General Magneto-Static Model

TL;DR

This paper develops a comprehensive nonlinear magneto-static model for complex multiply-connected domains, analyzing the influence of topology and boundary conditions on the existence of solutions.

Contribution

It introduces a general magneto-static model incorporating domain topology effects and proves existence results for solutions under various boundary conditions.

Findings

Existence of solutions depends on domain topology.

The model accounts for nonlinear B-H relations.

Boundary conditions significantly influence solution existence.

Abstract

In this paper we study a nonlinear magneto-static model on a general domain which is multiply-connected and has $m$ holes, and under a nonlinear relation between magnetic induction $\bold B$ and magnetic field $\bold H$. The equation contains a Neumann field $\bold h_1\in\Bbb H_1(\Omega)$ and a Dirichlet field $\bold h_2\in\Bbb H_2(\Omega)$, which represent the effects of domain topology. For a general electric current, the equation contains an unknown gradient $\nabla p(x)$, which represents the electric field. Existence results of solutions of boundary value problems of this model under various types of boundary conditions are proved, which exhibit the effects of domain topology.

Equations

$$\text{curl}[\mathcal{H}(x,\text{curl}u)]=\mathbb{J}(x),\text{div}u=0\text{in }\Omega ,$$

$$\text{curl}[\mathcal{H}(x,\text{curl}u)]=f(x,u)+\nabla p,\text{div}u=0\text{in }\Omega .$$

$$\mathbb{B}=\mathcal{B}(x,\mathbb{H}),\text{or}\mathbb{H}=\mathcal{H}(x,z).$$

$$\sigma \mathbb{E}=\nabla p+h_1,\mathbb{B}=\text{curl}u+h_2,$$

$$\{\begin{array}{rlr}& \text{curl}[\mathcal{H}(x,\text{curl}u+h_2(x))]=\mathbb{J}(x)+h_1(x)& \text{in }\Omega ,\\ & \text{div}u=0& \text{in }\Omega ;\end{array}$$

$$\{\begin{array}{rlr}& \text{curl}[\mathcal{H}(x,\text{curl}u+h_2(x))]=f(x,\text{curl}u+h_2)+h_1(x)+\nabla p& \text{in }\Omega ,\\ & \text{div}u=0& \text{in }\Omega .\end{array}$$

$$\nabla p=\sigma \mathbb{E}-h_1,$$

$${\mathbb{H}}_1(\Omega )=\{u\in L^2(\Omega ,{\mathbb{R}}^3):\text{curl}u=0\text{and}\text{div}u=0\text{in }\Omega ,\nu \cdot u=0\text{on }\partial \Omega \},$$

$${\mathbb{H}}_2(\Omega )=\{u\in L^2(\Omega ,{\mathbb{R}}^3):\text{curl}u=0\text{and}\text{div}u=0\text{in }\Omega ,u_T=0\text{on }\partial \Omega \}.$$

$$\Vert h_j{\Vert }_{H^1(\Omega )}\le C_j(\Omega )\Vert h_j{\Vert }_{L^2(\Omega )},\forall h_j\in {\mathbb{H}}_j(\Omega ),j=1,2.$$

$$X(\Omega ,\text{div}0)=\{u\in X(\Omega ):\text{div}u=0\text{in }\Omega \},$$

$$X(\Omega ,\text{curl}0)=\{u\in X(\Omega ):\text{curl}u=0\text{in }\Omega \},$$

$$X_{t0}(\Omega )=\{u\in X(\Omega ):u_T=0\text{on }\partial \Omega \},$$

$$X_{n0}(\Omega )=\{u\in X(\Omega ):\nu \cdot u=0\text{on }\partial \Omega \}.$$

$$\mathcal{H}(\Omega ,\text{curl},\text{div})=\{w\in L^2(\Omega ,{\mathbb{R}}^3):\text{curl}w\in L^2(\Omega ,{\mathbb{R}}^3),\text{div}w\in L^2(\Omega )\},$$

$$\mathcal{H}(\Omega ,\text{curl},\text{div}0)=\{w\in \mathcal{H}(\Omega ,\text{curl},\text{div}),\text{div}w=0\},$$

$$\mathcal{H}(\Omega ,\text{curl}0,\text{div}0)=\{w\in \mathcal{H}(\Omega ,\text{curl},\text{div}):\text{curl}w=0,\text{div}w=0\}.$$

$$\dot{Y}(\Omega )=\{\varphi \in X(\Omega ):{\int }_{\Omega }\varphi dx=0\}.$$

$$L_t^{2,-1/2}(\Omega ,{\mathbb{R}}^3)=\{w\in L^2(\Omega ,{\mathbb{R}}^3):\nu \times w\in H^{-1/2}(\partial \Omega ,{\mathbb{R}}^3)\},$$

$$L_{\nu }^{2,-1/2}(\Omega ,{\mathbb{R}}^3)=\{w\in L^2(\Omega ,{\mathbb{R}}^3):\nu \cdot w\in H^{-1/2}(\partial \Omega )\},$$

$$L_{t0}^{2,-1/2}(\Omega ,{\mathbb{R}}^3)=\{w\in L_t^{2,-1/2}(\Omega ,{\mathbb{R}}^3):\nu \times w=0\text{on }\partial \Omega \},$$

$$L_{\nu 0}^{2,-1/2}(\Omega ,{\mathbb{R}}^3)=\{w\in L_{\nu }^{2,-1/2}(\Omega ,{\mathbb{R}}^3):\nu \cdot w=0\text{on }\partial \Omega \}.$$

$$T{H}^{k+1/2}(\partial \Omega ,{\mathbb{R}}^3)=$$

$$T{C}^{k,\alpha }(\partial \Omega ,{\mathbb{R}}^3)=$$

$$⟨\varphi ,\eta {⟩}_{\partial \Omega ,1/2}=⟨\varphi ,\eta {⟩}_{H^{-1/2}(\partial \Omega ),H^{1/2}(\partial \Omega )},$$

$${\dot{H}}^{-1/2}(\partial \Omega )=\{\zeta \in H^{-1/2}(\partial \Omega ):⟨\zeta ,1{⟩}_{\partial \Omega ,1/2}=0\}.$$

$$L^2(\Omega ,{\mathbb{R}}^3)=\mathcal{H}(\Omega ,\text{curl}0){\oplus }_{L^2(\Omega )}\text{curl}{H}_{t0}^1(\Omega ,\text{div}0),$$

$$L^2(\Omega ,{\mathbb{R}}^n)=\mathcal{H}(\Omega ,\text{div}0){\oplus }_{L^2(\Omega )}\text{grad}{H}_0^1(\Omega ),$$

$$\text{curl}{H}^1(\Omega ,{\mathbb{R}}^3)=\text{curl}{H}_{n0}^1(\Omega ,\text{div}0)=$$

$$\text{curl}{H}_{t0}^1(\Omega ,\text{div}0)=$$

$${\mathcal{H}}^{\Gamma }(\Omega ,\text{div}0)=$$

$${\mathcal{H}}_0^{\Sigma }(\Omega ,\text{div}0)=$$

$$⟨u\cdot \nu ,1{⟩}_{H^{-1/2}({\Sigma }_i),H^{1/2}({\Sigma }_i)}=0,i=1,\cdots ,N\}.$$

$$\mathcal{H}(\Omega ,\text{curl}0)=$$

$$\mathcal{H}(\Omega ,\text{div}0)=$$

$${\mathcal{H}}_0(\Omega ,\text{div}0)=$$

$$L^2(\Omega ,{\mathbb{R}}^3)={\mathcal{H}}_0^{\Sigma }(\Omega ,\text{div}0){\oplus }_{L^2(\Omega )}{\mathbb{H}}_1(\Omega ){\oplus }_{L^2(\Omega )}\text{grad}{H}^1(\Omega ).$$

$$L^2(\Omega ,{\mathbb{R}}^3)={\mathcal{H}}^{\Gamma }(\Omega ,\text{div}0){\oplus }_{L^2(\Omega )}{\mathbb{H}}_2(\Omega ){\oplus }_{L^2(\Omega )}\text{grad}{H}_0^1(\Omega ).$$

$$c_1|z{|}^2-g_1(x)\le ⟨\mathcal{H}(x,z),z⟩,|\mathcal{H}(x,z)|\le c_2|z|+g_2(x).$$

Full text

1. Introduction

In this paper we study a nonlinear magneto-static model on a general domain in $\mathbb{R}^{3}$, which is multiply-connected and has $m$ holes (namely the boundary of $\partial\Omega$ consists of $m+1$ connected components). We are interested in the effects of domain topology on solvability of the boundary value problems.

In the recent years quasilinear magneto-static models in electromagnetism have been studied by many authors. A typical equation is the quasilinear Maxwell type equation of the following form:

[TABLE]

where $\mathbb{J}$ is a given vector field. This system is a time-independent version of the eddy current model for a magnetic and anisotropic material where the relation between magnetic induction field $\mathbb{B}$ and the magnetic field $\mathbb{H}$ is given by the BH-curve $\mathbb{H}=\mathcal{H}(\mathbb{B})$ which is usually nonlinear, see For instance, Milani and Picard [MP1, MP2], R. Picard [Pi], Jochmann [Jo], F. Bachinger, U. Langer and J. Schöberl [BLS], Jiang and Zheng [JZ1, JZ2], L. Yousept [Yo], Pan [P4], and the references therein.

It was shown in [P4] that for a system similar to (1.1) with $\mathbb{J}$ replaced by a nonlinear vector-valued function $\mathbb{f}(x,\mathbb{u})$, it is in general necessary to introduce a potential term into the equation and one needs to consider a Maxwell-Stokes type system

[TABLE]

It was also shown in [P4] that the type of boundary condition for $p$ should be determined according to the topology of $\Omega$. If $\Omega$ has no holes, then it is natural to consider Dirichlet boundary condition for $p$. If $\Omega$ has holes, then one has to consider Neumann type boundary condition for $p$. In this paper we will explain the physical significance of the term $\nabla p$.

In this paper we assume that the relation between magnetic induction $\mathbb{B}$ and magnetic field intensity $\mathbb{H}$ is nonlinear:

[TABLE]

To avoid the complexity of electric part, we assume that the relation between electric displacement $\mathbb{D}$ and electric field intensity $\mathbb{E}$ is linear: $\mathbb{D}=\varepsilon\mathbb{E},$ where $\varepsilon$ is scalar or matric valued function. Finally, we shall assume the electric current density $\mathbb{j}$ satisfies the Ohm’s law. We shall introduce a vector field $\mathbb{u}$ and a scalar function $p$, such that the electric field $\mathbb{E}$ and magnetic induction $\mathbb{B}$ can be represented by

[TABLE]

where $\mathbb{h}_{1}\in\mathbb{H}_{1}(\Omega)$, $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$, which describe the topology of the domain $\Omega$. Then we can derive the magneto-static model. If the electric current is given, then the equation is

[TABLE]

and if the electric current depends on the magnetic induction, then the equation is

[TABLE]

We shall examine solvability of (1.4) subjected to a boundary condition on $\mathbb{u}$. For (1.5) we need also a boundary condition for $p$. Following the observation in [P4], we shall consider Neumann type boundary condition for $p$. Regularity of weak solutions of BVPs of (1.4) and (1.5) can be obtained by the methods in [P4] and will not be discussed in this paper.

We mention that the gradient term $\nabla p$ in (1.5) has a clear physical meaning. In fact, from (1.3) we see that

[TABLE]

so $\nabla p$ represents both electric field $\mathbb{E}$ and domain topology. We also mention that both (1.4) and (1.5) contain the topological terms $\mathbb{h}_{1},\mathbb{h}_{2}$, and we will see that solvability of the boundary value problems (BVPs for short) for (1.4) and (1.5) depends on the choice of $\mathbb{h}_{1},\mathbb{h}_{2}$, which is an effect of domain topology on these equations.

In this paper we shall use various results on vector fields, including the divergence-curl-gradient inequalities, and regularity of div-curl system and of time-independent Maxwell’s system, which can be fund in literature, see for instance [DaL, GR, MMT, Sc, W, BW, NW, AS, AuA, CDN, KY] and the references therein. We mention that time-independent semilinear Maxwell system has also been studied by many authors, see for instance [Y, BF, JS, P3] and the references therein. We also mention that the Meissner states of type I​I superconductors can be described by a quasilinear system of the form similar to (1.1), see [Ch, Mon, BaP, P1, LP].

This paper is organized as follows. In section 2 we list notation for various spaces of vector fields, and collect some facts on these spaces that we will need. In section 3 we derive the magneto-static equations. Sections 4, 5, and 6 are devoted to study of existence of weak solutions to BVPs of (1.4) under various type of boundary conditions on $\mathbb{u}$. Solvability of BVPs for (1.5) is studied in section 7.

2. Preliminaries

2.1. Notation

Most of materials in this subsection can be fund in [DaL, GR]. Let $\Omega$ be a bounded domain in $\mathbb{R}^{3}$ and $\nu$ be the unit outer normal vector of the boundary $\partial\Omega$. We assume that $\Omega$ has the following properties ([DaL, p.217]):

• $(O_{1})$

$\Omega$ is a bounded domain in $\mathbb{R}^{3}$ with a $C^{r}$ boundary $\partial\Omega$, $r\geq 1$; $\Omega$ is locally situated on one side of $\partial\Omega$; $\partial\Omega$ has a finite number of connected components $\Gamma_{1},\cdots,\Gamma_{m+1}$, where $m\geq 0$ and $\Gamma_{m+1}$ denoting the boundary of the unbounded connected component of the set $\mathbb{R}^{3}\setminus\bar{\Omega}$.

• $(O_{2})$

There exist $N$ manifolds of dimension $2$ and of class $C^{r}$ denoted by $\Sigma_{1},\cdots,\Sigma_{N}$, $N\geq 0$, such that $\Sigma_{i}\cap\Sigma_{j}=\emptyset$ for $i\neq j$ and non-tangential to $\partial\Omega$, such that $\dot{\Omega}=\Omega\setminus(\sum_{j=1}^{N}\Sigma_{j})$ is simply-connected and Lipschitz.

2.1.1. The spaces of vector fields

As in [BaP, P1, P4], we denote the Sobolev spaces of scalar functions by $W^{k,p}(\Omega)$, $H^{k}(\Omega)$ and $H^{s}(\partial\Omega)$ etc, and denote the Sobolev spaces of vector fields by $W^{k,p}(\Omega,\mathbb{R}^{3})$, $H^{k}(\Omega,\mathbb{R}^{3})$, $H^{s}(\partial\Omega,\mathbb{R}^{3})$ etc. We use same notation for the norm of scalar functions and for vector fields. For instance, the norms in $H^{k}(\Omega)$ and in $H^{k}(\Omega,\mathbb{R}^{3})$ are both denoted by $\|\cdot\|_{H^{k}(\Omega)}$. We denote

[TABLE]

It is well-known that $\dim\mathbb{H}_{1}(\Omega)=N$ and $\dim\mathbb{H}_{2}(\Omega)=m$, where $m$ and $N$ are given in $(O_{1})$ and $(O_{2})$. We denote by $\mathbb{H}_{j}(\Omega)^{\perp}_{L^{2}(\Omega)}$ the orthogonal complementary of $\mathbb{H}_{j}(\Omega)$ in $L^{2}(\Omega,\mathbb{R}^{3})$. When $\partial\Omega$ is of $C^{2}$, using regularity of div-curl system we have

[TABLE]

If $X(\Omega)$ denotes a space of vector fields on $\Omega$, we write

[TABLE]

For instance,

[TABLE]

If $Y(\Omega)$ is a space of scalar functions on $\Omega$, we denote

[TABLE]

Denote

[TABLE]

We also need spaces of tangential vector fields:

[TABLE]

We denote

[TABLE]

2.1.2. Decomposition of $L^{2}(\Omega,\mathbb{R}^{3})$

Let us recall that (see [DaL, p.225-226]):

[TABLE]

and recall image of operator curl  (see [DaL, p.222, Proposition 3; p.226, Remark 5]):

[TABLE]

where

[TABLE]

These yield the following decompositions of the kernel of the operators curl  and div :

[TABLE]

It follows that $\mathbb{H}_{1}(\Omega)\subset\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ and $\mathbb{H}_{2}(\Omega)\perp_{L^{2}(\Omega)}\mathbb{H}_{1}(\Omega)$.

