrm{curl}\vec{w}]=T_{0,\Omega }[\mathrm{curl}\vec{w}]-{\overrightarrow{T}}_{1,\Omega }[\mathrm{div}\vec{w}]+{\overrightarrow{T}}_{2,\Omega }[\mathrm{curl}w],$$

$$T_{0,\Omega }[\mathrm{curl}\vec{w}]$$

$${\overrightarrow{T}}_{1,\Omega }[\mathrm{div}w]$$

$${\overrightarrow{T}}_{2,\Omega }[\mathrm{curl}\vec{w}]$$

$$\{\begin{array}{rl}\mathrm{curl}\vec{w}& =\vec{g},\\ \vec{w}{|}_{\partial \Omega }& =0,\end{array}$$

$$W^{p,\mathrm{div}}(\Omega ,{\mathbb{R}}^3)$$

$$W^{p,\mathrm{curl}}(\Omega ,{\mathbb{R}}^3)$$

$$\Vert \vec{u}{\Vert }_{W^{p,\mathrm{div}}}$$

$$\Vert \vec{u}{\Vert }_{W^{p,\mathrm{curl}}}$$

$$W^{p,\text{mbox}div-curl}(\Omega ,{\mathbb{R}}^3)=W^{p,\mathrm{div}}(\Omega ,{\mathbb{R}}^3)\cap W^{p,\mathrm{curl}}(\Omega ,{\mathbb{R}}^3).$$

$$⟨{\gamma }_{\mathbf{n}}(\vec{u}),\mathrm{tr}{v}_0{⟩}_{\partial \Omega }={\int }_{\Omega }\vec{u}\cdot \nabla v_{0}d\vec{y}+{\int }_{\Omega }(\mathrm{div}\vec{u})v_{0}d\vec{y},$$

$$W_0^{2,\mathrm{div}}(\Omega ,{\mathbb{R}}^3)=\{\vec{u}\in W^{p,\mathrm{div}}(\Omega ,{\mathbb{R}}^3):{\gamma }_{\mathbf{n}}(\vec{u})=0\}.$$

$$W_{\mathbf{n}}^{p,\mathrm{div}\text{-}\mathrm{curl}}(\Omega ,{\mathbb{R}}^3)=W_0^{p,\mathrm{div}}(\Omega ,{\mathbb{R}}^3)\cap W^{p,\mathrm{curl}}(\Omega ,{\mathbb{R}}^3),$$

$$\mathrm{div}\vec{w}$$

$$\mathrm{curl}\vec{w}$$

$${\int }_{\Omega }\vec{g}\cdot \nabla u_{0}d\vec{x}=0,$$

$$D=e_1\frac{\partial }{\partial x_1}+e_2\frac{\partial }{\partial x_2}+e_3\frac{\partial }{\partial x_3}.$$

$$D\vec{w}=-g_0+\vec{g}.$$

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

$$F_{\partial \Omega }[\phi ](\vec{x})$$

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

$$T_{0,\Omega }[\vec{w}](\vec{x})={\int }_{\Omega }E(\vec{y}-\vec{x})\cdot \vec{w}(\vec{y})d\vec{y},$$

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

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

$${\overrightarrow{U}}_{\Omega }[w_0](\vec{x})=\mathrm{Vec}\left({\int }_0^{1}t\vec{x}D{w}_0(t\vec{x})dt\right)={\int }_0^{1}t\vec{x}\times \nabla w_0(t\vec{x})dt,\vec{x}\in \Omega .$$

$$\vec{w}=-{\overrightarrow{T}}_{1,\Omega }[g_0]+{\overrightarrow{T}}_{2,\Omega }[\vec{g}]-{\overrightarrow{U}}_{\Omega }[T_{0,\Omega }[\vec{g}]]+\nabla h,$$

Full text

00footnotetext: Keywords: Div-curl system, Neumann problem, Dirichlet problem, Beltrami fields, Vekua-type problem, Maxwell’s equations 00footnotetext: Mathematics Subject Classification (2020): 35Q60, 30G20.

1 Introduction

One of the fundamental theorems in vector analysis is the very well-known “Helmholtz Decomposition Theorem”. This result states that any vector field in $\mathbb{R}^{3}$ can be completely characterized in terms of its divergence and curl (sometimes also called rotational or vorticity). This theorem was formulated by Hermann von Helmholtz [19], and it represents any vector field $\vec{w}$ in $\mathbb{R}^{3}$ as the sum of a divergence-free vector field $(\operatorname{curl}\vec{v})$ and an irrotational vector field $(\operatorname{grad}v_{0})$. More precisely, the following decomposition is satisfied:

[TABLE]

Here, the Helmholtz potentials $v_{0}$ and $\vec{v}$ are given (see [24, p. 166] and [18, Thm. 5.1.1]) by

[TABLE]

In turn, $L$ is the Newton potential is defined by

[TABLE]

and it is a right inverse of the Laplacian. Later on, the uniqueness of the Helmholtz decomposition (1.1) was proved under the assumption that the solution satisfies the asymptotic behavior $w(\vec{x})=o(1/|\vec{x}|)$, for $|\vec{x}|\rightarrow\infty$. In particular, it was established that

[TABLE]

is a right inverse of $\operatorname{curl}$ in the entire three-dimensional space. It is worth pointing out here that (1.5) is sometimes called the Biot–Savart operator.

In this work, we will use techniques employed in various works concerning the div-curl system in star-shaped domains [8], the div-curl system in Lipschitz domains [9], a perturbed div-curl system [11] and the div-curl system in exterior domains [12]. In all of those works, the quaternionic analysis played a fundamental role. In particular, we will show that any vector field $\vec{w}$ admits a Helmholtz-type decomposition in bounded star-shaped domains (see Proposition 3 below) as follows:

[TABLE]

Here, the potentials $v_{0}^{*}$ and $\vec{v}^{*}$ are defined over $\Omega$, and are given by

[TABLE]

Moreover,

[TABLE]

In addition, $\operatorname{div}\vec{v}^{*}$ is harmonic in $\Omega$. Some examples of this solution are computed implicitly in Example 13. Comparing (1.1) and (1.6), we can see that the potentials are now given as integrals over the domain $\Omega$ instead of over the entire three-dimensional space. Besides, our potential $\vec{v}^{*}$ has an extra term, we will analyze in detail the operator which represents it, as it is the key for the analysis of some boundary-value problems associated to the div-curl system.

Analogizing the operators involved in the Helmholtz-type decomposition (1.6)–(1.8) with one of the most important operators in quaternionic analysis —the Teodorescu transform $T_{\Omega}$—, the following is an appropriate decomposition:

[TABLE]

where

[TABLE]

and $v_{0}^{*}$, $\vec{v}^{*}$ and $\psi_{0}$ are as previously defined in (1.7) and (1.8), respectively.

One of the aims of this work is to study analytically the solutions of the div-curl system in star-shaped domains under appropriate boundary conditions. In that sense, this manuscript may be considered as a completion of the analysis carried out in [8]. It is important to point out here that a general solution of the div-curl system in star-shaped domains was obtained in [8] assuming no boundary conditions. The explicit solution of that problem is recalled below in Theorem 1, and it is given in terms of some integral operators appearing in quaternionic analysis as well as a monogenic completion operator. The solution of the problem under consideration in the present work hinges on embedding the vector-valued differential system into a quaternionic-valued one. A solution is found in the algebra $\mathbb{H}$ of quaternions, and a projection into $\mathbb{R}^{3}$ is performed then without affecting the equivalent system. As a consequence of the Helmholtz-type decomposition (1.6)-(1.8), we obtain a right inverse of $\operatorname{curl}$. The fact that this right inverse is not unique as well as its regularity properties will be used to satisfy some suitable boundary-value problems (BVPs). These results will be presented in Section 4. Throughout, we will work in the context of the spaces $W^{p,\mbox{\scriptsize div-curl}}(\Omega)$ and $W_{{}_{\mathbf{n}}}^{p,\mbox{\scriptsize div-curl}}(\Omega)$ in order to guarantee a natural regularity for all $L^{p}$-solutions.