From the first equality of (2.3) and the first equality of (2.5) we have

[TABLE]

From the second equality of (2.3) and the second equality of (2.5) we have

[TABLE]

2.2. Assumptions on $\mathcal{H}(x,\mathbb{z})$

When $\Omega$ is bounded, we say $\mathcal{H}\in C^{k,\alpha}_{\text{\rm loc}}(\bar{\Omega}\times\mathbb{R}^{3},\mathbb{R}^{3})$ if $\mathcal{H}\in C^{k,\alpha}_{\text{\rm loc}}(\bar{\Omega}\times K,\mathbb{R}^{3})$ for any bounded set $K\subset\mathbb{R}^{3}$. Now we list the conditions on $\mathcal{H}(x,\mathbb{z})$ that we may need.

• $(H_{1})$

$\mathcal{H}\in C^{0}_{\text{\rm loc}}(\bar{\Omega}\times\mathbb{R}^{3},\mathbb{R}^{3})$, and there exist positive constants $c_{1},c_{2}$, and functions $g_{1}\in L^{1}(\Omega)$, $g_{2}\in L^{2}(\Omega)$ such that, for all $x\in\bar{\Omega}$ and $\mathbb{z}\in\mathbb{R}^{3}$,

[TABLE]

• $(H_{2})$

$\mathcal{H}\in C^{1}_{\text{\rm loc}}(\bar{\Omega}\times\mathbb{R}^{3},\mathbb{R}^{3})$, and there exist positive constants $\mu,M_{1},M_{2}$ such that for all $x\in\bar{\Omega}$ and $\mathbb{z},\;\xi\in\mathbb{R}^{3}$,

[TABLE]

• $(H_{3})$

$\mathcal{H}(x,\mathbb{z})$ has an inverse $\mathcal{B}(x,\mathbb{w})$, namely, $\mathbb{z}=\mathcal{B}(x,\mathbb{w})$ if and only if $\mathbb{w}=\mathcal{H}(x,\mathbb{z})$.

Remark 2.1**.**

If $\mathcal{H}$ satisfies $(H_{2})$ then it is Lipschitz in $\mathbb{z}$, and has the following property:

• $(H_{4})$

There exists a constant $\mu>0$ such that

[TABLE]

We can show that $(H_{1}),(H_{2}),(H_{3})$ imply the following properties of $\mathcal{B}$:

• $(B_{1})$

$\mathcal{B}\in C^{0}_{\text{\rm loc}}(\bar{\Omega}\times\mathbb{R}^{3},\mathbb{R}^{3})$, and there exist positive constants $c_{3},c_{4}$, and functions $g_{3}\in L^{1}(\Omega)$, $g_{4}\in L^{2}(\Omega)$ such that, for all $x\in\bar{\Omega}$ and $\mathbb{w}\in\mathbb{R}^{3}$,

[TABLE]

• $(B_{2})$

$\mathcal{B}\in C^{1}_{\text{\rm loc}}(\bar{\Omega}\times\mathbb{R}^{3})$, and there exist positive constants $\lambda_{0},N_{1},N_{2}$ such that for all $x\in\bar{\Omega}$ and $\mathbb{w},\;\eta\in\mathbb{R}^{3}$,

[TABLE]

Remark 2.2**.**

• (i)

If $\mathbb{H}$ satisfies $(H_{1})$, $(H_{3})$, then $\mathcal{B}$ satisfies $(B_{1})$, with

[TABLE]

• (ii)

If $\mathbb{H}$ satisfies $(H_{2})$, $(H_{3})$, then $\mathcal{B}$ satisfies $(B_{2})$, where $\lambda_{0}$ and $N_{2}$ depend on $\mu$ and $M_{2}$, $N_{1}$ depends on $\mu,M_{1},M_{2}$. In fact,

[TABLE]

where $C_{0}$ is an absolute constant.

Remark 2.3**.**

If $(B_{2})$ holds, then for any $\mathbb{w}_{1},\mathbb{w}_{2}\in\mathbb{T}^{3}$ and $x\in\bar{\Omega}$,

[TABLE]

2.3. Assumption on $\mathbb{f}(x,\mathbb{z})$

• $(f_{1})$

$\mathbb{f}\in C^{1}_{\text{\rm loc}}(\bar{\Omega}\times\mathbb{R}^{3},\mathbb{R}^{3})$, and there exist positive constants $K_{0},K_{1},K_{2}$ and $f_{0}\in L^{2}(\Omega)$ such that, for all $x\in\bar{\Omega}$ and $\mathbb{z}\in\mathbb{R}^{3}$,

[TABLE]

2.4. The Projections $\mathcal{P}_{\nu}$ and $\mathcal{P}_{n}$

Definition 2.4**.**

We introduce the projection $\mathcal{P}_{\nu}$ as follows. If $\mathbb{w}\in L^{2}(\Omega,\mathbb{R}^{3})$, then $\mathcal{P}_{\nu}[\mathbb{w}]=\mathbb{w}+\nabla\phi_{\mathbb{w}}$, where $\phi_{\mathbb{w}}\in\dot{H}^{1}(\Omega)$ is the unique weak solution of

[TABLE]

namely, for any $\phi\in H^{1}(\Omega)$ and

[TABLE]

Remark 2.5**.**

Assume $\Omega$ is abounded domain in $\mathbb{R}^{3}$ with a $C^{2}$ boundary. Then $\mathcal{P}_{\nu}:L^{2}(\Omega,\mathbb{R}^{3})\mapsto\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ is a continuous projection.

Definition 2.6**.**

Assume $\nu\times\mathbb{H}^{0}$ satisfies (6.8). We introduce the Neumann projection $\mathcal{P}_{n}$ associated with $\nu\cdot\text{\rm curl\,}\mathbb{H}^{0}_{T}$ as follows. If $\mathbb{w}\in L^{2}(\Omega,\mathbb{R}^{3})$, then $\mathcal{P}_{n}[\mathbb{w}]=\mathbb{w}+\nabla\psi_{\mathbb{w}}$, where $\psi_{\mathbb{w}}\in\dot{H}^{1}(\Omega)$ is the unique weak solution of

[TABLE]

namely, for all $\eta\in H^{1}(\Omega)$ it holds that

[TABLE]

Lemma 2.7**.**

Assume $\Omega$ is a bounded domain in $\mathbb{R}^{3}$ with a $C^{2}$ boundary, and $\nu\times\mathbb{H}^{0}$ satisfies (6.8). Then $\mathcal{P}_{n}:L^{2}(\Omega,\mathbb{R}^{3})\to\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ is a continuous projection.

Proof.

Denote by $\mathbb{H}^{0}$ the divergence-free and tangential component preserving extension on $\mathbb{H}^{0}_{T}$. For any $\mathbb{w}\in L^{2}(\Omega,\mathbb{R}^{3})$, $\mathcal{P}_{n}[\mathbb{w}]=\mathbb{w}+\nabla\psi_{\mathbb{w}}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$. Hence $\nu\cdot\mathcal{P}_{n}[\mathbb{w}]\in H^{-1/2}(\partial\Omega)$. Denote the connected components of $\partial\Omega$ by $\Gamma_{j}$, $j=1,\cdots,m+1$, and let $\eta_{j}$ denote the harmonic function in $\Omega$ such that $\eta_{j}=\delta_{ij}$ on $\Gamma_{i}$. Then $\nu\times\nabla\eta_{j}=\mathbb{0}$ on $\partial\Omega$. So

[TABLE]

Hence $\mathcal{P}_{n}[\mathbb{w}]\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$. Taking $\psi_{\mathbb{w}}$ as a test function we have

[TABLE]

So

[TABLE]

Using the trace theorem and applying the Poincaré inequality to $\psi\in\dot{H}^{1}(\Omega)$, we get

[TABLE]

∎

3. Derivation of the Magneto-static Equations

3.1. Time-dependent problem: quasi-static magnetic fields

3.1.1. The Maxwell equations and our assumptions

We start with the general case of magnetic and anisotropic materials, occupying a bounded $C^{2}$ domain in $\mathbb{R}^{3}$, which is multiply-connected and has holes. The operators div , curl , $\nabla$ and $\Delta$ act on the space variables $x$. The Maxwell equations are of the following form

[TABLE]

where $\mathbb{E}$ is the electric field, $\mathbb{B}$ is the magnetic induction, $\mathbb{H}$ is the magnetic field, $\mathbb{j}$ is the current density, $\rho$ is the charge density, and $\varepsilon$ is the permittivity.111From the first and third equations in (3.1) we get

$\partial_{t}\rho+\text{\rm div\,}\mathbb{j}=0,$

which is the conservation law of electric charge.

Assumption 3.1**.**

(Assumption on the $\mathbb{H}$-$\mathbb{B}$ relation.) There exists a nonlinear vector-valued function $\mathcal{B}(x,\mathbb{z})$ such that magnetic induction $\mathbb{B}$ can be represented by magnetic field $\mathbb{H}$:

[TABLE]

$\mathcal{B}(x,\cdot)$* has an inverse $\mathcal{H}(x,\cdot)$, which means that*

[TABLE]

Assumption 3.2**.**

(Assumption on the magnetic induction $\mathbb{B}$.) The magnetic induction $\mathbb{B}(t,x)$ is such that

[TABLE]

Assumption 3.3**.**

(Assumption on the electric field $\mathbb{E}$.) The electric field $\mathbb{E}(t,x)$ is such that

[TABLE]

Later on we shall assume also the Ohm’s law holds (see for instance [LL, p.199-120]):

Assumption 3.4**.**

(Assumption on the current density $\mathbb{j}$.) The current density $\mathbb{j}$ satisfies the Ohm’s law

[TABLE]

where $\sigma\geq 0$ is the electric conductivity, and $\mathbb{j}_{a}$ is the applied current density.

3.1.2. Rewrite (3.1) in term of $\mathbb{E}$ and a vector potential $\mathbb{A}$

Step 1. General case. From (3.1)(b) we have $\text{\rm div\,}\mathbb{B}=0$. From the first line of (2.4) and second line of (2.5) we see that there exist vector fields $\mathbb{w}(t,x)$ and $\mathbb{h}_{2}(t,x)$, with $\mathbb{w}(t,\cdot)\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)$ and $\mathbb{h}_{2}(t,\cdot)\in\mathbb{H}_{2}(\Omega)$, such that

[TABLE]

Since $\mathbb{H}_{2}(\Omega)$ is of finite dimensions, and $\text{\rm curl\,}\mathbb{w}(t,\cdot)$ is orthogonal to $\mathbb{H}_{2}(\Omega)$, it follows that

[TABLE]

From (3.1)(d) and (3.7) we have

[TABLE]

From this, (3.5) and (3.8) we have, for any $t>0$,

[TABLE]

From this and the last equality in (3.8) we have $\partial_{t}\mathbb{h}_{2}=\mathbb{0}$, so $\text{\rm curl\,}(\mathbb{E}+\partial_{t}\mathbb{w})=-\partial_{t}\mathbb{h}_{2}=\mathbb{0}$. By the first line of (2.5), there exists a scalar function $\xi(t,x)$ and a vector field $\tilde{\mathbb{h}}_{1}(t,x)$, with $\xi(t,\cdot)\in H^{1}(\Omega)$, $\tilde{\mathbb{h}}_{1}(t,\cdot)\in\mathbb{H}_{1}(\Omega)$, such that

[TABLE]

Using this, (3.5) and (3.8), and since $\mathbb{H}_{1}(\Omega)$ is of finite dimensions and $\nabla\xi(t,\cdot)$ is orthogonal to $\mathbb{H}_{1}(\Omega)$, we can show that

[TABLE]

Introduce a vector field

[TABLE]

From the properties of $\mathbb{w}$ and $\xi$ we see that

[TABLE]

Now we can represent $\mathbb{E},\mathbb{B},\mathbb{H}$ by $\mathbb{A}$ (see (3.9), (3.7), (3.3)):

[TABLE]

Plugging the relation $\mathbb{H}=\mathcal{H}(x,\mathbb{B})$ into (3.1) (c) we get

[TABLE]

Finally from (3.1)(a) and (3.11) we have

[TABLE]

Step 2. Assume the Ohm’s law (3.6). Then from (3.1) (c) and (3.11) we have

[TABLE]

Plugging the relation $\mathbb{H}=\mathcal{H}(x,\mathbb{B})$ into the above equality we get

[TABLE]

3.1.3. Quasi-static approximation

Next we consider a quasi-static approximation of the Maxwell equations (3.1) (in the form of (3.12)) by neglecting the displacement current $\partial_{t}(\varepsilon\mathbb{E})$.

Step 1. General case. From (3.12) with the displacement current neglected we have:

[TABLE]

We introduce a function $\Phi$ such that for $t\geq 0$,

[TABLE]

Set

[TABLE]

From (3.10) we have

[TABLE]

Now Eq. (3.15) is transferred to

[TABLE]

here $\mathbb{u}$ and $\mathbb{j}$ may depend on $t$. Eq. (3.13) takes the form

[TABLE]

Step 2. Assume the Ohm’s law (3.6). Then a quasi-static approximation of the Maxwell equations (3.1) (in the form of (3.14)) gives

[TABLE]

After introducing $\Phi(t,x)$ by (3.16), and introducing $\mathbb{u},\mathbb{u}^{0},q$ by (3.17), we have (3.19), and (3.20) is transferred to

[TABLE]

In the special case when $\sigma$ and $\varepsilon$ are positive constants, from (3.19) and (3.21) we get

[TABLE]

3.2. Time-independent problems: magneto-static fields

Step 1. Assume that

[TABLE]

Then the steady state problem of (3.18) takes the following form:

[TABLE]

This equation is different to (3.18) in the sense that in (3.24) $\mathbb{u}$ and $\mathbb{j}$ are independent of $t$. (3.24) should be coupled with (3.19), which takes the form

[TABLE]

Step 2. Assume the Ohm’s law (3.6), and assume

[TABLE]

Let us introduce $p,\mathbb{h}_{1}$ by

[TABLE]

Since $\partial_{t}\Phi=-q(x)=-p(x)/\sigma$, there exists a function $\Phi_{0}(x)$ such that

[TABLE]

Now the relations between $\mathbb{A},\mathbb{E},\mathbb{B},\mathbb{H},q$ and $\mathbb{u},\mathbb{h}_{1},\mathbb{h}_{2},\Phi_{0},p$ are the following:

[TABLE]

The steady state problem of (3.21) takes the following form:

[TABLE]

Now the Gauss’s law of electricity (3.19) is reduced to (3.22).

[TABLE]

and (3.1) is reduced to (3.29)-(3.30). $\rho$ should not be arbitrarily given, instead, it is a function determined by $\mathbb{h}_{1}$ and $p$.

So we conclude that:

Lemma 3.5**.**

Assume the assumptions 3.1, 3.2, 3.3.

• (i)

After neglecting the displacement current $\partial_{t}(\varepsilon\mathbb{E})$ from the Maxwell equations (3.1), and introducing the magnetic potential $\mathbb{u}$, and assume $\mathbb{u}$ and $\mathbb{j}$ are independent of $t$, then (3.1) is reduced to (3.24)-(3.25).

• (ii)

Assume furthermore the Ohm’s law (3.6) holds, $\sigma$ is a positive constant, and assume $\mathbb{u},\tilde{\mathbb{h}}_{1},\mathbb{j}_{s},q,\varepsilon,\rho$ are independent of $t$.

• (a)

Solution $(\mathbb{E},\mathbb{B},\mathbb{H})$ of (3.1) is represented by (3.28), and (3.1) is reduced to (3.29)-(3.30).

• (b)

If $\mathbb{j}_{a}$ and $\rho$ are independent of $\mathbb{u}$, then $\rho$ is determined by $\mathbb{j}_{a}$ by (3.22), and the magneto-static problem is reduced to a single system (3.29).

Remark 3.6**.**

Assume the Ohm’s law (3.6) and assume (3.26).

• (i)

Equation (3.29) indicates the relation between magnetic field $\mathbb{H}$, electric field $\mathbb{E}$, and electric current $\mathbb{j}=\sigma\mathbb{E}+\mathbb{j}_{a}$:

[TABLE]

• (ii)

(About the topological effects on electromagnetism.) (3.28) and (3.29) show that the electromagnetic fields depend on $\mathbb{h}_{1},\mathbb{h}_{2}$ which represents topology of the domain. (3.28) indicates that $\mathbb{E}$ contains $\mathbb{h}_{1}$ and $\mathbb{B}$ contains $\mathbb{h}_{2}$ explicitly, which suggests that $\mathbb{E}$ is more directly linked to simply-connectedness of $\Omega$, and $\mathbb{H}$ is more directly linked to connectedness of $\partial\Omega$.

• (iii)

(About the physical meaning of $\nabla p$ in (3.29).) Mathematically we may say that, the gradient term $\nabla p$ is necessarily introduced into the equation in (3.29) to balance the equality, which is necessary for solvability of the boundary value problem. On the other hand, from (3.28) and (3.6) we see that

[TABLE]

so $\nabla p$ represents both electric field $\mathbb{E}$ and domain topology.

Remark 3.7**.**

In [P4] we considered the following system

[TABLE]

This is a particular case of (3.29) without the topological effect terms $\mathbb{h}_{1},\mathbb{h}_{2}$, and without the potential term $\nabla p$, where $\mathbb{j}_{a}$ is either given or $\mathbb{j}_{a}=\mathbb{f}(x,\mathbb{u})$. It was shown in [P4] that (3.31) under some Dirichlet boundary condition may not be well-posed, and a potential term should be introduced to balance the equation:

[TABLE]

The boundary condition for $p$ considered in [P4] is Dirichlet type if $\Omega$ has no holes, and Neumann type if $\Omega$ has holes. Now we can say that $\nabla p$ gives the electric field $\mathbb{E}$, hence a Dirichlet boundary condition on $p$ is to prescribe the tangential component of $\mathbb{E}$, and a Neumann boundary condition on $p$ is to prescribe the normal component of $\mathbb{E}$.

(3.29) is a better model than (3.32) in the following sense. (3.32) says that magnetic field (described by $\mathbb{u}$) and electric field (described by $\nabla p$) depend on each other, (3.29) says that magnetic field (described by $\mathbb{u}$) and electric field (described by $\nabla p$ and $\mathbb{h}_{1}$) depend on each other, and both depend on domain topology (described by $\mathbb{h}_{1},\mathbb{h}_{2}$).

3.3. BVPs where the current is given

Assume $\mathbb{j}_{a}(x)$ is given and independent of $\mathbb{u},\mathbb{h}_{1},\mathbb{h}_{2}$. As mentioned in Lemma 3.5, we should assume $\rho$ is determined by $p,\mathbb{h}_{1}$ by (3.30). Then (3.1) is reduced to a single system (3.29). If we write $\mathbb{j}_{a}(x)=\mathbb{J}(x),$ then the first equation in (3.29) can be written as follows:

[TABLE]

Under a suitable boundary value condition, the unknown $p(x)$ can be solved independent of $\mathbb{u}$. In this case, replacing $\mathbb{J}+\nabla p$ by $\mathbb{J}$, the equation is in the form of (1.4). Then we may pose one of the following boundary conditions for $\mathbb{u}$ on $\partial\Omega$:

• (1)

Dirichlet boundary condition $\mathbb{u}_{T}=\mathbb{u}^{0}_{T}$;

• (2)

Tangential curl boundary condition $\nu\times\text{\rm curl\,}\mathbb{u}=\nu\times\mathbb{B}^{0}$;

• (3)

Normal curl boundary condition $\nu\cdot\text{\rm curl\,}\mathbb{u}+\nu\cdot\mathbb{h}_{2}(x)=B^{0}_{n}$;

• (4)

Natural boundary condition $\nu\times\mathcal{H}(x,\text{\rm curl\,}\mathbb{u}+\mathbb{h}_{2}(x))=\nu\times\mathbb{H}^{0}$;

• (5)

Co-normal boundary condition $\nu\cdot\mathcal{H}(x,\text{\rm curl\,}\mathbb{u}+\mathbb{h}_{2}(x))=H^{0}_{n}$;

3.4. BVPs where the current depends on magnetic induction

Assume $\mathbb{j}_{a}$ depends on the magnetic induction $\mathbb{B}=\text{\rm curl\,}\mathbb{u}+\mathbb{h}_{2}.$ Then we may write $\mathbb{j}_{a}=\mathbb{f}(x,\text{\rm curl\,}\mathbb{u}+\mathbb{h}_{2}),$ and (3.29) takes the form of (1.5). We may pose one of the above mentioned boundary conditions for $\mathbb{u}$, and either Dirichlet or Neumann condition for $p$. Other type boundary conditions are also possible. However, following the discussions in [P4], we have the following observations.

• (1)

If $\Omega$ has no holes, then BVP of (1.5) with Dirichlet boundary condition for $p$ (and with a suitable boundary condition on $\mathbb{u}$) is well-posed.

• (2)

If $\Omega$ has holes, then BVP of (1.5) with Neumann boundary condition for $p$ (and with a suitable boundary condition on $\mathbb{u}$) is well-posed, but the problem with Dirichlet boundary condition for $p$ is not well-posed.

4. Dirichlet BVP of the Maxwell System with a Given Current

In this section we consider the Dirichlet BVP of the Maxwell system with a given current $\mathbb{J}$:

[TABLE]

In (4.1) there are no boundary conditions for $\mathbb{h}_{1}$ and $\mathbb{h}_{2}$, and we may view $\mathbb{h}_{1},\mathbb{h}_{2}$ as parameters. If $\mathbb{u}$ is a solution of (4.1), then for any $\mathbb{h}\in\mathbb{H}_{2}(\Omega)$, $\mathbb{u}+\mathbb{h}$ is also a solution of (4.1). To avoid non-uniqueness due to $\mathbb{H}_{2}(\Omega)$, we may require the solution $\mathbb{u}\perp_{L^{2}(\Omega)}\mathbb{H}_{2}(\Omega).$ As in [P3, Lemma 2.1] we can show that the condition $\mathbb{J}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ is necessary for (4.1) to have a weak solution. Throughout this section we assume that

[TABLE]

and

[TABLE]

(4.3) implies that there exists a divergence-free and curl-minimizing extension $\mathcal{U}$ of $\mathbb{u}^{0}_{T}$, see [P2]. Set $\mathbb{b}=\text{\rm curl\,}\mathcal{U}$ and $\mathbb{v}=\mathbb{u}-\mathcal{U}$. Then (4.1) can be written as follows:

[TABLE]

In the following we derive solvability of (4.1) by variational methods, monotone operator methods, and the reduction methods. See Proposition 5.7 in [P4]) for solvability by compact operator methods.

4.1. Solvability of (4.1) by variational methods

Proposition 4.1**.**

Let $\Omega,\mathbb{J},\mathbb{h}_{1},\mathbb{h}_{2},\mathbb{u}^{0}_{T}$ satisfy (4.2) and (4.3), $\mathcal{H}(x,\mathbb{z})=\nabla_{\mathbb{z}}P(x,\mathbb{z})$ for a function $P$ satisfying

• $(P_{1})$

$P\in C^{1,\alpha}_{\text{\rm loc}}(\bar{\Omega}\times\mathbb{R}^{3})$, $P(x,\mathbb{z})$ is strictly convex in $z$, and there exist positive constants $c_{1},c_{2},c_{3}$, functions $p_{1}\in L^{1}(\Omega)$ and $p_{2}\in L^{2}(\Omega)$ such that

[TABLE]

Then (4.1) has a solution $\mathbb{u}\in H^{1}(\Omega,\text{\rm div\,}0)$.

Proof.

Condition $(P_{1})$ here is slightly more general than the condition $(P)$ in [P4]. By the argument in the proof of [P4, Proposition 3.1] we know that the energy functional has a minimizer $\mathbb{u}$ in $H^{1}_{t0}(\Omega,\text{\rm div\,}0)\cap\mathbb{H}_{2}(\Omega)^{\perp}$. Hence for all $\mathbb{w}\in H^{1}_{t0}(\Omega,\text{\rm div\,}0)\cap\mathbb{H}_{2}(\Omega)^{\perp}$ it holds that

[TABLE]

This equality remains true for all $\mathbb{w}\in H^{1}_{t0}(\Omega,\text{\rm div\,}0)$ because $\mathbb{J}+\mathbb{h}_{1}$ is perpendicular to $\mathbb{H}_{2}(\Omega)$. So we can use a variant form of the De Rham lemma for functionals vanishing on $H^{1}_{t0}(\Omega,\text{\rm div\,}0)$ (see [P4, Lemma 2.2] and [P5, Lemma 4.2]) to conclude that there exists $p\in L^{2,-1/2}(\Omega)$ such that the equality

[TABLE]

holds as elements in $H^{1,*}_{t0}(\Omega,\mathbb{R}^{3})$, and $\gamma(p)=0$ on $\partial\Omega$. This means that

[TABLE]

Let $\zeta$ be such that

[TABLE]

Then $\mathbb{w}=\nabla\zeta\in H^{1}_{t0}(\Omega,\mathbb{R}^{3})$. Plugging it into the above equality we find

[TABLE]

because $\text{\rm div\,}(\mathbb{J}+\mathbb{h}_{1})=0$ in $\Omega$ and $\zeta=0$ on $\partial\Omega$. Hence $p=0$. So $\mathbb{v}$ is a weak solution of (4.4). ∎

Corollary 4.2**.**

Let $\Omega,\mathbb{J},\mathbb{h}_{1},\mathbb{h}_{2},\mathbb{u}^{0}_{T}$ satisfy (4.2) and (4.3), $\mathcal{H}(x,\mathbb{z})=a(x,|\mathbb{z}|^{2})\mathbb{z}$, where $a(s,x)$ satisfies the following condition:

• $(a)$

$a\in C^{1}_{\text{\rm loc}}(\bar{\Omega}\times\mathbb{R}^{3})$, $\nabla_{x}a\in C^{\alpha}_{\text{\rm loc}}(\bar{\Omega}\times\mathbb{R}^{3},\mathbb{R}^{3})$ with $0<\alpha<1$, and there exist positive constants $\lambda,\Lambda$ and $\delta\in(0,1)$ such that

[TABLE]

Then (4.1) has a weak solution $\mathbb{u}\in H^{1}(\Omega,\text{\rm div\,}0)$.

Proof.

The proof is similar to the proof of Corollary 3.2 in [P4]. Let $c(t,x)=\int_{0}^{|t|}a(s,x)ds$, $P(x,\mathbb{z})=c(x,|\mathbb{z}|^{2})$. From $(a)$ we see that $P(x,\mathbb{z})$ satisfies $(P_{1})$. In particular,

[TABLE]

So $P(x,\mathbb{z})$ is strictly convex in $\mathbb{z}$. Hence the conclusion follows from Proposition 4.1. ∎

4.2. Solvability of (4.1) by monotone operator methods

Theorem 4.3**.**

Assume $\Omega,\mathbb{J},\mathbb{h}_{1},\mathbb{h}_{2},\mathbb{u}^{0}_{T}$ satisfy (4.2), (4.3), and $\mathcal{H}$ satisfies $(H_{1}),(H_{2})$. Then (4.1) has a unique solution $\mathbb{u}\in H^{1}(\Omega,\text{\rm div\,}0)\cap\mathbb{H}_{2}(\Omega)^{\perp}_{L^{2}(\Omega)}$. Moreover, the solution map $(\mathbb{h}_{1},\mathbb{J})\mapsto\mathbb{u}$ is continuous from $\mathbb{H}_{1}(\Omega)\times\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ to $H^{1}(\Omega,\text{\rm div\,}0)$.

Proof.

Step 1. Define

[TABLE]

Endowed with the norm in $H^{1}(\Omega,\mathbb{R}^{3})$, $X$ is a separable Hilbert space. By the div-curl-gradient inequalities we know that

[TABLE]

Hence both $\|\mathbb{w}\|_{H^{1}(\Omega)}$ and $\|\text{\rm curl\,}\mathbb{w}\|_{L^{2}(\Omega)}$ are the equivalent norms in $X$. Denote by $X^{*}$ the dual space of $X$, and denote by $\langle\cdot,\cdot\rangle_{X^{*},X}$ the paring between $X^{*}$ and $X$.

For any given $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$, $\mathbb{w}\in X$, we define a functional $A_{\mathbb{h}_{2},\mathbb{w}}$ on $X$ by

[TABLE]

Using the condition $(H_{1})$ we see that $A_{\mathbb{h}_{2},\mathbb{w}}$ is a bounded and linear functional on $X$ and

[TABLE]

Thus there is an element in $X^{*}$, which is denoted by $\mathcal{A}_{\mathbb{h}_{2}}(\mathbb{w})$, such that

[TABLE]

The mapping $\mathbb{w}\mapsto\mathcal{A}_{\mathbb{h}_{2}}(\mathbb{w})$ defines an operator $\mathcal{A}_{\mathbb{h}_{2}}:X\to X^{*}$, and we have

[TABLE]

From $(H_{2})$ we know that $\mathcal{H}(x,\mathbb{z})$ is Lipschitz in $\mathbb{z}$, hence $\mathcal{A}_{\mathbb{h}_{2}}$ is continuous. On the other hand, $(H_{2})$ implies $(H_{4})$, hence

[TABLE]

Hence $\mathcal{A}_{\mathbb{h}_{2}}$ is strongly monotone from $X$ to $X^{*}$. By the Browder-Minty theorem (see for instance [Z2, p.557, Theorem 26.A]) we know that $\mathcal{A}_{\mathbb{h}_{2}}$ is surjective.

Step 2. Given any $\mathbb{u}\in L^{2}(\Omega,\mathbb{R}^{3})$ we can define a functional $L[\mathbb{u}]$ on $X$ by

[TABLE]

Then $L[\mathbb{u}]$ is a bounded and linear functional on $X$, and there is an element in $X^{*}$, which is denoted by $\mathcal{L}(\mathbb{u})$, such that

[TABLE]

Since $\mathcal{A}_{\mathbb{h}_{2}}$ is surjective and $\mathcal{L}(\mathbb{J}+\mathbb{h}_{1})\in X^{*}$, there exists $\mathbb{w}\in X$ such that $\mathcal{A}_{\mathbb{h}_{2}}(\mathbb{w})=\mathcal{L}(\mathbb{J}+\mathbb{h}_{1})$, that is,

[TABLE]

From (4.8) and since $\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)\subset\mathbb{H}_{2}(\Omega)^{\perp}_{L^{2}(\Omega)}$ we have

[TABLE]

From these and (4.9) we have

[TABLE]

Hence by the De Rham lemma on $H^{1}_{t0}(\Omega,\text{\rm div\,}0)$ (see [P4, Lemma 2.2] and [P5, Lemma 4.2]) we know that there exists $p\in L^{2,-1/2}_{0}(\Omega)$ such that, in the sense of functionals in $H^{1}_{t0}(\Omega,\text{\rm div\,}0)$ we have

[TABLE]

Using the argument in the proof of Proposition 4.1 we can show that $p=0$. Thus $\mathbb{w}\in H^{1}_{t0}(\Omega,\text{\rm div\,}0)$ is a weak solution of (4.4). By the strongly monotonicity of $\mathcal{A}_{\mathbb{h}_{2}}$ we know that for any $\mathbb{J}+\mathbb{h}_{1}$, the weak solution $\mathbb{w}$ is unique, and the map $\mathbb{J}+\mathbb{h}_{1}\mapsto\mathbb{w}$ is continuous. ∎

4.3. Solvability of (4.1) by the reduction method

A description of the reduction method can be found in [P3, P4]. Since $\mathbb{J}+\mathbb{h}_{1}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$, there exists $\mathbb{j}_{0}\in H^{1}(\Omega,\mathbb{R}^{3})$ such that

[TABLE]

such a $\mathbb{j}_{0}$ is unique. The following lemma is obvious.