It is important to mention that there are several results in the literature on the existence of an inverse of $\operatorname{curl}$. To that end, authors usually impose some suitable additional boundary conditions in bounded domains[14, 39]. In particular, the existence of a compact inverse operator of $\operatorname{curl}$ in a subspace of the divergence-free vector fields with vanishing normal trace was proved in [39]. The boundary conditions used in that work assured that $\operatorname{curl}$ and $\operatorname{curl}^{-1}$ were self-adjoint operators in their respective domains of definition. In the present work, we will provide an explicit expression of a right inverse of $\operatorname{curl}$ from the subspace of divergence-free vector fields to the subspace of divergence-free vector fields with vanishing normal trace. The right inverse preserves the property of being compact (see Proposition 5 below). It is worth mentioning that the domains of definition of $\operatorname{curl}$ and its right inverse will be larger than that considered in [39].

On the other hand, the BVP

[TABLE]

has been extensively studied, see for instance [3, 15, 37]. In particular, the author of [15] provided necessary and sufficient conditions for the existence of a unique solution which depends continuously on $\vec{g}$ in bounded domains of $\mathbb{R}^{n}$. More recently, the authors of [2] found a solution in star-shaped domains with respect to a ball, under the assumption that $\vec{g}$ is a divergence-free Dini-continuous function. That solution was expressed in terms of an integral representation formula which, in turn, was obtained in [15]. In our construction of a right inverse of $\operatorname{curl}$ that vanishes on the boundary, the monogenic completion process involved in the construction of the right inverse of $\operatorname{curl}$ without boundary conditions is straightforward because the scalar part of the Teodorescu transform applied to $\vec{g}$ vanishes. In the alternative proof that we present herein, we will follow some ideas used in [3]. As an application of our right inverse of the $\operatorname{curl}$ operator (namely $R_{\Omega}$, which is bounded in $L^{p}$ and divergence-free invariant), we will construct Beltrami fields with coefficient $\alpha_{0}$ through a Neumann series that converges uniformly for certain values of $\alpha_{0}$ satisfying $|\alpha_{0}|<1/\|R_{\Omega}\|$ (see Theorem 12).

The outline of this manuscript is as follows. In Section 2, we will present some preliminaries needed to construct the general solution of the div-curl system. In Section 3.2, we will provide a Helmholtz-type decomposition in bounded star-shaped domains of $\mathbb{R}^{3}$, which generalizes the classical Helmholtz decomposition (1.1). Moreover, a right inverse of $\operatorname{curl}$ without boundary conditions is derived and we analyze some properties inherited from the Teodorescu transform, such as boundedness and compactness. In Subsection 4, we will investigate BVPs for the equation $\operatorname{curl}\vec{w}=\vec{g}$, considering the Neumann boundary condition $\vec{w}|_{\partial\Omega}\cdot\eta=0$ or, alternatively, the Dirichlet boundary condition $\vec{w}|_{\partial\Omega}=0$. Section 5 provides an application of the bounded right inverse of $\operatorname{curl}$ to the construction of Beltrami fields. More precisely, we analyze the eigenvalue problem $\operatorname{curl}\vec{w}=\alpha_{0}\vec{w}$ without boundary conditions, and then with the boundary condition $\vec{w}|_{\partial\Omega}\cdot\eta=a_{0}$ as well (see Propositions 12 and 14, respectively). The last section of this work will be devoted to applying our results to several related problems, including some generalized Vekua-type problems and the Maxwell system in inhomogeneous media.

2 Preliminaries

2.1 Function spaces

In this work, we will employ the usual function spaces associated to the operators $\operatorname{div}$ and $\operatorname{curl}$, which appear in many problems on electromagnetism [7, 14]. More precisely, we will let

[TABLE]

It is well known that $W^{p,\operatorname{div}}(\Omega,\mathbb{R}^{3})$ and $W^{p,\operatorname{curl}}(\Omega,\mathbb{R}^{3})$ are Banach spaces with respect to the norms

[TABLE]

The classical Sobolev space $W^{1,p}(\Omega,\mathbb{R}^{3})$ is a proper subset of the intersection

[TABLE]

The normal trace operator in $W^{p,\operatorname{div}}(\Omega,\mathbb{R}^{3})$ is the function $\gamma_{\mathbf{n}}\colon W^{p,\operatorname{div}}(\Omega,\mathbb{R}^{3})\to(W^{1-1/q,q}(\partial\Omega,\mathbb{R}))^{*}$ defined by $\gamma_{\mathbf{n}}(\vec{u})=\vec{u}|_{\partial\Omega}\cdot\eta$. By the Divergence Theorem, the normal trace operator is weakly defined by

[TABLE]

for each $\vec{u}\in W^{p,\operatorname{div}}(\Omega,\mathbb{R}^{3})$ and each $v_{0}\in W^{1,q}(\Omega,\mathbb{R})$. Here, the symbol $\langle\cdot,\cdot\rangle_{\partial\Omega}$ denotes the duality pairing between $W^{1-1/q,q}(\partial\Omega)$ and its dual space $(W^{1-1/q,q}(\partial\Omega))^{*}$. When $p=2$, we will denote $W^{1/2,2}(\partial\Omega)$ and $(W^{1/2,2}(\partial\Omega))^{*}$ by $H^{1/2}(\partial\Omega)$ and $H^{-1/2}(\partial\Omega)$, respectively. Let $W_{0}^{p,\operatorname{div}}(\Omega,\mathbb{R}^{3})$ be the kernel of the normal trace operator $\gamma_{\mathbf{n}}$. That is, let

[TABLE]

Define also the space

[TABLE]

and endow it with the norm $\|\vec{u}\|_{L^{p}}+\|\operatorname{div}\vec{u}\|_{L^{p}}+\|\operatorname{curl}\vec{u}\|_{L^{p}}$.

2.2 Construction of solution

In this stage of our work, we will recall some results reported in [8, 10]. In particular, we will employ the constructive solution obtained in those works, for the div-curl system in bounded star-shaped domains $\Omega\subseteq\mathbb{R}^{3}$. Consider the div-curl system without any boundary conditions

[TABLE]

where $g_{0}\in L^{p}(\Omega,\mathbb{R})$, $\vec{g}\in L^{p}(\Omega,\mathbb{R}^{3})$ and $\operatorname{div}\vec{g}=0$. Here, $\vec{g}$ is required to be weakly solenoidal, that is,

[TABLE]

for all test functions $u_{0}\in W_{0}^{1,q}(\Omega,\mathbb{R})$ and $\frac{1}{p}+\frac{1}{q}=1$.

Recall that the Moisil–Teodorescu differential operator is defined by

[TABLE]

We say that $w\in C^{1}(\Omega,\mathbb{H})$ is left-monogenic (respectively, right-monogenic) in $\Omega$ when $Dw=0$ (respectively, $wD=0$). In the following, we convey that the term ‘monogenic function’ will refer to left-monogenic functions. It is well-known that $\Delta=-D^{2}$. As a consequence, if $w$ is (left- or right-) monogenic then every component function $w_{i}$ is harmonic, for $i=0$, $1$, $2$, $3$. The application of $D$ to differentiable functions of the form $w=w_{0}+\vec{w}$ yields $Dw=-\operatorname{div}\vec{w}+\operatorname{grad}w_{0}+\operatorname{curl}\vec{w}$. One of the fundamental features to derive the solution of (2.22) in [8] was that (2.22) can be rewritten in terms of the Moisil–Teodorescu operator as a quaternionic formula, namely,

[TABLE]

Likewise, the use of quaternionic integral operators as well as a monogenic completion procedure were crucial in deriving the solution reported in [8]. Given a scalar harmonic function $u_{0}$, this completion process consisted in finding a purely vector harmonic function $\vec{u}$, such that $D(u_{0}+\vec{u})=0$.

The Teodorescu transform and the Cauchy operator are defined respectively by

[TABLE]

where $E(\vec{x})=-\vec{x}/(4\pi|\vec{x}|^{3})$, $\vec{x}\in\mathbb{R}^{3}\setminus\{0\}$ is the Cauchy kernel and $\vec{\eta}$ is the outward normal vector to $\partial\Omega$. We will usually work with $w\in L^{p}(\Omega)$ and $\varphi\in L^{p}(\partial\Omega)$. Following the notation of the decomposition used in [8], we denote the component operators of the Teodorescu transform as follows:

[TABLE]

Here, the scalar part is given by

[TABLE]

and its vector part is divided for strategic reasons as

[TABLE]

The monogenic completion operator $\overrightarrow{U}_{\!\!\Omega}\colon\text{Har}(\Omega,\mathbb{R})\to\text{Har}(\Omega,\mathbb{R}^{3})$ is given by

[TABLE]

This function sends real-valued harmonic functions into vector-valued harmonic functions. It is defined on star-shaped open sets $\Omega$ with respect to the origin. When $\Omega$ is star-shaped with respect to some other point, the definition of $\overrightarrow{U}_{\!\!\Omega}$ is adjusted by shifting the values of $\vec{x}$ accordingly. It has been established that $w_{0}+\overrightarrow{U}_{\!\!\Omega}[w_{0}]$ is monogenic, for any real-valued harmonic function $w_{0}$ (see [8, Prop. 2.3]).

The following result was proved in [8, Thm. 4.4].

Theorem 1** (Delgado and Porter [8]).**

Let $\Omega$ be a star-shaped open set. Let $g=g_{0}+\vec{g}\in L^{p}(\Omega)$ and $\operatorname{div}\vec{g}=0$. Then a general weak solution of the div-curl system (2.22) is given by

[TABLE]

where $h\in\text{Har}(\Omega,\mathbb{R})$ is arbitrary.

Remark 2**.**

Notice that the operator $\overrightarrow{U}_{\!\!\Omega}$ in (2.33) acts over $T_{0,\Omega}[\vec{g}]$, which is harmonic. Indeed, observe that

[TABLE]

Taking its scalar part, we check that $T_{0,\Omega}[\vec{g}]$ is harmonic if and only if $\operatorname{div}\vec{g}=0$. This fact was crucial in the construction of this general solution. An alternative construction was given in [9, Appendix], removing the restriction of star-shapedness. In fact, this alternative general solution is valid for bounded Lipschitz domains in $\mathbb{R}^{3}$ with weaker topological constraints. The difficulty in this case lies in the inversion of a layer potential that appears in its expression [9, Th. A.1]. It is worth mentioning that this problem is not present in the derivation of (2.33).

3 Methods

3.1 Helmholtz decomposition

The div-curl system (2.22) has been studied from various points of view in light of its fundamental importance in physics. Unfortunately, the solution (1.1) is provided by integral operators which are defined over all the three-dimensional space. This feature of that solution represents a serious limitation for many applications. In the following, we will obtain a Helmholtz-type decomposition for the solution given in Theorem 1. As observed in [8, Prop. 3.2], the components of the Teodorescu operator $T_{\Omega}$ can be rewritten in terms of the Newton potential $L$ over $\Omega$ as

[TABLE]

where

[TABLE]

Analogously, a similar decomposition was pointed out for the operator $\overrightarrow{U}_{\!\!\Omega}$ (see [8, Cor. 4.3]), namely,

[TABLE]

The solution of the div-curl system can be rewritten in a way similar to the classic Helmholtz decomposition theorem:

Proposition 3**.**

Let $\Omega$ be a star-shaped open set, and let $g=g_{0}+\vec{g}\in L^{p}(\Omega)$ with $\operatorname{div}\vec{g}=0$. Then the solution (2.33) admits a Helmholtz-type decomposition

[TABLE]

where $\vec{v}^{*}$ is given by

[TABLE]

Moreover, $\operatorname{div}\vec{v}^{*}$ is harmonic in $\Omega$.

Proof.

Replacing (3.35)–(3.37) in (2.33), we obtain (3.38). Using (3.35) and the harmonicity of $T_{0,\Omega}[\vec{g}]$ yields

[TABLE]

Note that $\operatorname{div}\vec{v}^{*}$ is harmonic if and only if the second term at the right-hand side of (3.40) is harmonic. Next, we will see that $\int_{0}^{1}t\vec{x}\cdot\operatorname{grad}T_{0,\Omega}[\vec{g}](t\vec{x})\,dt$ is harmonic. By the proof of [8, Prop. 2.3], $D(\nabla T_{0,\Omega}[\vec{g}](t\vec{x})\vec{x})=DT_{0,\Omega}[\vec{g}](t\vec{x})$, which implies $\Delta(\nabla T_{0,\Omega}[\vec{g}](t\vec{x})\vec{x})=0$. As a consequence,

[TABLE]

is harmonic, which guarantees that its scalar part is also harmonic. The result now follows from (3.40) ∎

The similarity between the decomposition (3.38) and the classical Helmholtz decomposition (1.1) is evident when $\Omega$ is the entire three-dimensional space. One difference between these two solutions is that the vector Helmholtz potential $\vec{v}$ of (1.1) is divergence-free, that is, $\operatorname{div}\vec{v}=0=T_{0,\mathbb{R}^{3}}[\vec{g}]$. In other words, the scalar part of the Teodorescu transform $T_{0,\mathbb{R}^{3}}$ defined in all $\mathbb{R}^{3}$ vanishes. On the other hand, $\operatorname{div}\vec{v}^{*}$ is harmonic in (3.38). In particular, the first term of $\vec{v}^{*}$ satisfies $\operatorname{div}L[\vec{g}]=T_{0,\Omega}[\vec{g}]$, so it is harmonic, and this property is enough to perform the monogenic completion process in the proof of Theorem 1. The kernel of the scalar operator $T_{0,\Omega}$ will be studied in Proposition 7.

3.2 Right inverse of $\operatorname{curl}$

It has been proved [39, Lem. 1] that the $\operatorname{curl}$ operator has a compact inverse from $L_{\Sigma}^{2}(\Omega)$ to $L_{\Sigma,\operatorname{curl}}^{2}(\Omega)$. Here,

[TABLE]

The proof is based on the following orthogonal decomposition of the space $L^{2}(\Omega)$:

[TABLE]

Here, $\operatorname{Ker}\operatorname{curl}=\{\vec{u}\in L^{2}(\Omega):\operatorname{curl}\vec{u}=0\}$. Let $\operatorname{Sol}^{p}(\Omega)$ be the subspace of $L^{p}$ consisting of all divergence-free functions (sometimes also called solenoidal vector fields), and let $\operatorname{Sol}_{0}^{p}(\Omega)\subset\operatorname{Sol}^{p}(\Omega)$ be the subspace of solenoidal vector fields with vanishing normal trace. In notation,

[TABLE]

In general, $L^{2}_{\Sigma}(\Omega)$ is a subset of $\operatorname{Sol}_{0}^{2}(\Omega)$. However, if $\Omega$ is simply connected, then the domain of definition of the compact inverse operator of $\operatorname{curl}$ studied in [39] reduces to $L_{\Sigma}^{2}(\Omega)=\operatorname{Sol}_{0}^{2}(\Omega)$. We refer the reader to [39, Th. 1] for mode details on the self-adjointess of $\operatorname{curl}$ on $L_{\Sigma}^{2}(\Omega)$ and its spectral theory. It is important to mention that the $\operatorname{curl}$ operator is self-adjoint when it acts over vector fields with vanished tangential trace, its symmetry in this domain of definition is illustrated by the well-known Green’s formula

[TABLE]

where $\Omega$ has a sufficiently regular boundary $\partial\Omega$ and $\vec{u},\vec{v}\in C^{1}(\overline{\Omega})$. Additionally, the reader may check [20] and references therein for an analysis of self-adjoint $\operatorname{curl}$ operators.