Lemma 4.4**.**

Assume $\Omega,\mathbb{J},\mathbb{h}_{1},\mathbb{h}_{2}$ satisfy (4.2) and (4.3), and $\mathbb{j}_{0}$ is the solution of (4.11).

• (i)

System (4.4) has a weak solution $\mathbb{v}\in H^{1}_{t0}(\Omega,\text{\rm div\,}0)$ if and only if there exists $\phi\in H^{1}(\Omega)$ and $\mathbb{h}_{1}^{\prime}\in\mathbb{H}_{1}(\Omega)$ such that the following system has a solution $\mathbb{v}$:

[TABLE]

• (ii)

If furthermore $\mathcal{H}$ satisfies $(H_{3})$, then (4.12) is equivalent to the following

[TABLE]

Lemma 4.5**.**

Given $\mathbb{h}_{1}^{\prime}\in\mathbb{H}_{1}(\Omega)$, system (4.13) has a solution $\mathbb{v}\in H^{1}_{t0}(\Omega,\text{\rm div\,}0)$ if and only if the following equation

[TABLE]

has a solution $\phi\in H^{1}(\Omega)$ which satisfies the following orthogonality condition

[TABLE]

Proof.

Recall that (4.13) has a solution if and only if

[TABLE]

On the other hand, $\phi$ satisfies (4.14) if and only if

[TABLE]

Here we have used the equality $\nu\cdot\text{\rm curl\,}\mathcal{U}=\nu\cdot\text{\rm curl\,}\mathcal{U}_{T}=\nu\cdot\text{\rm curl\,}\mathbb{u}^{0}_{T}.$ Hence (4.16) holds if and only if $\phi$ satisfies (4.14) and

[TABLE]

Since $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)\subset\mathbb{H}_{1}(\Omega)^{\perp}_{L^{2}(\Omega)}$, the above orthogonality is equivalent to (4.15). ∎

Lemma 4.6**.**

Assume $\mathcal{H}$ satisfies $(H_{3})$ and $(H_{4})$. Then for any given $\mathbb{j}_{0}$, $\mathcal{U}$ and $\mathbb{h}_{2}$, there exists at most one $(\phi,\mathbb{h}_{1}^{\prime})\in\dot{H}^{1}(\Omega)\times\mathbb{H}_{1}(\Omega)$ such that (4.13) is solvable.

Proof.

Suppose there exist $\phi_{j}\in\dot{H}^{1}(\Omega)$ and $\mathbb{h}^{\prime}_{1,j}\in\mathbb{H}_{1}(\Omega)$, $j=1,2$, such that (4.13) with $\phi=\phi_{j}$ and $\mathbb{h}_{1}^{\prime}=\mathbb{h}_{1,j}^{\prime}$ has a solution $\mathbb{v}_{j}$. We may assume $\mathbb{v}_{j}\in\mathbb{H}_{2}(\Omega)^{\perp}_{L^{2}(\Omega)}$. Write $\mathbb{u}_{j}=\mathbb{v}_{j}+\mathcal{U}$. Then

[TABLE]

Multiplying it by $\text{\rm curl\,}(\mathbb{u}_{2}-\mathbb{u}_{1})$ and integrating, and using condition $(H_{4})$ we have

[TABLE]

In the last line we used the fact that $\nu\times\mathbb{u}_{2T}=\nu\times\mathbb{u}_{1T}=\nu\times\mathbb{u}^{0}_{T}$. So $\text{\rm curl\,}(\mathbb{u}_{2}-\mathbb{u}_{1})=\mathbb{0}$, hence $\mathbb{u}_{2}-\mathbb{u}_{1}\in\mathbb{H}_{2}(\Omega)$. This together with the fact $\mathbb{u}_{2}-\mathbb{u}_{1}=\mathbb{v}_{2}-\mathbb{v}_{1}\in\mathbb{H}_{2}(\Omega)^{\perp}_{L^{2}(\Omega)}$ implies that $\mathbb{u}_{2}-\mathbb{u}_{1}=\mathbb{0}$, so $\mathbb{v}_{2}-\mathbb{v}_{1}=\mathbb{0}$.

Finally, since $\mathbb{u}_{2}=\mathbb{u}_{1}$,

[TABLE]

Since $\mathbb{h}_{1,2}^{\prime}-\mathbb{h}_{1,1}^{\prime}\in\mathbb{H}_{1}(\Omega)$, and $\nabla(\phi_{2}-\phi_{1})\in\nabla H^{1}(\Omega)$ which is orthogonal to $\mathbb{H}_{1}(\Omega)$, we conclude that $\mathbb{h}_{1,2}^{\prime}-\mathbb{h}_{1,1}^{\prime}=\mathbb{0}$ and $\nabla(\phi_{2}-\phi_{1})=\mathbb{0}$. ∎

4.3.1. Existence of solutions of (4.14) by the monotone operator method

(4.14) is a quasilinear co-normal problem. Existence and regularity of weak solutions to such type problems have been well studied, see for instance [LU, Chapter 10], [Li, Theorem 11.20].

Proposition 4.7**.**

Assume that $\Omega,\mathcal{H},\mathbb{h}_{2},\mathbb{u}^{0}_{T}$ satisfy (4.2) and $\mathbb{j}_{0}\in H^{1}(\Omega,\mathbb{R}^{3})$.

• (i)

For any $\mathbb{h}_{1}^{\prime}\in\mathbb{H}_{1}(\Omega)$, (4.14) has a unique solution $\phi\in\dot{H}^{1}(\Omega)$.

• (ii)

There exists $\mathbb{h}_{1}^{\prime}\in\mathbb{H}_{1}(\Omega)$ such that (4.14) has a solution $\phi\in\dot{H}^{1}(\Omega)$ which satisfies (4.15).

Proof.

(i). As in the proof of Theorem 4.3, we fix $\mathbb{h}_{1}^{\prime}\in\mathbb{H}_{1}(\Omega)$ and define a map $\mathcal{T}$ such that

[TABLE]

Using condition $(B_{1})$, $(B_{2})$ we can show that $\mathcal{T}:\dot{H}^{1}(\Omega)\mapsto\dot{H}^{1}(\Omega)^{*}$ is strongly monotone. By the Browder-Minty theorem $\mathcal{T}$ is surjective, hence there exists a unique $\phi\in\dot{H}^{1}(\Omega)$ such that, for any $\eta\in\dot{H}^{1}(\Omega)$,

[TABLE]

Then $\phi$ is the weak solution of (4.14).

(ii). Choose an orthonormal basis $\{\mathbb{e}_{1},\cdots,\mathbb{e}_{N}\}$ of $\mathbb{H}_{1}(\Omega)$, and write $\xi=(\xi_{1},\cdots,\xi_{N})^{t}$, $\mathbb{h}_{1}^{\prime}=\mathbb{h}^{\prime}_{1,\xi}=\sum_{i=1}^{N}\xi_{i}\mathbb{e}_{i}$, and write the unique solution of (4.14) associated with $\mathbb{h}^{\prime}_{1,\xi}$ by $\phi=\phi_{\xi}$.

Step 2.1. We show that the map $\xi\mapsto\phi_{\xi}$ is continuous from $\mathbb{R}^{N}$ into $\dot{H}^{1}(\Omega)$. To prove, let $\xi_{0}\in\mathbb{R}^{N}$ and assume $\xi_{j}\to\xi_{0}$. Since $\mathbb{H}_{1}(\Omega)$ is of finite dimensions, and $\xi_{j}\to\xi_{0}$, so $\mathbb{v}_{j}=\mathbb{h}^{\prime}_{1,\xi_{j}}-\mathbb{h}^{\prime}_{1,\xi_{0}}\to 0$ in $H^{1}(\Omega,\mathbb{R}^{3})$. Let us denote

[TABLE]

Set $\psi_{j}=\phi_{\xi_{j}}-\phi_{\xi_{0}}$, $\mathbb{v}_{j}=\mathbb{h}^{\prime}_{1,\xi_{j}}-\mathbb{h}^{\prime}_{1,\xi_{0}}$, $\mathbb{w}_{0}=\mathbb{h}^{\prime}_{1,\xi_{0}}+\nabla\phi_{\xi_{0}}$. We have

[TABLE]

Hence $\|\nabla\psi_{j}\|_{L^{2}(\Omega)}\leq{N\over\lambda_{0}}\|\mathbb{v}_{j}\|_{L^{2}(\Omega)}\to 0.$

Step 2.2. We can further show that there exists a constant $C(\Omega)$, which depends also on the constants in $(B_{1}),(B_{2})$, such that

[TABLE]

In fact, using the definition of weak solutions to (4.14) we have

[TABLE]

Using $(B_{1})$ and $(B_{2})$ we get from the above inequality that

[TABLE]

From this and $(B_{1})$ we get (4.18).

Step 2.3. We claim that

[TABLE]

Since $\mathbb{h}^{\prime}_{1,\xi}\in\mathbb{H}^{1}(\Omega)$, so $\mathbb{h}^{\prime}_{1,\xi}$ and $\nabla\phi_{\xi}$ are orthogonal to each other with respect to the $L^{2}$ inner product. So

[TABLE]

So (4.19) is true.

Step 2.4. Using the basis $\{\mathbb{e}_{1},\cdots,\mathbb{e}_{N}\}$ of $\mathbb{H}_{1}(\Omega)$, (4.15) can be written as

[TABLE]

Let us write

[TABLE]

Then (4.20) can be written as

[TABLE]

Since the map $\xi\mapsto\phi_{\xi}$ is continuous from $\mathbb{R}^{N}$ to $\dot{H}^{1}(\Omega)$, and the map $\mathbb{w}\mapsto\mathcal{B}(x,\mathbb{w}(x))$ is continuous from $L^{2}(\Omega,\mathbb{R}^{3})$ to $L^{2}(\Omega,\mathbb{R}^{3})$, so the map $\xi\mapsto\mathcal{B}_{1}(\mathbb{h}^{\prime}_{1,\xi}+\nabla\phi_{\xi})$ is continuous from $\mathbb{R}^{N}$ to $L^{2}(\Omega,\mathbb{R}^{3})$. Thus $\mathbb{f}(\xi)$ is continuous from $\mathbb{R}^{N}$ to $\mathbb{R}^{N}$.

Step 2.5. For any $\xi\in\mathbb{R}^{N}$ we have

[TABLE]

where $\mathbb{B}_{0}=\mathcal{B}(x,\mathbb{j}_{0})$, $C_{1}=\|\mathbb{B}_{0}\|_{L^{2}(\Omega)}$ and $C_{2}=\|\nu\cdot(\mathbb{h}_{2}+\text{\rm curl\,}\mathbb{u}^{0}_{T})\|_{H^{-1/2}(\partial\Omega)}$. From this, (4.18), (4.19) and the Cauchy inequality we get

[TABLE]

Using (4.19) and (4.22) we conclude that there exists $R_{0}>0$ such that

[TABLE]

It follows that there exists $R>0$ such that the following acute angle condition is satisfied:

[TABLE]

Here we denote $S_{R}=\{\xi\in\mathbb{R}^{N}:|\xi|=R\}$ and $B_{R}=\{\xi\in\mathbb{R}^{N}:|\xi|\leq R\}$. It then follows that there exists $\xi_{0}\in B_{R}$ such that $\mathbb{f}(\xi_{0})-\mathbb{c}=\mathbb{0}$ (see Proposition 2.8 of [Z1], and Theorem 1.11 of [G]). This $\xi_{0}$ solves equation (4.21).

For this $\xi_{0}$, let $\phi_{\xi_{0}}$ be the solution of (4.14) associated with $\mathbb{h}^{\prime}_{1,\xi_{0}}$. Then $(\mathbb{h}^{\prime}_{1,\xi_{0}},\phi_{\xi_{0}})$ solves problem (4.14)-(4.15). ∎

Summarizing the above discussions we get the following

Theorem 4.8**.**

Assume (4.2) and (4.3). Then (4.1) has a weak solution $\mathbb{u}\in H^{1}(\Omega,\text{\rm div\,}0)$.

Proof.

Let $\mathbb{j}_{0}$ be determined by $\mathbb{J}$ and $\mathbb{h}_{1}$ by (4.11). By Proposition 4.7 (ii) we know that we can choose $\mathbb{h}_{1}^{\prime}$ (depending on $\mathbb{h}_{1},\mathbb{h}_{2}$) such that (4.14) has a solution $\phi=\phi_{\mathbb{h}_{1}^{\prime}}\in\dot{H}^{1}(\Omega)$ which satisfies the orthogonality condition (4.15). Then by Lemmas 4.4 and 4.5 we know that for the given $\mathbb{h}_{2}$ and for the above $\mathbb{h}_{1}^{\prime}$, (4.13) has a solution $\mathbb{v}\in H^{1}_{t0}(\Omega,\text{\rm div\,}0)$, hence (4.12) has a solution $\mathbb{u}\in H^{1}(\Omega,\text{\rm div\,}0)$, and $\mathbb{u}$ is actually a solution of (4.1). ∎

5. Tangential Curl BVP of the Maxwell System with a Given Current

In this section we consider the tangential curl BVP of the Maxwell system with a given current $\mathbb{J}$:

[TABLE]

We assume condition (4.2) and

[TABLE]

We shall look for weak solutions $\mathbb{u}\in\mathcal{H}(\Omega,\text{\rm curl\,},\text{\rm div\,}0)$. If $\mathbb{u}_{0}$ is a solution, then the general solution can be written as $\mathbb{u}=\mathbb{u}_{0}+\mathbb{h}_{1}^{\prime}+\nabla\phi$ for any $\mathbb{h}_{1}^{\prime}\in\mathbb{H}_{1}(\Omega)$ and $\phi\in H^{1}(\Omega)\cap\text{ker}(\Delta)$. To avoid non-uniqueness we may look for a solution in $H^{1}_{n0}(\Omega,\text{\rm div\,}0)\cap\mathbb{H}_{1}(\Omega)^{\perp}_{L^{2}(\Omega)}$.

Denote by $\gamma_{\tau}$ the tangential trace map $\mathbb{u}\mapsto\nu\times\mathbb{u}$. It yields a tangential trace map

[TABLE]

A necessary condition for (5.1) to have a weak solution is $\mathbb{J}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ and

[TABLE]

In fact, if (5.1) has a weak solution $\mathbb{u}\in H^{1}(\Omega,\mathbb{R}^{3})$, then $\text{\rm curl\,}\mathbb{u}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$, hence $\nu\times\mathbb{B}^{0}=\nu\times\text{\rm curl\,}\mathbb{u}=T_{1}(\text{\rm curl\,}\mathbb{u})\in T_{1}[\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)].$

Remark 5.1**.**

(i)* If $\nu\times\mathbb{B}^{0}$ has a divergence-free $L^{2}$ extension to $\Omega$, namely if there exists $\mathbb{B}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ such that $\nu\times\mathbb{B}=\nu\times\mathbb{B}^{0}$ on $\partial\Omega$, then (5.4) is satisfied.*

(ii)* If $\Omega$ is a bounded domain in $\mathbb{R}^{3}$ with a $C^{2}$ boundary and $\nu\times\mathbb{B}^{0}\in T\!H^{1/2}(\partial\Omega,\mathbb{R}^{3})$, then (5.4) is satisfied.*

Proof.