It is worthwhile mentioning that an explicit expression for a right inverse of $\operatorname{curl}$ was reported as [8, Th. 4.1]. The result was derived from the decomposition of (2.33) as the sum $\vec{v}_{1}+\vec{v}_{2}$, where the summands satisfy $\vec{v}_{1}=-{\overrightarrow{T}}_{\!\!1,\Omega}[\vec{g}]$ and $\vec{v}_{2}=\overrightarrow{T}_{\!\!2,\Omega}[\vec{g}]-\overrightarrow{U}_{\!\!\Omega}T_{0,\Omega}[\vec{g}]$ in $\Omega$. Moreover, the following identities are satisfied in $\Omega$:

[TABLE]

Therefore, $-{\overrightarrow{T}}_{\!\!1,\Omega}$ is a right inverse for the $\operatorname{div}$ operator, and

[TABLE]

is a right inverse for the $\operatorname{curl}$ operator in the space of divergence-free functions. Moreover, $R_{\Omega}\colon\operatorname{Sol}^{p}(\Omega)\to\operatorname{Sol}^{p}(\Omega)$ is an invariant operator.

Proposition 4**.**

Let $\Omega$ be a star-shaped domain. The right inverse $R_{\Omega}$ for the curl operator (3.49) is bounded in $L^{p}(\Omega)$. Moreover, $R_{\Omega}:\operatorname{Sol}^{p}(\Omega)\to W^{p,\mbox{\scriptsize\rm div-curl}}(\Omega)$ is bounded.

Proof.

Recall that the Teodorescu operator $T_{\Omega}\colon L^{p}(\Omega)\to W^{1,p}(\Omega)$ is bounded [17, Th. 8.4], and that $|\operatorname{Sc}a|\leq|a|$ and $|\operatorname{Vec}a|\leq|a|$ hold, for each $a\in\mathbb{H}$. As a consequence,

[TABLE]

So, we only need to bound $\overrightarrow{U}_{\!\!\Omega}T_{0,\Omega}[\vec{g}]$. Notice that

[TABLE]

Integrating over $\Omega$, we readily obtain that $\|\overrightarrow{U}_{\!\!\Omega}T_{0,\Omega}[\vec{g}]\|_{L^{p}}\leq C_{\Omega}\|\vec{g}\|_{L^{p}}$. As a consequence of (3.49), it follows that $\|R_{\Omega}[\vec{g}]\|_{L^{p}}\leq C^{*}_{\Omega}\|\vec{g}\|_{L^{p}}$, as desired. Finally, the boundedness of $R_{\Omega}:\operatorname{Sol}^{p}(\Omega)\to W^{p,\mbox{\scriptsize\rm div-curl}}(\Omega)$ is a direct consequence of the fact that $\|R_{\Omega}[\vec{g}]\|_{W^{p,\mbox{\scriptsize div-curl}}}=\|R_{\Omega}[\vec{g}]\|_{L^{p}}+\|\vec{g}\|_{L^{p}}$. ∎

We can obtain now a right inverse operator for $\operatorname{curl}\operatorname{curl}$ by taking $g_{0}=0$ in (3.38). Indeed, let us define the operator $S_{\Omega}\colon\operatorname{Sol}^{p}(\Omega)\to W^{1,p}(\Omega)$ by

[TABLE]

As a consequence, given $\vec{g}\in\operatorname{Sol}^{p}(\Omega)$, there exists $S_{\Omega}[\vec{g}]\in W^{1,p}(\Omega)$ with the property that

[TABLE]

On the other hand, observe that our restriction to star-shaped domains in the construction of the operators $R_{\Omega}$ and $S_{\Omega}$ implies that the domains must be simply connected.

4 Results

4.1 Homogeneous Neumann condition

In the present and the next subsection, we will consider respectively Neumann and Dirichlet BVP associated to the $\operatorname{curl}$ operator. In the present stage, we will require that the normal vector be defined almost everywhere at the boundary of $\Omega$. Obviously, this requirement is satisfied in the case when the domain is Lipschitz. For a fixed $\alpha>0$ and $\vec{x}\in\partial\Omega$, the region of non-tangential approach with vertex at $\vec{x}$ is given by

[TABLE]

The function $\mathcal{N}_{\alpha}W\colon\partial\Omega\to[0,\infty]$ is the non-tangential maximal function given by

[TABLE]

When measuring the growth of $w$, the choice of $\alpha$ is largely irrelevant. In light of this remark, one simply writes $\mathcal{N}$ instead of $\mathcal{N}_{\alpha}$. Let $1<p<\infty$. The Hardy space $H^{p}(\Omega)$ consists of all monogenic functions $w$ in $\Omega$ whose non-tangential maximal function $\mathcal{N}w$ belongs to $L^{p}(\partial\Omega,\mathbb{R})$, that is,

[TABLE]

Following a compact-embedding argument, we can modify the operator $R_{\Omega}$ of (3.49) in order to obtain a compact operator. First, notice that the degree of freedom of the right inverse operator $R_{\Omega}$ is unique up to the sum of the gradient of a scalar function. That is, $R_{\Omega}[\vec{g}]+\nabla h$ is still a right inverse of $\operatorname{curl}$. However, we also want that the modified right inverse of $\operatorname{curl}$ continue leaving invariant the subspace $\operatorname{Sol}^{p}(\Omega)$ and have normal trace equal zero. To that end, let us define

[TABLE]

where $h$ is a scalar harmonic functions satisfying the Neumann problem

[TABLE]

Some works in the literature have studied this type of Neumann BVPs [6, 13, 22]. In the present report, we will employ the result in [6] for Lipschitz domains with connected boundary, which establishes that there exists a unique harmonic function $h$ in $\Omega$ which is unique up to constants, such that

[TABLE]

Here, $1<p<2+\epsilon$, and $\epsilon$ is the Lipschitz characteristic of the domain.

As a consequence of these discussion, $R_{\Omega,_{\mathbf{n}}}:\operatorname{Sol}^{p}(\Omega)\to\operatorname{Sol}_{0}^{p}(\Omega)$. Moreover, if $\Omega$ is a star-shaped domain with Lipschitz boundary, then $\vec{w}=R_{\Omega,_{\mathbf{n}}}[\vec{g}]$ provides a unique weak solution to the first order system

[TABLE]

for all $\vec{g}\in\operatorname{Sol}^{p}(\Omega)$ with $1<p<2+\epsilon$. In fact, a difference of two solutions of (4.60) belongs to

[TABLE]

which has finite dimension, and is trivial when $\Omega$ is simply connected [7, Ch. 9, Cor. 2].

Theorem 5**.**

Let $\Omega$ be a star-shaped domain with Lipschitz boundary. The right inverse $R_{\Omega,_{\mathbf{n}}}$ for the $\operatorname{curl}$ operator defined in (4.57)–(4.58) is compact in $L^{2}(\Omega)$.

Proof.