(i) If $\nu\times\mathbb{B}^{0}$ has an $L^{2}$ extension $\mathbb{B}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$, we can write $\mathbb{B}=\mathbb{B}_{1}+\mathbb{h}$ with $\mathbb{B}_{1}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ and $\mathbb{h}\in\mathbb{H}_{2}(\Omega)$. Then $\nu\times\mathbb{B}^{0}=\nu\times\mathbb{B}=\nu\times\mathbb{B}_{1}\in T_{1}[\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)].$

(ii) If $\partial\Omega\in C^{2}$ and $\nu\times\mathbb{B}^{0}\in T\!H^{1/2}(\partial\Omega,\mathbb{R}^{3})$, then $\mathbb{B}^{0}_{T}=(\nu\times\mathbb{B}^{0})\times\nu\in T\!H^{1/2}(\partial\Omega,\mathbb{R}^{3})$. So there exists a tangential component-preserving and divergence-free extension $\mathbb{B}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ such that $\mathbb{B}_{T}=\mathbb{B}^{0}_{T}$, see [P2]. Then $\nu\times\mathbb{B}=\nu\times\mathbb{B}_{T}=\nu\times\mathbb{B}^{0}_{T}=\nu\times\mathbb{B}^{0}$. Then from (i) we see that $\nu\times\mathbb{B}^{0}$ satisfies (5.4). ∎

Assume $\nu\times\mathbb{B}^{0}$ satisfies (5.4). Then there exists $\mathbb{b}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ such that $\nu\times\mathbb{b}=\nu\times\mathbb{B}^{0}$, and we can find $\mathbb{U}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)$ such that $\text{\rm curl\,}\mathbb{U}=\mathbb{b}$. Let $\mathbb{w}=\mathbb{u}-\mathbb{U}$. Then (5.1) is reduced to the following system

[TABLE]

5.1. Reduce (5.1) to a nonlinear first order div-curl system (5.6)-(5.7)

Lemma 5.2**.**

Let $\Omega,\mathbb{J},\mathbb{h}_{1},\mathbb{h}_{2}$ satisfy (4.2) and $\mathbb{b}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$.

• (i)

If $\mathbb{w}\in H^{1}(\Omega,\text{\rm div\,}0)$ is a weak solution of (5.5), and let $\mathbb{Z}=\text{\rm curl\,}\mathbb{w}$, then $\mathbb{Z}$ is a weak solution of

[TABLE]

and

[TABLE]

• (ii)

If (5.6) has a solution $\mathbb{Z}$ with

[TABLE]

then there exist $\hat{\mathbb{h}}_{2}\in\mathbb{H}_{2}(\Omega)$ and $\mathbb{w}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)$, such that $\text{\rm curl\,}\mathbb{w}=\mathbb{Z}-\hat{\mathbb{h}}_{2}$, and $\mathbb{w}$ is a weak solution of (5.5) with $\mathbb{h}_{2}$ replaced by $\mathbb{h}_{2}+\hat{\mathbb{h}}_{2}$. If furthermore $\mathbb{Z}$ satisfies (5.7), then $\hat{\mathbb{h}}_{2}=\mathbb{0}$, hence $\mathbb{w}$ is a solution of (5.5) with the given $\mathbb{h}_{2}$.

Proof.

(i) is obvious. We prove (ii). If (5.6) has a solution $\mathbb{Z}$ satisfying (5.8), then since $\text{\rm div\,}\mathbb{Z}=0$, using the second equality of (2.5) we can write $\mathbb{Z}=\mathbb{g}+\hat{\mathbb{h}}_{2}$ with $\mathbb{g}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ and $\hat{\mathbb{h}}_{2}\in\mathbb{H}_{2}(\Omega)$. Take $\mathbb{w}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)$ such that $\text{\rm curl\,}\mathbb{w}=\mathbb{g}$. Then $\mathbb{w}$ satisfies

[TABLE]

where $\mathbb{h}_{2}^{\prime}=\mathbb{h}_{2}+\hat{\mathbb{h}}_{2}\in\mathbb{H}_{2}(\Omega)$. Thus, (5.1) with the given $\mathbb{h}_{2}$ replaced by $\mathbb{h}_{2}^{\prime}$ has a solution $\mathbb{u}=\mathbb{w}+\mathbb{U}$. If $\mathbb{Z}$ satisfies (5.7), then $\hat{\mathbb{h}}_{2}=\mathbb{0}$, and $\text{\rm curl\,}\mathbb{w}=\mathbb{Z}$. Hence $\mathbb{u}=\mathbb{w}+\mathbb{U}$ is a solution of (5.5) for the given $\mathbb{h}_{2}$. ∎

Lemma 5.3**.**

Assume $\Omega,\mathbb{J},\mathbb{h}_{1},\mathbb{h}_{2}$ satisfy (4.2), $\mathcal{H}(x,\mathbb{z})$ is of $C^{1}$ and satisfies $(H_{3})$, and $\mathbb{b}\in C^{1}(\bar{\Omega},\mathbb{R}^{3})\cap\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ satisfying $\nu\times\mathbb{b}=\nu\times\mathbb{B}^{0}$.

• (i)

If $\mathbb{Z}$ is a $C^{1}$ solution of (5.6) and set $\mathbb{Y}(x)=\mathcal{H}(x,\mathbb{Z}(x)+\mathbb{b}(x)+\mathbb{h}_{2}(x))$, then $\mathbb{Y}(x)$ is a $C^{1}$ solution of

[TABLE]

• (ii)

If $\mathbb{Y}(x)$ is a $C^{1}$ solution of (5.9) and set

[TABLE]

then $\mathbb{Z}$ is a $C^{1}$ solution of (5.6). Moreover, $\mathbb{Z}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ if and only if

[TABLE]

Proof.

(i) is obvious and we prove (ii). If $\mathbb{Y}$ is a solution of (5.9) and set $\mathbb{Z}$ by (5.10), then $\mathbb{Z}$ is of $C^{1}$ and $\mathcal{H}(x,\mathbb{Z}(x)+\mathbb{b}(x)+\mathbb{h}_{2}(x))=\mathbb{Y}$. Using this and (5.9) we see that $\mathbb{Z}$ is a solution of (5.6). Since $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$, from the second equality of (2.5) we see that, $\mathbb{Z}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ if and only if (5.11) holds. ∎

5.2. Reduce (5.6) to a second order scalar equation (5.14)

Lemma 5.4**.**

Assume $\Omega,\mathbb{J},\mathbb{h}_{1},\mathbb{h}_{2}$ satisfy (4.2), $\mathcal{H}(x,\mathbb{z})$ satisfies $(H_{3})$ and $\mathbb{b}\in C^{1}(\bar{\Omega},\mathbb{R}^{3})\cap\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ satisfying $\nu\times\mathbb{b}=\nu\times\mathbb{B}^{0}$. Then we can find $\mathbb{Y}^{0}$ satisfying

[TABLE]

• (i)

If (5.9) has a solution $\mathbb{Y}\in\mathcal{H}(\Omega,\text{\rm curl\,})$, then $\mathbb{Y}$ can be written as

[TABLE]

and $\phi$ is a weak solution of the following problem

[TABLE]

• (ii)

If there exists $\mathbb{h}_{1}^{0}\in\mathbb{H}_{1}(\Omega)$ such that (5.14) has a solution $\phi\in H^{1}(\Omega)$, and set $\mathbb{Y}$ by (5.13), then $\mathbb{Y}$ is a solution of (5.9).

Proof.

We only need to prove (i). Assume (5.9) has a solution $\mathbb{Y}\in\mathcal{H}(\Omega,\text{\rm curl\,})$. Then

[TABLE]

So there exists a unique $\mathbb{Y}^{0}$ satisfying (5.12). Since $\text{\rm curl\,}(\mathbb{Y}-\mathbb{Y}^{0})=\mathbb{0}$, from the first equality in (2.5) we get (5.13). Since $\text{\rm div\,}(\mathbb{b}+\mathbb{h}_{2})=0$ in $\Omega$ and $\nu\times(\mathbb{b}+\mathbb{h}_{2})=\nu\times\mathbb{B}^{0}$ on $\partial\Omega$, from (5.9) we see that $\phi$ satisfies (5.14). ∎

5.3. Reduce (5.14) to Dirichlet problem (5.15)

We assume that $\mathcal{H}(x,\mathbb{z})$ satisfies $(H_{3})$ and the following condition:

• $(H_{3T})$

For any $x\in\partial\Omega$ and $\mathbb{y},\mathbb{z}\in\mathbb{R}^{3}$, the equality $\mathbb{z}_{T}=\mathcal{B}_{T}(x,\mathbb{y})$ holds if and only if $\mathbb{y}_{T}=\mathcal{H}_{T}(x,\mathbb{z}_{T})$.

Here we use the notation

[TABLE]

We shall show that, BVP (5.14) is equivalent to the following Dirichlet problem

[TABLE]

where $\phi_{0}$ is any function on $\partial\Omega$ satisfying

[TABLE]

For a function $\phi$ defined on $\partial\Omega$, we denote by $\nabla_{T}\phi$ the tangential gradient of $\phi$. If $\phi$ is extended over $\bar{\Omega}$, then

[TABLE]

Recall (see [NW, Lemma 2.5]) that, given a vector field $\mathbb{w}\in TC^{1}(\partial\Omega,\mathbb{R}^{3})$, a necessary and sufficient condition for existence of a function $\phi$ satisfying $\nabla_{T}\phi=\mathbb{w}$ is

[TABLE]

Proposition 5.5**.**

Assume $\Omega,\mathbb{J},\mathbb{h}_{1}$ satisfy (4.2), $\nu\times\mathbb{B}^{0}$ satisfies (5.2), $\mathcal{H}$ satisfies $(H_{3})$ and $(H_{3T})$.

• (i)

Given $\mathbb{Y}^{0}$ satisfying (5.12) and $\mathbb{h}_{1}^{0}\in\mathbb{H}_{1}(\Omega)$, a necessary condition for solvability of (5.14) is

[TABLE]

• (ii)

If $\mathbb{B}^{0}_{T}$ satisfies (5.18) and $\mathbb{h}^{0}_{1}\in\mathbb{H}_{1}(\Omega)$, then there exists a function $\phi_{0}$ on $\partial\Omega$, which depends on $\mathbb{h}^{0}_{1}$, such that (5.16) holds, and BVP (5.14) is reduced to the Dirichlet BVP (5.15).

• (iii)

In addition to the assumption in (ii), if furthermore $\mathcal{H}(x,\mathbb{z})$ satisfies $(H_{1})$ with $g_{2}\in L^{2}(\Omega)\cap L^{2}(\partial\Omega)$, then we can choose $\phi_{0}\in H^{1}(\partial\Omega)$ such that

[TABLE]

Moreover, if $\partial\Omega$ is of $C^{2,\alpha}$, $\mathbb{J}\in C^{\alpha}(\bar{\Omega},\mathbb{R}^{3})$, and if the right side of (5.16) is of $C^{1,\alpha}$, then we can choose $\phi_{0}\in C^{2,\alpha}(\partial\Omega)$ such that

[TABLE]

Proof.

(i). By $(H_{3T})$, boundary condition in (5.14) is equivalent to

[TABLE]

(5.19) implies that the vector field $\mathcal{H}_{T}(x,\mathbb{B}^{0}_{T})-\mathbb{Y}^{0}_{T}-\mathbb{h}^{0}_{1,T}$ is a tangential component of a gradient, which is possible if and only if (see (5.17))

[TABLE]

This condition is equivalent to (5.18) because

[TABLE]

(ii). Assume $\mathbb{B}^{0}_{T}$ satisfies (5.18). Then we can find $\phi_{0}\in H^{1}(\Omega)$ satisfying (5.16), such that (5.19) can be written as $(\nabla\phi)_{T}=(\nabla\phi_{0})_{T}$ on $\partial\Omega$. So we can reduce (5.14) to (5.15).

(iii) From $(H_{1})$ we have

[TABLE]

This implies that $\phi_{0}\in H^{1}(\partial\Omega)$. From (5.12) we have

[TABLE]

Using this and (2.1) we have

[TABLE]

Next assume (5.18) holds, $\partial\Omega$ is of $C^{2,\alpha}$, and the right side of (5.16) is of $C^{1,\alpha}$. Then $(\nabla\phi_{0})_{T}$ is of $C^{1,\alpha}$, so $\phi_{0}\in C^{2,\alpha}(\partial\Omega)$. Since $\mathbb{Y}^{0}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)\cap\mathbb{H}_{1}(\Omega)^{\perp}_{L^{2}(\Omega)}$ and $\text{\rm curl\,}\mathbb{Y}^{0}=\mathbb{J}+\mathbb{h}_{1}\in C^{\alpha}(\bar{\Omega},\mathbb{R}^{3})$, by the Schauder regularity of the div-curl system we have $\mathbb{Y}^{0}\in C^{1,\alpha}(\bar{\Omega},\mathbb{R}^{3})$, and

[TABLE]

Here we used the fact that $\mathbb{h}_{1},\mathbb{h}^{0}_{1}\in\mathbb{H}_{1}(\Omega)$, hence

[TABLE]

Finally from (5.19) we have

[TABLE]

∎

5.4. Solvability of (5.15)

The Dirichlet problem (5.15) has been very well studied. We list the following existence result for (5.15), see for instance [LU, Chpater 4, Theorem 8.3] and [GT, Theorem 15.11].

Lemma 5.6**.**

Assume $0<\alpha<1$, $\Omega$ is a bounded domain in $\mathbb{R}^{3}$ with a $C^{2,\alpha}$ boundary, $\mathcal{B}\in C^{1,\alpha}_{\text{\rm loc}}(\bar{\Omega}\times\mathbb{R}^{3},\mathbb{R}^{3})$ satisfies $(B_{1}),(B_{2})$, $\mathbb{Y}^{0},\mathbb{h}^{0}_{1}\in C^{1,\alpha}(\bar{\Omega},\mathbb{R}^{3})$ and $\phi_{0}\in C^{2,\alpha}(\partial\Omega)$. Then (5.15) has a unique solution $\phi\in C^{2,\alpha}(\bar{\Omega})$.

Proof.

Given $\phi_{0}\in C^{2,\alpha}(\partial\Omega)$, we can extend $\phi_{0}$ to a $C^{2,\alpha}$ function on $\bar{\Omega}$ and set $\psi=\phi-\phi_{0}$. Then $\phi$ solves (5.15) if and only if $\psi$ solves

[TABLE]

As in the proof of Proposition 4.7 (i), we define a map $\mathcal{T}$ such that

[TABLE]

Using conditions $(B_{1})$ and $(B_{2})$ we can show that $\mathcal{T}:H^{1}_{0}(\Omega)\to H^{-1}(\Omega)$ is hemi-continuous and strongly monotone, hence it is surjective. So there exists a unique $\psi\in H^{1}_{0}(\Omega)$ such that $\mathcal{T}(\psi)=0,$ Thus $\psi$ is the unique solution of (5.20) in $H^{1}_{0}(\Omega)$.