We will check firstly that $R_{\Omega,_{\mathbf{n}}}$ is bounded in $L^{2}(\Omega)$. By Proposition 4, it is enough to bound the last term $\nabla h$ in the expression (4.57). Due to $\nabla h$ being monogenic and by (4.59), it follows that $\nabla h\in H^{2}(\Omega)$. The equivalences provided by [31, Thm. 4.1] establish that $\nabla h=F_{\partial\Omega}[\nabla h]$. Using the continuity of the operator $F_{\partial\Omega}\colon L^{p}(\partial\Omega)\to L^{p}(\Omega)$, it follows that the next inequalities are satisfied:

[TABLE]

In this inequalities, the last one is a consequence of the definition of the non-tangential maximal function $\mathcal{N}$ in (4.56). On the other hand, the inequality (4.59) and the boundedness of the normal trace operator $\gamma_{\mathbf{n}}\colon W^{2,\mbox{\scriptsize div}}(\Omega)\to H^{-1/2}(\partial\Omega)$ (see [7, Ch. 9, Thm. 1]) guarantee that

[TABLE]

Notice that the conclusion readily follows now from the compactness of the embedding of $W_{\mathbf{n}}^{2,\operatorname{div}\text{-}\operatorname{curl}}(\Omega)$ into $L^{2}(\Omega)$ (see [1, Th. 2.8] and [38]). ∎

Proposition 6**.**

Let $\Omega$ be a star-shaped domain with Lipschitz boundary. The space $L^{2}(\Omega)$ allows the decomposition

[TABLE]

under the scalar product $\langle\vec{u},\vec{v}\rangle_{L^{2}}=\int_{\Omega}\vec{u}\cdot\vec{v}$.

Proof.

The proof follows from the decomposition $L^{2}(\Omega)=\operatorname{Sol}^{2}(\Omega)\oplus\operatorname{grad}W_{0}^{1,2}(\Omega)$ (see [35]) as well as from the facts that $R_{\Omega,_{\mathbf{n}}}\colon\operatorname{Sol}^{2}(\Omega)\to\operatorname{Sol}_{0}^{2}(\Omega)$ and $\vec{g}=\operatorname{curl}R_{\Omega,_{\mathbf{n}}}[\vec{g}]$, for all $\vec{g}\in\operatorname{Sol}^{2}(\Omega)$. ∎

The single-layer potential [29] is defined by

[TABLE]

Meanwhile, the boundary single-layer operator $\operatorname{\rm tr}M$ is obtained by evaluating the integral in (4.65) for $x\in\partial\Omega$. In such way, the single-layer potential $M$ is extended to all of $\mathbb{R}^{3}$.

To further investigate the operators $R_{\Omega}$ and $R_{\Omega,_{\mathbf{n}}}$, we will characterize the kernel of the component operator $T_{0,\Omega}$ involved in their constructions, as described by (3.49) and (4.57), respectively. We will restrict the domain of $T_{0,\Omega}$ to the class of divergence-free functions $\operatorname{Sol}^{p}(\Omega)$. Under these circumstances, the following question emerged in [8] and was left as an open question in that report: under which conditions does the general solution (2.33) coincide with (1.1), which is the solution given by the classical Helmholtz Decomposition Theorem? An affirmative answer to that question is provided in the following result.

Proposition 7**.**

The kernel of the scalar integral operator $T_{0,\Omega}$ in $\operatorname{Sol}^{2}(\Omega)$ is the subspace $\operatorname{Sol}_{0}^{2}(\Omega)$.

Proof.

It was noted in [8] that the scalar component $T_{0,\Omega}$ can be written as $T_{0,\Omega}[\vec{g}]=M[\vec{g}|_{\partial\Omega}\cdot\eta]$, for all $\vec{g}\in\operatorname{Sol}^{p}(\Omega)$. By [36, Th. 3.3], $\operatorname{\rm tr}M\colon L^{2}(\partial\Omega)\to H^{1/2}(\partial\Omega)$ is invertible or, in the generalized sense, $\operatorname{\rm tr}M:H^{-1/2}(\partial\Omega)\to H^{1/2}(\partial\Omega)$ (see [29, Thm. 6.12]). Conclude that $T_{0,\Omega}[\vec{g}]=0$ if and only if $\vec{g}$ has zero normal trace. ∎

Remark 8**.**

As a consequence of this proposition, if $\vec{g}\in\operatorname{Sol}_{0}^{2}(\Omega)$, then the general weak solution (2.33) reduces to

[TABLE]

Moreover, the right inverse of $\operatorname{curl}R_{\Omega}$ defined in (3.49) reduces to $R_{\Omega}=\overrightarrow{T}_{\!\!2,\Omega}$ (the Biot–Savart operator (1.5) over $\Omega$), and the modified right inverse $R_{\Omega,_{\mathbf{n}}}$ reduces to $R_{\Omega,_{\mathbf{n}}}=\overrightarrow{T}_{\!\!2,\Omega}+\nabla h$, with $h$ the solution of the Neumann problem (4.58).

4.2 Homogeneous Dirichlet condition

In the present stage of our work, we are interested in the analysis of the BVP with homogeneous Dirichlet condition

[TABLE]

By the well-known Helmholtz decomposition for vector fields in simply connected domains, it follows that

[TABLE]

It was noted in [15] that (4.67) has a unique solution when $\vec{g}\in\{\nabla h\colon h\in h\in W^{1,2}(\Omega)\}$. It is important to point out that the authors of [4, Cor. 8’] proved a result on the existence of a right inverse of $\operatorname{curl}$. They also established some estimates using results on differential forms with coefficients in Sobolev spaces.

We claim that $\vec{g}\in\operatorname{Sol}^{p}(\Omega)$ in (4.67) will necessarily have vanishing normal trace. Indeed, notice that

[TABLE]

where $\operatorname{Div}$ is the surface divergence. Moreover,

Proposition 9**.**

If $\vec{g}\in\operatorname{Sol}^{2}(\Omega)$ and $\vec{w}\in W^{2,\mbox{\scriptsize div}}(\Omega)$ be a solution of the system (4.67), then $\vec{g}\in\operatorname{Sol}_{0}^{2}(\Omega).$

Proof.

Let $\vec{w}\in W^{2,\mbox{\scriptsize div}}(\Omega)$ is a solution of $\operatorname{curl}\vec{w}=\vec{g}$ with $\vec{w}|_{\partial\Omega}=0$. Friedrichs’ inequalities imply that $\vec{w}\in W^{1,2}(\Omega)$ (see [7, Ch. 9, Cor. 1]). The Borel–Pompeiu formula [17] and the decomposition (2.28) yield that

[TABLE]

As a consequence, the scalar part on the right-hand side of this identity vanishes. Proposition 7 shows now that $T_{0,\Omega}[\vec{g}]=0$ if and only if $\vec{g}\in\operatorname{Sol}_{0}^{2}(\Omega)$, as desired. ∎

Using some properties of the Teodorescu transform $T_{\Omega}$, it is possible to prove that this class of solutions vanishes not only at the boundary but also in the entire exterior of the domain which is the set $\Omega^{-}=\mathbb{R}^{3}\setminus\overline{\Omega}$. In other words, we have the following result.

Proposition 10**.**

Let $\Omega$ be a star-shaped domain. Let $\vec{g}\in\operatorname{Sol}^{2}(\Omega)$. Then the solutions of (4.67) belonging to $W^{2,\mbox{\scriptsize div}}(\Omega)$ vanishes in the exterior domain $\Omega^{-}$.

Proof.

Propositions 7 and 9 yield $T_{0,\Omega}[\vec{g}]=0$. Using then the Borel–Pompeiu formula, we obtain that $\vec{w}=T_{\Omega}[D\vec{w}]$ and $\vec{w}\in W_{\mathbf{n}}^{2,\operatorname{div}\text{-}\operatorname{curl}}(\Omega)$. Use now the fact that $DT_{\Omega}=T_{\Omega}D$ holds in $W_{0}^{1,2}(\Omega)$ and that the Teodorescu transform $T_{\Omega}$ is monogenic in $\Omega^{-}$ (see [17, Prop. 8.1]) to establish that $\vec{w}=T_{\Omega}[D\vec{w}]=DT_{\Omega}[\vec{w}]=0$ in $\Omega^{-}$. ∎

The novelty of this right inverse operator $R_{\Omega}$ is precisely the term that involves the radial operator $\overrightarrow{U}_{\!\!\Omega}$ acting on $T_{0,\Omega}$. However, Propositions 9 and 7 show that $\overrightarrow{U}_{\!\!\Omega}T_{0,\Omega}[\vec{g}]$ vanishes.

On the other hand, if we modify the operator $R_{\Omega}$ to find a solution of (4.67) as we did in Section 4.1, the modified operator will be similar to that constructed in [3, Cor. 2.3]. Using that $T_{\Omega}$ is monogenic in $\Omega^{-}$, that $T_{0,\Omega}[\vec{g}]=0$ in $\Omega$ and the maximum Principle, we obtain that $T_{0,\Omega}[\vec{g}]\equiv 0$ in all $\mathbb{R}^{3}$. So $\operatorname{curl}\overrightarrow{T}_{\!\!2,\Omega}[\vec{g}]=0$ in $\Omega^{-}$. In the following, we let $p^{*}\in W^{2,p}_{\text{loc}}(\Omega^{-})$ be the antigradient of $\overrightarrow{T}_{\!\!2,\Omega}[\vec{g}]$, that is, let $\overrightarrow{T}_{\!\!2,\Omega}[\vec{g}]=\nabla p^{*}$ in $\Omega^{-}$. Let us define

[TABLE]

where $p$ is the biharmonic function satisfying the Dirichlet boundary value problem (see [32])

[TABLE]

Moreover, $p\in W^{2,p}(\Omega)$.

Proposition 11**.**

Let $\Omega$ be a $C^{2}$ bounded simply connected domain with connected boundary. Then the operator $R_{\Omega,0}\colon\operatorname{Sol}_{0}^{p}(\Omega)\to W_{0}^{1,2}(\Omega)$ defined in (4.71) and (4.72) is a right inverse of $\operatorname{curl}$ with vanishing Dirichlet condition. In other words, $\vec{w}=R_{\Omega,0}[\vec{g}]$ provides a weak solution of (4.67), with $\Delta\operatorname{div}\vec{w}=0$.∎

5 Beltrami fields

This section is devoted to the construction of Beltrami fields through an uniformly convergent Neumann series in terms of the inverse curl operator $R_{\Omega}$ analyzed in this work. It is worth mentioning that quaternionic analysis techniques have been used previously for the generation of Beltrami fields, a recent example is [27].

Recall that a Beltrami field in $\Omega$ is a vector field $\vec{w}\colon\Omega\to\mathbb{R}^{3}$ satisfying the equation

[TABLE]

where the potential $\alpha_{0}$ is a real-valued function (see [5, 23]). Observe that if $\alpha_{0}=c_{0}\in\mathbb{R}$ is a constant, then $\operatorname{div}\vec{w}=0$. On the other hand, if $\alpha_{0}$ is a differentiable function, then the compatibility conditions transform into

[TABLE]

Let $\alpha_{0}\in L^{\infty}(\Omega)$. By Proposition 4, the operator $R_{\Omega}$ is bounded in $L^{p}(\Omega)$. Let $\|R_{\Omega}\|$ its norm operator from $L^{p}(\Omega)$ to itself. If $\|R_{\Omega}\alpha_{0}I\|<1$ or $\|\alpha_{0}\|_{L^{\infty}}<1/\|R_{\Omega}\|$, then $I-R_{\Omega}\alpha_{0}I$ has a bounded inverse. Moreover, the Neumann series of the operator $(I-R_{\Omega}\alpha_{0}I)^{-1}$ (see [28, Th. 1.3] and [17, Ex. 4.15]) is given by the following uniformly convergent series:

[TABLE]

where $R_{\Omega}\colon\operatorname{Sol}^{p}(\Omega)\to\operatorname{Sol}^{p}(\Omega)$ is a right inverse operator of $\operatorname{curl}$ previously defined in (3.49). Let us denote the class of irrotational vector fields as $\operatorname{Irr}^{p}(\Omega)=\{\vec{u}\in L^{p}(\Omega):\operatorname{curl}\vec{u}=0\}.$ The next result shows a way to generate Beltrami fields with $\alpha_{0}\in\mathbb{R}$ constant, using Neumann expansion series.

Theorem 12**.**

Let $\Omega$ be a star-shaped domain. Let $1<p<\infty$, $\vec{g}\in\operatorname{Sol}^{p}(\Omega)\cap\operatorname{Irr}^{p}(\Omega)$ and $\alpha=\alpha_{0}\in\mathbb{R}$. If $\|R_{\Omega}\alpha_{0}I\|<1$ or $|\alpha_{0}|<1/\|R_{\Omega}\|$, then

[TABLE]

is a Beltrami field in $\Omega$.

Proof.

Beforehand, notice that the iterated application of the operator

$R_{\Omega}\colon\operatorname{Sol}^{p}(\Omega)\to\operatorname{Sol}^{p}(\Omega)$ is feasible in view that $\operatorname{div}(\alpha_{0}\vec{g})=\alpha_{0}\operatorname{div}\vec{g}=0$ and $\operatorname{div}(\alpha_{0}R_{\Omega}[\cdot])=\alpha_{0}\operatorname{div}R_{\Omega}[\cdot]=0$. Applying the curl operator to both ends of the Neumann series (5.76), we obtain that

[TABLE]

It follows that $\operatorname{curl}\vec{w}=\alpha_{0}\vec{w}$ in $\Omega$ and $\operatorname{div}\vec{w}=\operatorname{div}\vec{g}=0$, as we desired. ∎

Example 13**.**

Let $\Omega$ be the unit ball in $\mathbb{R}^{3}$, and let $\alpha_{0}=c_{0}\in\mathbb{R}$ and $\vec{g}=\vec{c}\in\mathbb{R}^{3}$ satisfy the hypotheses of Theorem 12. The explicit formulas for the Teodorescu transform obtained in [16, App. A] read as follows:

[TABLE]

Using these expressions, it is easy to check that

[TABLE]

As a consequence, the modified right inverse of $\operatorname{curl}$ has the same expression. That is, $R_{\Omega,\eta}[\vec{c}]=R_{\Omega}[\vec{c}]=-(1/2)\vec{x}\times\vec{c}$, due to $\eta(\vec{x})=\vec{x}$. Observe that we have constructed implicitly a general solution $\vec{v}=-\frac{1}{2}\vec{x}\times\vec{c}+\nabla h$ for the div-curl system $\operatorname{div}\vec{v}=0$, $\operatorname{curl}\vec{v}=\vec{c}$. Here, $h$ is an arbitrary harmonic function (we refer to [8, Ex. 4.5] for another example with non-constant $\operatorname{curl}$). To compute $T_{\Omega}[\vec{x}\times\vec{c}]$ in this iterative process, observe that

[TABLE]

is purely vectorial. According to Proposition 7, $T_{0,\Omega}[\vec{x}\times\vec{c}]=0$. Using the Borel–Pompeiu formula, it is possible to check that ${\overrightarrow{T}}_{\!\!1,\Omega}[\vec{x}\cdot\vec{c}]=(1/2)(c_{1}(x_{1}^{2}-1),c_{2}(x_{2}^{2}-1),c_{3}(x_{3}^{2}-1))$. Thus,

[TABLE]

Finally, the first few terms of the Neumann series of the Beltrami field $\vec{w}$ constructed in this example are given by

[TABLE]

We will describe next a method to generate Beltrami fields subject to a Neumann condition. To that end, we will use the modified right inverse of $\operatorname{curl}$ $R_{\Omega,_{\mathbf{n}}}\colon\operatorname{Sol}^{p}(\Omega)\to\operatorname{Sol}_{0}^{p}(\Omega)$ defined in (4.57). This operator is not only bounded operator in $L^{p}(\Omega)$, but it is also compact (see Theorem 5).