Then using the conditions on $\mathcal{B}$ and by the Schauder regularity theory of elliptic equations we see that $\psi\in C^{2,\alpha}(\bar{\Omega})$. ∎

5.5. Solvability of (5.1)

Using Lemma 5.6 we have existence for (5.1) under the following assumption

[TABLE]

Theorem 5.7**.**

Assume $\Omega,\mathcal{H},\nu\times\mathbb{B}^{0},\mathbb{J},\mathbb{h}_{1}$ satisfy (5.21), $\mathbb{Y}^{0}$ is determined by $\mathbb{J}$ and $\mathbb{h}_{1}$ by (5.12). For any $\mathbb{h}^{0}_{1}\in\mathbb{H}_{1}(\Omega)$ such that (5.18) holds, there exists an $\mathbb{h}_{2}^{0}\in\mathbb{H}_{2}(\Omega)$ which depends on $\mathbb{h}_{1}^{0}$, such that (5.1) with $\mathbb{h}_{2}=\mathbb{h}_{2}^{0}$ has a solution $\mathbb{u}\in C^{2,\alpha}(\bar{\Omega},\mathbb{R}^{3})$.

In particular, if $\Omega$ has no holes, and if there exists $\mathbb{h}^{0}_{1}\in\mathbb{H}_{1}(\Omega)$ such that (5.18) holds, then (5.1) has a solution.

Proof.

Since $\nu\times\mathbb{B}^{0}\in T\!C^{1,\alpha}(\partial\Omega,\mathbb{R}^{3})$, from Remark 5.1 (ii) we know that $\nu\times\mathbb{B}^{0}\in T_{1}[C^{1,\alpha}(\bar{\Omega},\mathbb{R}^{3})\cap\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)]$. Hence there exists $\mathbb{b}\in C^{1,\alpha}(\bar{\Omega},\mathbb{R}^{3})\cap\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ such that $\nu\times\mathbb{b}=\nu\times\mathbb{B}^{0}$ on $\partial\Omega$. Then $\mathbb{b}=\text{\rm curl\,}\mathbb{U}$ for some $\mathbb{U}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)\cap C^{2,\alpha}(\bar{\Omega},\mathbb{R}^{3})$, and we reduce (5.1) to (5.5).

Since $\partial\Omega$ is of $C^{2,\alpha}$, so $\mathbb{h}_{1},\mathbb{h}^{0}_{1}\in\mathbb{H}_{1}(\Omega)\subset C^{1,\alpha}(\bar{\Omega},\mathbb{R}^{3})\cap\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$. Since $\mathbb{J}\in C^{\alpha}(\bar{\Omega},\mathbb{R}^{3})\cap\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$, then $\mathbb{J}+\mathbb{h}_{1}\in C^{1,\alpha}(\bar{\Omega},\mathbb{R}^{3})\cap\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$. So we can find a unique $\mathbb{Y}^{0}$ satisfying (5.12), and by regularity of div-curl system we have $\mathbb{Y}^{0}\in C^{1,\alpha}(\bar{\Omega},\mathbb{R}^{3})$.

Since $\mathbb{B}^{0}_{T},\mathbb{Y}^{0},\mathbb{h}^{0}_{1}$ satisfy (5.18) and $\mathcal{H}(x,\mathbb{z})$ satisfies $(H_{3T})$, from Proposition 5.5 we can find a function $\phi_{0}$ defined on $\partial\Omega$ such that (5.16) holds. Since $\mathcal{H}$ is of $C^{1,\alpha}$, $\mathbb{B}^{0}_{T}$ and $\mathbb{h}^{0}_{1}$ are of $C^{1,\alpha}(\partial\Omega,\mathbb{R}^{3})$, so the right side of (5.16) is of $C^{1,\alpha}$, hence $\phi_{0}$ can be chosen such that $\phi_{0}\in C^{2,\alpha}(\partial\Omega)$. By Lemma 5.6 the Dirichlet problem (5.15) has a solution $\phi_{\mathbb{h}^{0}_{1}}\in C^{2,\alpha}(\bar{\Omega})$, and by Proposition 5.5, $\phi_{\mathbb{h}^{0}_{1}}$ is a solution of (5.14). Set

[TABLE]

By Lemma 5.4 $\mathbb{Y}_{\mathbb{h}_{1}^{0}}$ is a $C^{1,\alpha}$ solution of (5.9).

Now we show that, for the above given $\mathbb{h}^{0}_{1}$, we can find a unique $\mathbb{h}_{2}^{0}\in\mathbb{H}_{2}(\Omega)$ (and $\mathbb{h}_{2}^{0}$ depends on $\mathbb{h}^{0}_{1}$), such that $\mathcal{B}(x,\mathbb{Y}_{\mathbb{h}_{1}^{0}}(x))-\mathbb{b}(x)-\mathbb{h}_{2}^{0}(x)$ satisfies (5.11). To prove, let us write the orthonormal basis of $\mathbb{H}_{2}(\Omega)$ as $\{\mathbb{y}^{1},\cdots,\mathbb{y}^{m}\}$, and write the orthogonality condition (5.11) for $\mathbb{B}(x,\mathbb{Y}_{\mathbb{h}_{1}^{0}})$ as follows:

[TABLE]

Obviously, for the given $\mathbb{h}^{0}_{1}\in\mathbb{H}_{1}(\Omega)$ we can find a unique $\mathbb{h}_{2}^{0}\in\mathbb{H}_{2}(\Omega)$ such that the above equalities hold. In fact we can choose

[TABLE]

Then (5.11) holds.

For the above choice of $\mathbb{h}_{2}^{0}$, let

[TABLE]

From Lemmas 5.2, 5.3, $\mathbb{Z}=\mathbb{Z}_{\mathbb{h}_{1}^{0},\mathbb{h}_{2}}$ is a $C^{1,\alpha}$ solution of (5.6) and satisfies (5.7). Hence $\mathbb{Z}_{\mathbb{h}^{0}_{1},\mathbb{h}^{0}_{2}}=\text{\rm curl\,}\mathbb{w}$ for some $\mathbb{w}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)$. Then $\mathbb{w}$ is a weak solution of (5.5) with $\mathbb{h}_{2}=\mathbb{h}_{2}^{0}$. Since $\mathbb{Z}_{\mathbb{h}^{0}_{1},\mathbb{h}^{0}_{2}}\in C^{1,\alpha}(\bar{\Omega},\mathbb{R}^{3})$, we see that $\mathbb{w}\in C^{2,\alpha}(\bar{\Omega},\mathbb{R}^{3})$. Let $\mathbb{u}=\mathbb{U}+\mathbb{w}$. Then $\mathbb{u}\in C^{2,\alpha}(\bar{\Omega},\mathbb{R}^{3})$ is a solution of (5.1) with $\mathbb{h}_{2}=\mathbb{h}^{0}_{2}$. ∎

6. Other BVPs of the Maxwell System with a Given Current

6.1. The normal curl BVP of the Maxwell System

In this subsection we consider the following problem

[TABLE]

We assume (4.2) and

[TABLE]

It is natural to consider weak solutions of (6.1) in $\mathcal{H}(\Omega,\text{\rm curl\,},\text{\rm div\,}0)$. A necessary condition for (6.1) to have a weak solution is $\int_{\partial\Omega}B_{n}^{0}dS=0$ and $\mathbb{J}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$.

Lemma 6.1**.**

Assume $\mathbb{J},\mathbb{h}_{1}$ satisfy (4.2) and $B^{0}_{n}$ satisfies (6.2). Then (6.1) is solvable in $\mathcal{H}(\Omega,\text{\rm curl\,})$ for some $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$ if and only if the following system has a solution $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$:

[TABLE]

Proof.

If $\mathbb{u}\in\mathcal{H}(\Omega,\text{\rm curl\,})$ is a solution of (6.1) and let $\mathbb{Z}=\text{\rm curl\,}\mathbb{u}+\mathbb{h}_{2}$, then $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ is a solution of (6.3). On the other hand, if (6.3) has a solution $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$, then we can always take $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$ (which depends on $\mathbb{Z}$) such that $\mathbb{Z}-\mathbb{h}_{2}\perp_{L^{2}(\Omega)}\mathbb{H}_{2}(\Omega)$. Then $\mathbb{Z}-\mathbb{h}_{2}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$, so there exists $\mathbb{u}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)$ such that $\text{\rm curl\,}\mathbb{u}=\mathbb{Z}-\mathbb{h}_{2}$. Then $\mathbb{u}$ is a solution of (6.1) corresponding to this $\mathbb{h}_{2}$. ∎

Lemma 6.2**.**

Assume that $\mathcal{H}$ satisfies $(H_{1}),(H_{3})$, $\mathbb{J},\mathbb{h}_{1}$ satisfy (4.2) and $B^{0}_{n}$ satisfies (6.2). Let $\mathbb{Y}^{0}$ be determined by $\mathbb{J}$ and $\mathbb{h}_{1}$ by (5.12).

• (i)

If there exists $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$ such that (6.1) has a solution $\mathbb{u}\in\mathcal{H}(\Omega,\text{\rm curl\,})$, then there exists $\mathbb{h}^{0}_{1}\in\mathbb{H}_{1}(\Omega)$ (depending on $\mathbb{h}_{2}$) such that the following problem

[TABLE]

has a solution $\phi\in H^{1}(\Omega)$ which satisfies the orthogonality condition

[TABLE]

• (ii)

If there exists $\mathbb{h}^{0}_{1}\in\mathbb{H}_{1}(\Omega)$ such that (6.4) has a solution $\phi$, then there exists $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$ (depending on $\mathbb{h}_{1}^{0}$) such that $(\phi,\mathbb{h}_{2})$ satisfies (6.5), and hence (6.1) has a solution $\mathbb{u}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)$.

Proof.

$\mathbb{Z}$ is a solution of (6.3) if and only if $\mathbb{Y}(x)=\mathcal{H}(x,\mathbb{Z}(x))$ is a solution of

[TABLE]

Similar to the proof of Lemma 5.4 we can show that (6.6) is equivalent to (6.4). ∎

Theorem 6.3**.**

Assume $\Omega,\mathcal{H}(x,\mathbb{z}),\mathbb{J},\mathbb{h}_{1}$ satisfy (4.2) and $B^{0}_{n}$ satisfies (6.2). For any $\mathbb{h}_{1}^{0}\in\mathbb{H}_{1}(\Omega)$, there exists $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$ (depending $\mathbb{h}_{1},\mathbb{h}_{1}^{0}$) such that (6.1) has a solution.

Proof.

Let $\mathbb{Y}^{0}$ be determined by $\mathbb{J}$ and $\mathbb{h}_{1}$ by (5.12), and let $\mathbb{h}_{1}^{0}\in\mathbb{H}_{1}(\Omega)$. As in the proof of Proposition 4.7 (i), we define a map $\mathcal{T}$ by

[TABLE]

Conditions $(B_{1})$, $(B_{2})$ imply that $\mathcal{T}$ is hemi-continuous and strongly monotone from $\dot{H}^{1}(\Omega)$ to $\dot{H}^{1}(\Omega)^{*}$, hence $\mathcal{T}$ is surjective. So there exists a unique $\phi\in\dot{H}^{1}(\Omega)$ such that

[TABLE]

$\phi$ is the unique weak solution of (6.4) in $\dot{H}^{1}(\Omega)$. Note that $\phi$ depends on $\mathbb{h}_{1},\mathbb{h}_{1}^{0}$. By choosing $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$ depending on $\mathbb{h}_{1},\mathbb{h}_{1}^{0}$ such that (6.5) holds, and using Lemma 6.2 (ii), we conclude that Eq. (6.1) with this choice of $\mathbb{h}_{2}$ has a weak solution $\mathbb{u}$. ∎

6.2. The natural BVP of the Maxwell System

In this section we consider the natural BVP of the Maxwell system with a given current222Let us mention that in [MP1], the electro-static problem is reduced to a problem of the form (6.7).:

[TABLE]

We assume (4.2) and

[TABLE]

Let us denote by $\pi_{\tau}$ the tangential trace map $\mathbb{w}\mapsto\mathbb{w}_{T}$ from $\mathcal{H}(\Omega,\text{\rm curl\,})$ to $H^{-1/2}(\partial\Omega,\mathbb{R}^{3})$, where $\mathbb{w}_{T}=(\nu\times\mathbb{w})\times\nu$. In particular $\mathbb{H}^{0}_{T}=(\nu\times\mathbb{H}^{0})\times\nu$.

Definition 6.4**.**

Assume $\Omega,\mathbb{J},\mathbb{h}_{1},\mathbb{h}_{2}$ satisfy (4.2) and $\nu\times\mathbb{H}^{0}$ satisfies (6.8). $\mathbb{u}$ is called a weak solution of (6.7) if $\mathbb{u}\in\mathcal{H}(\Omega,\text{\rm curl\,},\text{\rm div\,}0)$, $\mathcal{H}(x,\text{\rm curl\,}\mathbb{u}(x)+\mathbb{h}_{2}(x))\in L^{2,-1/2}_{t}(\Omega,\mathbb{R}^{3})$, and for any $\mathbb{w}\in\mathcal{H}(\Omega,\text{\rm curl\,})$ it holds that

[TABLE]

$\mathbb{u}$* is called an $H^{1}$-weak solution of (6.7) if $\mathbb{u}$ is a weak solution and $\mathbb{u}\in H^{1}(\Omega,\mathbb{R}^{3})$.*

If $\mathbb{u}$ is a solution of (6.7), then for any $\mathbb{v}\in\mathcal{H}(\Omega,\text{\rm curl\,}0,\text{\rm div\,}0)$, $\mathbb{u}+\mathbb{v}$ is also a weak solution of (6.7). Therefore solvability of (6.1) in $\mathcal{H}(\Omega,\text{\rm curl\,},\text{\rm div\,}0)$ is equivalent to solvability of $H^{1}$-weak solutions in $H^{1}_{n0}(\Omega,\text{\rm div\,}0)\cap\mathbb{H}_{1}(\Omega)^{\perp}_{L^{2}(\Omega)}$.

Lemma 6.5**.**

Assume $\Omega,\mathbb{J},\mathbb{h}_{1},\mathbb{h}_{2}$ satisfy (4.2) and $\nu\times\mathbb{H}^{0}$ satisfies (6.8). If (6.7) has a weak solution $\mathbb{u}\in\mathcal{H}(\Omega,\text{\rm curl\,},\text{\rm div\,}0)$, then

[TABLE]

Proof.

Assume (6.7) has a weak solution $\mathbb{u}\in H^{1}(\Omega,\mathbb{R}^{3})$. Then

[TABLE]

Taking $\mathbb{w}=\mathbb{h}\in\mathbb{H}_{1}(\Omega)$ in (6.9) we get the last equality in (6.10). ∎

Remark 6.6**.**

If $\mathbb{J}\in L^{2}(\Omega,\mathbb{R}^{3})$, $\nu\times\mathbb{H}^{0}\in T\!H^{1/2}(\partial\Omega,\mathbb{R}^{3})$ and they satisfy the first equality in (6.10), then

[TABLE]

Proof.

Let $\mathbb{H}^{0}$ be a divergence-free and tangential component preserving extension of $\mathbb{H}^{0}_{T}$ on to $\Omega$. From the first equality in (6.10), for any $\phi\in H^{1}(\Omega)$ we have

[TABLE]

∎

We have the following two existence results.

Proposition 6.7**.**

Assume $\Omega,\mathbb{J},\mathbb{h}_{1},\mathbb{h}_{2},\nu\times\mathbb{H}^{0}$ satisfy (4.2), (6.8) and (6.10), and $\mathcal{H}(x,\mathbb{z})=\nabla_{\mathbb{z}}P(x,\mathbb{z})$ for a function $P$ satisfying $(P_{1})$. Then (6.7) has an $H^{1}$-weak solution $\mathbb{u}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)\cap\mathbb{H}_{1}(\Omega)^{\perp}_{L^{2}(\Omega)}$.

Proof.