Proposition 14**.**

Let $\Omega$ be a star-shaped domain with Lipschitz boundary. Let $\vec{g}\in\operatorname{Sol}^{2}(\Omega)\cap\operatorname{Irr}^{2}(\Omega)$. Suppose that $\|R_{\Omega,_{\mathbf{n}}}\alpha_{0}I\|<1$ or $|\alpha_{0}|<1/\|R_{\Omega,_{\mathbf{n}}}\|$. Then

[TABLE]

is a Beltrami field in $\Omega$ satisfying the Neumann boundary condition $\vec{w}|_{\partial\Omega}\cdot\eta=a_{0}$ if and only if $\vec{g}=\nabla h$, where $h$ a solution of the Neumann BVP

[TABLE]

Observe that this construction of Beltrami fields relies on the condition $\|R_{\Omega}\alpha_{0}I\|<1$ (or $|\alpha_{0}|<1/\|R_{\Omega}\|$). The authors of this manuscript are aware that a sharper bound in terms of the operator norm of $R_{\Omega}$ is needed. In the following, we will give a bound for $|\alpha_{0}|$ which depends on $\Omega$ and the operator norm of the Teodorescu operator $T_{\Omega}$. Indeed, it is easy to compute an upper bound for the $\overrightarrow{U}_{\!\!\Omega}\circ T_{0,\Omega}$ operator. By (3.2), we readily obtain

[TABLE]

Therefore, integrating over $\Omega$ and using the boundedness of the Teodorescu transform from $L^{p}(\Omega)$ to $W^{1,p}(\Omega)$, we have

[TABLE]

where $\|T_{\Omega}\|_{L^{p}\to W^{1,p}}$ is the operator norm from $L^{p}(\Omega)$ to $W^{1,p}(\Omega)$. Consequently, $\dfrac{\|T_{\Omega}\|_{L^{p}\to W^{1,p}}}{(q+1)^{1/q}}\text{Vol }(\Omega)\,\text{diam }(\Omega)$ is an upper bound for $\|\overrightarrow{U}_{\!\!\Omega}\circ T_{0,\Omega}\|$ from $L^{p}(\Omega)$ to itself. Due to $R_{\Omega}[\vec{g}]=\overrightarrow{T}_{\!\!2,\Omega}[\vec{g}]-\overrightarrow{U}_{\!\!\Omega}[T_{0,\Omega}[\vec{g}]]$, then

[TABLE]

Taking

[TABLE]

we ensure that the norm of the composition of operators $R_{\Omega}\alpha_{0}I$ is strictly less than one, as required by the Neumann series (5.75).

It is easy to compute an upper bound for the norm of the $\overrightarrow{U}_{\!\!\Omega}^{*}$ operator is easy to compute. By (3.2), we readily obtain

[TABLE]

Therefore, integrating over $\Omega$:

[TABLE]

Consequently, taking $|\alpha_{0}|\leq\dfrac{(q+1)^{1/q}}{\text{Vol }(\Omega)\,\text{diam }(\Omega)}$, we ensure that the norm of the composition of operators $U_{\Omega}\alpha_{0}I$ is strictly less than one, as required by the Neuman series.

6 Vekua-type problems and its applications to the inhomogeneous Maxwell equations

6.1 The operator $D-\vec{\alpha}$

The purpose of this section is to analyze the system

[TABLE]

where $g\in L^{p}(\Omega)$ and $D$ is the Moisil–Teodorescu operator defined in (2.24). The identity $D\vec{w}=-\operatorname{div}\vec{w}+\operatorname{curl}\vec{w}$ guarantees that (6.89) is equivalent to the following div-curl system:

[TABLE]

Taking the divergence in the second equation and using that $\operatorname{curl}\vec{\alpha}=0$, we obtain

[TABLE]

Let $\Omega$ be star-shaped and take $a=(a_{1},a_{2},a_{3})\in\Omega$. Define the antigradient operator $\mathcal{A}$ by

[TABLE]

where $\vec{u}$ is any vector field in the class $\operatorname{Irr}^{p}(\Omega)$. Since $|\mathcal{A}[\vec{u}](\vec{x})|\leq\max_{i}|x_{i}-a_{i}|\max_{i}\|u_{i}\|_{L^{p}(\Omega)}$ and $\nabla\mathcal{A}[\vec{u}]=\vec{u}$, we obtain that $\mathcal{A}:\operatorname{Irr}^{p}(\Omega)\to W^{1,p}(\Omega)$, for all $1\leq p\leq\infty$.

Remark 15**.**

If $\Omega$ is a star-shaped domain and $\vec{\alpha}\in\operatorname{Irr}^{p}(\Omega)$, then it is always possible to construct a positive scalar function $\varphi\in W^{1,p}(\Omega)$ such that $\vec{\alpha}=\nabla\varphi/\varphi$, namely,

[TABLE]

Another important feature about this class of irrotational vector fields was illustrated by the example in [33]. In that work, the authors considered

[TABLE]

where $\vec{x}^{(i)}\in\mathbb{R}^{3}\setminus\overline{\Omega}$ is dense in a closed surface outside $\Omega$. Since the system $\left\{\frac{\vec{x}-\vec{x}^{i}}{|\vec{x}-\vec{x}^{i}|^{3}}\right\}_{i}$ is complete in $L^{2}(\Omega)\cap\text{Ker }D$ (see [17, Th. 10.4]), it follows that this class of irrotational vector fields is quite large.

On the other hand, the analysis of the operator $D-\vec{\alpha}$ was trivialized in [33] assuming that $\vec{\alpha}$ has the form $\nabla\varphi/\varphi$. Indeed, in that case, the factorization $D-\vec{\alpha}=\varphi D\varphi^{-1}$ holds. This factorization allowed to find a straightforward right inverse of $\varphi D\varphi^{-1}$ in terms of the classical Teodorescu transform $T_{\Omega}$ [33, 34]: if $\varphi^{-1}w\in L^{p}(\Omega)$, then

[TABLE]

Therefore, an immediate consequence is that (see [33, Lemma 1])

[TABLE]

We will see how the factorization $\varphi D\varphi^{-1}$ allows us to give an explicit solution to the system (6.89) in terms of our solution to the div-curl system (2.33).

Theorem 16**.**

Let $\Omega$ be a star-shaped domain, and let $\vec{\alpha}\in L^{\infty}(\Omega)$ be such that $\operatorname{curl}\vec{\alpha}=0$. Suppose that $g\in L^{p}(\Omega)$ satisfies the compatibility condition $\operatorname{div}\vec{g}=\vec{\alpha}\cdot\vec{g}$. Then a weak solution of (6.89) is given by

[TABLE]

where $\varphi$ is constructed as in (6.93), $R_{\Omega}$ is the right inverse of the $\operatorname{curl}$ operator defined in (3.49) and $h$ is an arbitrary harmonic function.