Set $P_{2}(x,\mathbb{z})=P(x,\mathbb{z}+\mathbb{h}_{2}(x))$. Then $\nabla_{\mathbb{z}}P_{2}(x,\mathbb{z})=\mathcal{H}(x,\mathbb{z}+\mathbb{h}_{2}(x))$, and $P_{2}(x,\mathbb{z})$ also satisfies the condition $(P_{1})$ (with different constants and functions $g_{1},g_{2}$). Define

[TABLE]

and

[TABLE]

By the div-curl-gradient inequalities we have

[TABLE]

As in the proof of Proposition 3.1 in [P4] we can show that $\mathcal{E}$ is coercive in $Y$, and $\mathcal{E}$ has a minimizer $\mathbb{w}_{0}\in Y$. From (6.10) and (6.11) we have

[TABLE]

Hence $\mathcal{E}[\mathbb{w}+\mathbb{h}+\nabla\phi]=\mathcal{E}[\mathbb{w}]$ for all $\mathbb{w}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)$, $\mathbb{h}\in\mathbb{H}_{1}(\Omega)$ and $\phi\in H^{1}(\Omega)$. Recall that $\mathcal{H}(\Omega,\text{\rm curl\,})=H^{1}_{n0}(\Omega,\text{\rm div\,}0)\oplus\text{\rm grad\,}H^{1}(\Omega).$ Therefore the minimizer $\mathbb{w}_{0}$ of $\mathcal{E}$ on $Y$ is also a minimizer of $\mathcal{E}$ on $\mathcal{H}(\Omega,\text{\rm curl\,})$. ∎

Theorem 6.8**.**

Assume $\Omega,\mathbb{J},\mathbb{h}_{1},\mathbb{h}_{2},\nu\times\mathbb{H}^{0}$ satisfy (4.2), (6.8) and (6.10), and $\mathcal{H}$ satisfies $(H_{1}),(H_{2})$. Then (6.7) has an $H^{1}$-weak solution $\mathbb{u}$, and $\mathbb{u}$ is unique in $H^{1}_{n0}(\Omega,\text{\rm div\,}0)\cap\mathbb{H}_{1}(\Omega)^{\perp}_{L^{2}(\Omega)}$. The solution map $(\mathbb{h}_{1},\mathbb{J})\mapsto\mathbb{u}$ is continuous from $\mathbb{H}_{1}(\Omega)\times\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ to $H^{1}_{n0}(\Omega,\text{\rm div\,}0)$.

Proof.

The proof is similar to the proof of Theorem 4.3. Define $Y$ as in (6.12). For any given $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$, we can define a map $\mathcal{A}_{\mathbb{h}_{2}}:Y\to\mathbb{Y}^{*}$ such that

[TABLE]

We can show that $\mathcal{A}_{\mathbb{h}_{2}}:Y\to Y^{*}$ is hemi-continuous and strongly monotone, hence surjective. We can also define an operator $\mathcal{L}:L^{2}(\Omega,\mathbb{R}^{3})\to Y^{*}$, such that

[TABLE]

Since $\mathcal{A}_{\mathbb{h}_{2}}$ is surjective, there exists $\mathbb{u}\in Y$ such that $\mathcal{A}_{\mathbb{h}_{2}}(\mathbb{u})=\mathcal{L}(\mathbb{J}+\mathbb{h}_{1}),$ that is,

[TABLE]

For all $\mathbb{h}\in\mathbb{H}_{1}(\Omega)$ and $\phi\in H^{1}(\Omega)$ we have

[TABLE]

On the other hand, from (6.10) we have

[TABLE]

From these and (6.13) we see that (6.9) holds for all $\mathbb{w}=\mathbb{v}+\mathbb{h}+\nabla\phi$, where $\mathbb{v}\in Y$, $\mathbb{h}\in\mathbb{H}_{1}(\Omega)$ and $\phi\in H^{1}(\Omega)$. So (6.9) holds for all $\mathbb{w}\in\mathcal{H}(\Omega,\text{\rm curl\,})$. Hence $\mathbb{u}\in Y$ is a weak solution of (6.7). By the strongly monotonicity of $\mathcal{A}_{\mathbb{h}_{2}}$ we know that the weak solution $\mathbb{u}$ is unique in $Y$, and the map $\mathbb{J}+\mathbb{h}_{1}\mapsto\mathbb{u}$ is continuous from $Y^{*}$ to $Y$. ∎

6.3. The co-normal BVP of the Maxwell System

In this section we consider the following

[TABLE]

Definition 6.9**.**

Assume $\Omega,\mathbb{J},\mathbb{h}_{1},\mathbb{h}_{2}$ satisfy (4.2) and $H^{0}_{n}\in H^{-1/2}(\partial\Omega)$. We say $\mathbb{u}$ is a weak solution of (6.14) if

• (i)

$\mathbb{u}\in\mathcal{H}(\Omega,\text{\rm curl\,},\text{\rm div\,}0)$, $\mathcal{H}(x,\text{\rm curl\,}\mathbb{u}(x)+\mathbb{h}_{2}(x))\in L^{2,-1/2}_{\nu}(\Omega,\mathbb{R}^{3})$;

• (ii)

the equality $\nu\cdot\mathcal{H}(x,\text{\rm curl\,}\mathbb{u}(x)+\mathbb{h}_{2}(x))=H^{0}_{n}$ holds in the $H^{-1/2}(\partial\Omega)$ sense;

• (iii)

for all $\mathbb{v}\in H^{1}_{t0}(\Omega,\mathbb{R}^{3})$ it holds that

[TABLE]

To prove existence of solutions to (6.14), in addition to (4.2), we assume the following conditions:

• $(H_{3N})$

For any $x\in\partial\Omega$ and $\mathbb{z},\mathbb{w}\in\mathbb{R}^{3}$, the equality $\nu\cdot\mathbb{z}=\nu\cdot\mathcal{B}(x,\mathbb{w})$ holds if and only if $\nu\cdot\mathbb{w}=\nu\cdot\mathcal{H}(x,\mathbb{z})$.

• ($H^{0}$)

$H^{0}_{n}\in H^{1/2}(\partial\Omega)$, and there exists $\mathbb{H}^{0}\in H^{1/2}(\partial\Omega,\mathbb{R}^{3})$ such that $\nu\cdot\mathbb{H}^{0}=H^{0}_{n}$.

Lemma 6.10**.**

Assume that $\Omega,\mathbb{J},\mathbb{h}_{1}$ satisfy (4.2), $\mathcal{H}$ satisfies $(H_{1}),(H_{3}),(H_{3N})$, $H^{0}_{n}$ satisfies $(H^{0}$). Let $\mathbb{Y}^{0}$ be determined by $\mathbb{J}$ and $\mathbb{h}_{1}$ by (5.12).

• (i)

Given $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$, if (6.14) has a weak solution $\mathbb{u}\in\mathcal{H}(\Omega,\text{\rm curl\,})$, then there exists $\mathbb{h}^{0}_{1}\in\mathbb{H}_{1}(\Omega)$ (depending on $\mathbb{h}_{1},\mathbb{h}_{2}$) such that the following problem

[TABLE]

has a weak solution $\phi$, and $\phi$ satisfies orthogonality condition (6.5).

• (ii)

If there exists $\mathbb{h}^{0}_{1}\in\mathbb{H}_{1}(\Omega)$ such that (6.15) has a weak solution $\phi\in H^{1}(\Omega)$, and let $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$ (depending on $\mathbb{h}_{1},\mathbb{h}_{1}^{0}$) be such that (6.5) holds, then (6.14) has a weak solution $\mathbb{u}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)$.

Proof.

Step 1. If $\mathbb{u}\in\mathcal{H}(\Omega,\text{\rm curl\,})$ is a weak solution of (6.14), let $\mathbb{Z}=\text{\rm curl\,}\mathbb{u}$. From the first equation in (6.14) we see that there exist $\mathbb{h}^{0}_{1}\in\mathbb{H}_{1}(\Omega)$ and $\phi\in H^{1}(\Omega)$ such that $\mathcal{H}(x,\mathbb{Z}+\mathbb{h}_{2})=\mathbb{Y}^{0}+\mathbb{h}^{0}_{1}+\nabla\phi$. By $(H_{3})$ we have $\mathbb{Z}=\mathcal{B}(x,\mathbb{Y}^{0}+\mathbb{h}^{0}_{1}+\nabla\phi)-\mathbb{h}_{2},$ and from the second and third equalities in (6.14) we have

[TABLE]

Since $\mathbb{Z}\in\text{\rm curl\,}\mathcal{H}(\Omega,\text{\rm curl\,},0)=\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$, it satisfies the orthogonality condition (6.5). From $(H_{3N})$ and $(H^{0})$ we see that the boundary condition in (6.16) is equivalent to the boundary condition in (6.15).

Step 2. Assume $(\mathbb{h}^{0}_{1},\phi)$ solves (6.15) and there exists $\mathbb{h}_{2}$ such that (6.5) holds. Then

[TABLE]

Hence there exists $\mathbb{u}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)$ such that $\text{\rm curl\,}\mathbb{u}=\mathbb{Z}$. Then

[TABLE]

By $(H_{3})$, $(H_{3N})$, $(H^{0})$ and (6.15) we see that $\mathbb{u}$ is a weak solution of (6.14) for this $\mathbb{h}_{2}$. ∎

Theorem 6.11**.**

Assume $\Omega,\mathbb{J},\mathbb{h}_{1},\mathcal{H}(x,\mathbb{z})$ satisfy (4.2), $(H_{3N})$, and $H^{0}_{n}$ satisfies $(H^{0})$. For any $\mathbb{h}_{1}^{0}\in\mathbb{H}_{1}(\Omega)$, there exists $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$ (depending $\mathbb{h}_{1},\mathbb{h}_{1}^{0}$) such that (6.14) has a weak solution $\mathbb{u}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)$.

Proof.

Given any $\mathbb{h}_{1}$, let $\mathbb{Y}^{0}$ be determined by $\mathbb{J}$ and $\mathbb{h}_{1}$ by (5.12). Then $\mathbb{Y}^{0}\in H^{1}(\Omega,\mathbb{R}^{3})$. By using the monotone operator method (see the proofs of Proposition 4.7 (i), Lemma 5.6), we can show that for any $\mathbb{h}_{1}^{0}\in\mathbb{H}_{1}(\Omega)$, (6.15) has a unique solution $\phi\in\dot{H}^{1}(\Omega)$ (and $\phi$ depends on $\mathbb{h}_{1},\mathbb{h}_{1}^{0}$). We choose $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$ which depends on $\mathbb{h}_{1},\mathbb{h}_{1}^{0}$ such that (6.5) holds. Then from Lemma 6.10, for this choice of $\mathbb{h}_{2}$, (6.14) has a weak solution $\mathbb{u}$. ∎

7. The Maxwell-Stokes System with Solution-dependent Current

In this section we consider the Maxwell-Stokes system where the current is of the form $\mathbb{f}(x,\text{\rm curl\,}\mathbb{u}+\mathbb{h}_{2})$:

[TABLE]

Throughout this section we assume that

[TABLE]

It was observed in [P4] that, if (7.1) has a solution $\mathbb{u}$, then $\mathbb{f}(x,\text{\rm curl\,}\mathbb{u}+\mathbb{h}_{2})+\mathbb{h}_{1}+\nabla p\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0),$ which holds if and only if

[TABLE]

It suggests $p$ satisfies Neumann type boundary condition. In this section, in addition to a boundary condition for $\mathbb{u}$.333The case where $\mathbb{u}$ satisfies a Dirichlet boundary condition $\mathbb{u}_{T}=\mathbb{u}^{0}_{T}$ can be treated by the methods in [P4], hence will not be discussed here. We also mention that most of analyzes in this section are valid for the nonlinear term of the form $\mathbb{f}(x,\mathbb{u}+\mathbb{h}_{2})$. we assume $p$ satisfies the following boundary condition

[TABLE]

7.1. The normal curl BVP

In this subsection we consider the following problem

[TABLE]

We assume (7.2) and $B^{0}_{n}$ satisfies (6.2).

Definition 7.1**.**

We say that $(\mathbb{u},p)$ is a weak solution of (7.3) if $\mathbb{u}\in\mathcal{H}(\Omega,\text{\rm curl\,},\text{\rm div\,}0)$ and $p\in H^{1}(\Omega)$ such that

• (i)

$\mathcal{H}(x,\text{\rm curl\,}\mathbb{u}(x)+\mathbb{h}_{2}(x))\in L^{2}(\Omega,\mathbb{R}^{3})$, $\mathbb{f}(x,\text{\rm curl\,}\mathbb{u}(x)+\mathbb{h}_{2}(x))\in L^{2,-1/2}(\Omega,\mathbb{R}^{3})$;

• (ii)

the equality $\nu\cdot\text{\rm curl\,}\mathbb{u}(x)+\nu\cdot\mathbb{h}_{2}(x)=B^{0}_{n}$ holds in $H^{-1/2}(\partial\Omega)$;

• (iii)

[TABLE]

[TABLE]

We shall show the relation between (7.3) with the $Z$-system

[TABLE]

and we shall show the equivalence of $Z$-system (7.6) and the $Y$-system

[TABLE]

Here $\mathcal{P}_{\nu}$ is the projection operator onto $\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$, see section 2.4.

Definition 7.2**.**

(i)* We say $\mathbb{Z}$ is a weak solution of (7.6) if $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ such that*

[TABLE]

(ii)* We say $\mathbb{Y}$ is a weak solution of (7.7) if $\mathbb{Y}\in L^{2}(\Omega,\mathbb{R}^{3})$ such that*

[TABLE]

Lemma 7.3**.**

Assume (7.2) and (6.2).

• (i)

If $\mathbb{Y}\in L^{2}(\Omega,\mathbb{R}^{3})$ is a weak solution of (7.7), then $\mathbb{Y}\in\mathcal{H}(\Omega,\text{\rm curl\,})$, and the first equality in (7.7) holds a.e. in $\Omega$.

• (ii)

If $(\mathbb{u},p)\in\mathcal{H}(\Omega,\text{\rm curl\,},\text{\rm div\,}0)\times H^{1}(\Omega)$ is a weak solution of (7.3) and letting $\mathbb{Z}=\text{\rm curl\,}\mathbb{u}$, then $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ and it is a weak solution of (7.6). On the other hand, if $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ is a weak solution of (7.6), then there exists $\mathbb{h}_{2}^{0}\in\mathbb{H}_{2}(\Omega)$ such that (7.3) with $\mathbb{h}_{2}$ replaced by $\mathbb{h}_{2}+\mathbb{h}_{2}^{0}$ has a weak solution $(\mathbb{u},p)\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)\times H^{1}(\Omega)$, and $\text{\rm curl\,}\mathbb{u}=\mathbb{Z}-\mathbb{h}_{2}^{0}$. If furthermore $\mathbb{Z}\perp_{L^{2}(\Omega)}\mathbb{H}_{2}(\Omega)$, then $\mathbb{h}^{0}_{2}=\mathbb{0}$.

• (iii)

If $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ is a weak solution of (7.6) and letting $\mathbb{Y}=\mathcal{H}(x,\mathbb{Z}(x)+\mathbb{h}_{2}(x)),$ then $\mathbb{Y}\in\mathcal{H}(\Omega,\text{\rm curl\,})$ and it is a weak solution of (7.7). On the other hand, if $\mathbb{Y}\in L^{2}(\Omega,\mathbb{R}^{3})$ is a weak solution of (7.7) and letting $\mathbb{Z}=\mathcal{B}(x,\mathbb{Y}(x))-\mathbb{h}_{2}(x),$ then $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ and it is a weak solution of (7.6).

Proof.