6.2 The operator $D+M^{\vec{\alpha}}$

Let $M^{\alpha}$ be the right-hand side multiplication operator by the function $\alpha$, usually we will employ bounded functions. To start with, notice that we can readily observe that we can develop a solution method of the equation

[TABLE]

for the class of $p$-integrable irrotational vector fields $\vec{\alpha}$. That is, $\vec{\alpha}\in\operatorname{Irr}^{p}(\Omega)$. This method hinges on the combination of the theory developed here to solve the div-curl system in star-shaped domain in $\mathbb{R}^{3}$ and some variational methods employed in the theory of elliptic partial differential equations. Unlike the operator $D+\vec{\alpha}$ for which there exists a complete functional theory if $\vec{\alpha}=\nabla\varphi/\varphi$ (generalized Teodorescu and Cauchy operators, Borel–Pompeiu formula, Plemelj–Shokotski, etc.), there is no integral operator theory for the operators $D+\alpha_{0}$ and $D+M^{\vec{\alpha}}$, when $\alpha_{0}$ and $\vec{\alpha}$ are non-constant scalar and vector functions respectively.

Notice that the system (6.100) is equivalent to the following type div-curl system:

[TABLE]

Comparing the systems (6.101) and (6.90) we can observe that the second equations in both systems corresponding to the vector part is equal. Therefore, under the hypothesis $\operatorname{curl}\vec{\alpha}=0$, we obtain the same compatibility condition as that obtained for $D-\vec{\alpha}$ in Section 6.1, namely, $\operatorname{div}\vec{g}=\vec{\alpha}\cdot\vec{g}$.

Theorem 17**.**

Let $\Omega$ be a star-shaped domain, and let $\vec{\alpha}\in L^{\infty}(\Omega)$ be such that $\operatorname{curl}\vec{\alpha}=0$. Suppose that $g\in L^{2}(\Omega)$ satisfies the compatibility condition $\operatorname{div}\vec{g}=\vec{\alpha}\cdot\vec{g}$. Then a weak solution of (6.100) is given by

[TABLE]

where $\varphi$ is constructed as in (6.93), $w_{0}$ is a solution of the conductivity equation $\operatorname{div}(\varphi^{2}\nabla w_{0})=\varphi g_{0}+2\nabla\varphi\cdot\vec{w}^{*}$, $\vec{w}^{*}=\varphi R_{\Omega}[\vec{g}/\varphi]$, and $\vec{u}$ is an arbitrary vector field belonging to the kernel of $D+M^{\vec{\alpha}}$.

Proof.

Using Remark 15, there exists a scalar function $\varphi\in W^{1,\infty}(\Omega)$ such that $\vec{\alpha}=\nabla\varphi/\varphi$. This implies that the equivalent system (6.101) can be expressed alternatively as

[TABLE]

Notice that the right inverse of $\operatorname{curl}$ $R_{\Omega}\colon\operatorname{Sol}^{p}(\Omega)\to\operatorname{Sol}^{p}(\Omega)$ in (3.49) allows us to obtain a solution of the second equation of (6.103), though some adjustments are required to satisfy also the first. To that end, let

[TABLE]

The application of the operator $R_{\Omega}$ is well-defined since $\operatorname{div}(\vec{g}/\varphi)=\varphi^{-1}(\operatorname{div}\vec{g}-\vec{\alpha}\cdot\vec{g})=0$ holds by the compatibility condition. As a consequence, $\operatorname{div}(\vec{w}^{*}/\varphi)=0$ and $\operatorname{curl}(\vec{w}^{*}/\varphi)=\vec{g}/\varphi$. Let us define

[TABLE]

where $w_{0}$ is a solution of the elliptic conductivity equation

[TABLE]

The existence of a solution of (6.106) is well-known, and it is based on the use of variational methods (see [21, Theorem 4.1] and [30, Theorem 10]). More precisely, we need to minimize the following functional $\epsilon$ in $W^{1,2}(\Omega)$:

[TABLE]

Moreover, the uniqueness of the minimum is guaranteed under some boundary Dirichlet condition. Without loss of generality, let us suppose that $w_{0}|_{\partial\Omega}=0$. We only need to verify that $\varphi g_{0}+2\nabla\varphi\cdot w^{*}\in L^{2}(\Omega)$, but this follows from the fact that $\varphi\in W^{1,\infty}(\Omega)$, $g=g_{0}+\vec{g}\in L^{2}(\Omega)$ and $R_{\Omega}[\vec{g}/\varphi]\in L^{2}(\Omega)$. We conclude that $\vec{w}$ satisfies (6.100). ∎

6.3 Time-indedendent Maxwell system in inhomogeneous media

Finally, let us consider the Maxwell system in inhomogeneous media, where the permittivity and permeability are bounded scalar functions in $\Omega$, $\epsilon=\epsilon(\vec{x})$, $\mu=\mu(\vec{x})\in L^{\infty}(\Omega)$. More precisely, assume that the following hold:

[TABLE]

Here, the charge and the current densities are related by the identity $\rho=\frac{1}{i\omega}\operatorname{div}\vec{j}$. Following [26] and [25, Ch. 4], if the electric and magnetic fields are time-independent, then we can rewrite (6.108) in terms of the $D+M^{\vec{\alpha}}$ operator as

[TABLE]

where $\vec{\mathcal{E}}=\sqrt{\epsilon}\vec{E}$ and $\vec{\mathcal{H}}=\sqrt{\mu}\vec{H}$. The new vector fields that appear by the right-hand multiplication operators $M^{\vec{\epsilon}}$ and $M^{\vec{\mu}}$ are given respectively by

[TABLE]

Obviously, (6.108) and (6.109) are equivalents.

It has been noticed [25] that scalar fundamental solutions of the Schrödinger operator with potential $\Delta\varphi/\varphi=c^{2}$ (where $c$ is a constant) generate purely vector fundamental solutions of the operator $D+M^{\vec{\alpha}}$, where $\vec{\alpha}=\nabla\varphi/\varphi$. Unfortunately, we cannot used this procedure to generate a fundamental solution in the present case. The advantage to know a fundamental solution of $D+M^{\vec{\alpha}}$ is that we could adapt the solution method presented in Section 6.1. Instead of that, we will apply Theorem 17 in order to give an explicit solution of the time-independent Maxwell system in inhomogeneous media (6.108).

Theorem 18**.**

Let $\Omega$ be a star-shaped domain. Let $\mu,\epsilon\in W^{1,\infty}(\Omega)$ be non vanishing scalar functions and $\rho,\vec{j}\in L^{p}(\Omega)$. Then a weak general solution of (6.109) is given by

[TABLE]

where $\vec{u}_{1}$ and $\vec{u}_{2}$ are arbitrary vector fields in the kernel of $D+M^{\vec{\epsilon}}$ and $D+M^{\vec{\mu}}$, respectively. Moreover, $h_{1}$ and $h_{2}$ are respectively solutions of the conductivity equations

[TABLE]

Proof.

By (6.108), it readily follows that $\operatorname{div}\vec{j}=0$ in the time-independent case. We will verify that the right-hand sides of the equations in (6.109) satisfy the hypotheses of Theorem 17. The fact that $\vec{\epsilon}$ and $\vec{\mu}$ are irrotational vector fields is straightforward, and the compatibility condition $\operatorname{div}(\sqrt{\mu}\vec{j})=\vec{\mu}\cdot\sqrt{\mu}\vec{j}$ holds. Applying Theorem 17, we have

[TABLE]

where $h_{1}$ and $h_{2}$ are solutions of the conductivity equations

[TABLE]

Without loss of generality, suppose that $h_{i}|_{\partial\Omega}=0$ has zero trace for $i=1,2$. By the non-uniqueness of the solutions established in Theorem 17, let $\vec{u}_{i}$ be such that $(D+M^{\epsilon})\vec{u}_{1}=0$ and $(D+M^{\mu})\vec{u}_{2}=0$, respectively. Finally, the last expression comes from the fact that $\vec{\mathcal{E}}=\sqrt{\epsilon}\vec{E}$ and $\vec{\mathcal{H}}=\sqrt{\mu}\vec{H}$. ∎