We only prove (i). Assume $\mathbb{Y}\in L^{2}(\Omega,\mathbb{R}^{3})$ is a weak solution of (7.7). By Definition 7.2 (ii) and by $(f_{1})$ we see that, for all $\mathbb{v}\in H^{1}_{t0}(\Omega,\mathbb{R}^{3})$ we have

[TABLE]

So $\text{\rm curl\,}\mathbb{Y}$ exists as a functional on $H^{1}_{t0}(\Omega,\mathbb{R}^{3})$. For any $\mathbb{w}\in\mathcal{D}(\Omega,\mathbb{R}^{3})$ we have

[TABLE]

Hence $\text{\rm curl\,}\mathbb{Y}\in L^{2}(\Omega,\mathbb{R}^{3})$, and $\text{\rm curl\,}\mathbb{Y}=\mathcal{P}_{\nu}[\mathbb{f}(x,\mathcal{B}(x,\mathbb{Y}))]+\mathbb{h}_{1}.$ ∎

To solve (7.7), we use (2.6) to write $\mathbb{Y}=\mathbb{w}+\hat{\mathbb{h}}_{1}+\nabla\psi$, with $\mathbb{w}\in\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$, $\hat{\mathbb{h}}_{1}\in\mathbb{H}_{1}(\Omega)$, $\psi\in H^{1}(\Omega)$. If $\mathbb{Y}$ satisfies (7.7) then $\psi$ satisfies

[TABLE]

Proposition 7.4**.**

Assume $\Omega,\mathcal{H}(x,\mathbb{z}),B_{n}^{0}$ satisfy (7.2) and (6.2), $\mathbb{w}\in\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$ and $\hat{\mathbb{h}}_{1}\in\mathbb{H}_{1}(\Omega)$. Then (7.8) has a unique solution $\psi=\psi_{\mathbb{w},\hat{\mathbb{h}}_{1}}\in\dot{H}^{1}(\Omega)$. The map $(\mathbb{w},\hat{\mathbb{h}}_{1})\mapsto\nabla\psi_{\mathbb{w},\hat{\mathbb{h}}_{1}}$ is a locally Lipschitz and bounded map from $\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)\times\mathbb{H}_{1}(\Omega)$ into $\text{grad}H^{1}(\Omega)$.

Proof.

As in the proof of Proposition 4.7 (i), we can define a map $\mathcal{T}$ such that

[TABLE]

Using $(B_{1})$ and $(B_{2})$ we can show $\mathcal{T}:\dot{H}^{1}(\Omega)\mapsto\dot{H}^{1}(\Omega)^{*}$ is Lipschitz and strongly monotone, hence $\mathcal{T}$ is surjective. Hence there exists a unique $\psi\in\dot{H}^{1}(\Omega)$ such that

[TABLE]

From this and $(B_{1})$ we have

[TABLE]

here we have used the Poincaré inequality for $\psi_{\mathbb{w},\hat{\mathbb{h}}_{1}}\in\dot{H}^{1}(\Omega)$. Hence

[TABLE]

where $C$ depends only on the constants and $g_{3},g_{4}$ in $(B_{1})$. Thus $(\mathbb{w},\hat{\mathbb{h}}_{1})\mapsto\nabla\psi_{\mathbb{w},\hat{\mathbb{h}}_{1}}$ is a bounded map with respect to the $L^{2}$ norms.

Using $(B_{2})$ and Remark 2.3 we have

[TABLE]

From this and using $(B_{1})$ we have

[TABLE]

Here we have used (7.9). It follows that

[TABLE]

where the constant $C$ is linearly increases in $\|\mathbb{w}+\hat{\mathbb{h}}_{1}\|_{L^{2}(\Omega)}+\|\mathbb{w}^{\prime}+\hat{\mathbb{h}}_{1}^{\prime}\|_{L^{2}(\Omega)}$. It follows that the map $(\mathbb{w},\hat{\mathbb{h}}_{1})\mapsto\nabla\psi_{\mathbb{w},\hat{\mathbb{h}}_{1}}$ is locally Lipschitz in $L^{2}(\Omega,\mathbb{R}^{3})$. ∎

For any $\mathbb{w}\in\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$, $\hat{\mathbb{h}}_{1}\in\mathbb{H}_{1}(\Omega)$, we denote

[TABLE]

We look for $\mathbb{v}\in\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$ such that

[TABLE]

Lemma 7.5**.**

Assume (7.2) and (6.2). Let $\hat{\mathbb{h}}_{1}\in\mathbb{H}_{1}(\Omega)$.

• (i)

For any $\mathbb{w}\in\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$ and (7.10) has a unique solution $\mathbb{v}=\mathbb{V}_{\hat{\mathbb{h}}_{1}}[\mathbb{w}]\in H^{1}(\Omega,\mathbb{R}^{3})\cap\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$.

[TABLE]

where $C$ depends on $\Omega,\mathcal{H}(x,\mathbb{z})$ and $\mathbb{f}(x,\mathbb{z})$. The map $\mathbb{V}_{\hat{\mathbb{h}}_{1}}$ is a compact map in $\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$.

• (ii)

If in addition $\mathbb{f}(x,\mathbb{z})$ satisfies

[TABLE]

then $\mathbb{V}_{\hat{\mathbb{h}}_{1}}$ has a fixed point $\mathbb{w}_{\hat{\mathbb{h}}_{1}}$.

Proof.

By the definition of $\mathcal{P}_{\nu}$, we know that the right side of the first equation of (7.10) lies in $\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$. Hence (7.10) has a unique solution $\mathbb{v}\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)\cap\mathbb{H}_{1}(\Omega)^{\perp}_{L^{2}(\Omega)}=H_{1}(\Omega,\mathbb{R}^{3})\cap\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$. We have

[TABLE]

Using $(B_{1})$, (7.12), (7.9), and Lemma 2.7, we get (7.11). Hence $\mathbb{V}_{\hat{\mathbb{h}}_{1}}$ is continuous and bounded from $\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$ into $H^{1}(\Omega,\mathbb{R}^{3})\cap\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$. Then by the Sobolev imbedding theorem we know that $\mathbb{V}_{\hat{\mathbb{h}}_{1}}$ is compact from $\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$ into itself.

If $\mathbb{f}(x,\mathbb{z})$ satisfies (7.12), then there exists $R>0$ such that $\mathbb{V}_{\hat{\mathbb{h}}_{1}}$ maps the closed ball $B_{R}$ with center $\mathbb{0}$ and radius $R$ in $\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$ into $B_{R}$ itself. By Schauder’s fixed point theorem we conclude that $\mathbb{V}_{\hat{\mathbb{h}}_{1}}$ has a fixed point in $B_{R}$. ∎

Theorem 7.6**.**

Assume (7.2), (6.2) and (7.12). Then for any $\hat{\mathbb{h}}_{1}\in\mathbb{H}_{1}(\Omega)$ there exists $\mathbb{h}_{2}=\mathbb{h}_{2}(\mathbb{h}_{1},\hat{\mathbb{h}}_{1})$ such that (7.3) with $\mathbb{h}_{2}=\mathbb{h}_{2}(\mathbb{h}_{1},\hat{\mathbb{h}}_{1})$ has a weak solution $(\mathbb{u},p)\in H^{1}(\Omega,\mathbb{R}^{3})\times H^{1}(\Omega)$.

Proof.

Let $\mathbb{w}_{\hat{\mathbb{h}}_{1}}$ denote the fixed point of the map $\mathbb{V}_{\hat{\mathbb{h}}_{1}}$ in $\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$, and let

[TABLE]

$\mathbb{w}_{\hat{\mathbb{h}}_{1}}$ satisfies

[TABLE]

Then $\mathbb{Y}_{\hat{\mathbb{h}}_{1}}$ is a solution of (7.7). By Lemma 7.3 (iii), $\mathbb{Z}_{\hat{\mathbb{h}}_{1}}$ is a solution of (7.6). Then there exists $\mathbb{h}_{2}=\mathbb{h}_{2}(\mathbb{h}_{1},\hat{\mathbb{h}}_{1})$ depending on $\mathbb{h}_{1}$ and $\hat{\mathbb{h}}_{1}$, such that $\mathbb{Z}_{\hat{\mathbb{h}}_{1}}\perp_{L^{2}(\Omega)}\mathbb{H}_{2}(\Omega)$, namely

[TABLE]

Then $\mathbb{Z}_{\hat{\mathbb{h}}_{1}}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$. By Lemma 7.3 (ii), there exists $(\mathbb{u},p)\in H_{n0}^{1}(\Omega,\text{\rm div\,}0)\times H^{1}(\Omega)$ which solves (7.3) for the given $\mathbb{h}_{1}$. ∎

7.2. The natural-Neumann BVP

In this section we consider the natural BVP of the Maxwell-Stokes system

[TABLE]

We assume (7.2) and

[TABLE]

where $\mathbb{H}^{0}_{T}=(\nu\times\mathbb{H}^{0})\times\nu$.

Definition 7.7**.**

We say that $(\mathbb{u},p)$ is a weak solution of (7.15) if $\mathbb{u}\in\mathcal{H}(\Omega,\text{\rm curl\,},\text{\rm div\,}0)$, $p\in H^{1}(\Omega)$, such that

• (i)

$\mathcal{H}(x,\text{\rm curl\,}\mathbb{u}(x)+\mathbb{h}_{2}(x))\in L^{2,-1/2}_{t}(\Omega,\mathbb{R}^{3})$, $\mathbb{f}(x,\text{\rm curl\,}\mathbb{u}(x)+\mathbb{h}_{2}(x))\in L^{2}(\Omega,\mathbb{R}^{3})$;

• (ii)

[TABLE]

[TABLE]

Regarding (7.15) we have the following $Z$-system

[TABLE]

and $Y$-system

[TABLE]

where $\mathcal{P}_{n}$ is the Neumann projection associated with $\nu\cdot\text{\rm curl\,}\mathbb{H}^{0}_{T}$, see section 2.4.

Definition 7.8**.**

(i)* We say $\mathbb{Z}$ is a weak solution of (7.19) if $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ such that*

[TABLE]

and for all $\mathbb{v}\in H^{1}(\Omega,\mathbb{R}^{3})$ it holds that

[TABLE]

(ii)* We say $\mathbb{Y}$ is a weak solution of (7.20) if $\mathbb{Y}\in L^{2,-1/2}_{t}(\Omega,\mathbb{R}^{3})$ such that*

[TABLE]

and for all $\mathbb{v}\in H^{1}(\Omega,\mathbb{R}^{3})$ it holds that

[TABLE]

Similar to Lemma 7.3 we have the following

Lemma 7.9**.**

Assume (7.2) and (7.16).

• (i)

If $\mathbb{Y}\in L^{2,-1/2}_{t}(\Omega,\mathbb{R}^{3})$ is a weak solution of (7.20), then $\mathbb{Y}\in\mathcal{H}(\Omega,\text{\rm curl\,})$, the first equality in (7.20) holds a.e. in $\Omega$, and the equality $\nu\times\mathbb{Y}=\nu\times\mathbb{H}^{0}$ holds in the sense of trace in $H^{-1/2}(\partial\Omega,\mathbb{R}^{3})$.

• (ii)

If $(\mathbb{u},p)\in\mathcal{H}(\Omega,\text{\rm curl\,},\text{\rm div\,}0)\times H^{1}(\Omega)$ is a weak solution of (7.15) and letting $\mathbb{Z}=\text{\rm curl\,}\mathbb{u}$, then $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ and it is a weak solution of (7.19). On the other hand, if $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ is a weak solution of (7.19), then there exists $\mathbb{h}_{2}^{0}\in\mathbb{H}_{2}(\Omega)$ such that (7.15) with $\mathbb{h}_{2}$ replaced by $\mathbb{h}_{2}+\mathbb{h}_{2}^{0}$ has a weak solution $(\mathbb{u},p)\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)\times H^{1}(\Omega)$, and $\text{\rm curl\,}\mathbb{u}=\mathbb{Z}-\mathbb{h}_{2}^{0}$. If furthermore $\mathbb{Z}\perp_{L^{2}(\Omega)}\mathbb{H}_{2}(\Omega)$, then $\mathbb{h}^{0}_{2}=\mathbb{0}$.

• (iii)

If $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ is a weak solution of (7.19) and letting $\mathbb{Y}=\mathcal{H}(x,\mathbb{Z}(x)+\mathbb{h}_{2}(x)),$ then $\mathbb{Y}\in\mathcal{H}(\Omega,\text{\rm curl\,})$ and it is a weak solution of (7.20). On the other hand, if $\mathbb{Y}\in L^{2,-1/2}_{t}(\Omega,\mathbb{R}^{3})$ is a weak solution of (7.20) and letting $\mathbb{Z}=\mathcal{B}(x,\mathbb{Y}(x))-\mathbb{h}_{2}(x),$ then $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ and it is a weak solution of (7.19).

Theorem 7.10**.**

Assume (7.2), (7.16) and (7.12). Then for any given $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$, there exists $\mathbb{h}_{1}\in\mathbb{H}_{1}(\Omega)$ which depends on $\mathbb{h}_{2}$ such that (7.15) has a weak solution $(\mathbb{u},p)\in H^{1}(\Omega,\mathbb{R}^{3})\times H^{1}(\Omega)$.

Proof.

Step 1. To solve (7.20), we use (2.7) to write $\mathbb{Y}=\mathbb{w}+\hat{\mathbb{h}}_{2}+\nabla\psi$, with $\mathbb{w}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$, $\hat{\mathbb{h}}_{2}\in\mathbb{H}_{2}(\Omega)$ and $\psi\in H^{1}_{0}(\Omega)$. If $\mathbb{Y}$ satisfies (7.20) then $\psi$ satisfies

[TABLE]

As in the proof of Proposition 7.4 (also see Proposition 4.7 (i)), we can show that the map $\mathcal{T}:H^{1}_{0}(\Omega)\to H^{-1}(\Omega)$ defined by

[TABLE]

is hemi-continuous and strongly monotone. Then there exists a unique $\psi=\psi_{\mathbb{w},\hat{\mathbb{h}}_{2}}\in H^{1}_{0}(\Omega)$ such that $\mathcal{T}[\psi_{\mathbb{w},\hat{\mathbb{h}}_{2}}]=0$. The map $(\mathbb{w},\hat{\mathbb{h}}_{2})\mapsto\nabla\psi_{\mathbb{w},\hat{\mathbb{h}}_{2}}$ is a locally Lipschitz and bounded map from $\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)\times\mathbb{H}_{2}(\Omega)$ into $\text{grad}H^{1}_{0}(\Omega)$.

Step 2. Denote

[TABLE]

We look for $\mathbb{v}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ such that

[TABLE]

We shall show that, there exists a unique $\mathbb{h}_{1}=\mathbb{h}_{1,\mathbb{w},\hat{\mathbb{h}}_{2}}\in\mathbb{H}_{1}(\Omega)$ such that (7.22) has a unique solution $\mathbb{v}=\mathbb{V}_{\hat{\mathbb{h}}_{2}}[\mathbb{w}]\in H^{1}(\Omega,\mathbb{R}^{3})\cap\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$, and

[TABLE]

where $C$ depends on $\Omega,\mathcal{H}(x,\mathbb{z})$ and $\mathbb{f}(x,\mathbb{z})$.

To prove, we take a tangential component preserving extension of $\mathbb{H}^{0}_{T}$, which is also denoted by $\mathbb{H}^{0}$, see [P2]. Then $\mathbb{H}^{0}\in H^{1}(\Omega,\text{\rm div\,}0)$. We can choose $\mathbb{H}^{0}$ such that

[TABLE]

Let $\mathbb{z}=\mathbb{v}-\mathbb{H}^{0}$. Then (7.22) is reduced to

[TABLE]

Let us denote by $\mathbb{g}$ the right side of the first equation in (7.24). Then (7.24) is solvable in $H^{1}_{t0}(\Omega,\text{\rm div\,}0)$ if and only if $\mathbb{g}\in\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$.

By the definition of $\mathcal{P}_{n}$ we easily see that $\mathbb{g}\in\mathcal{H}_{0}(\Omega,\text{\rm div\,}0)$. On the other hand, there exists a unique $\mathbb{h}_{1}=\mathbb{h}_{1,\mathbb{w},\hat{\mathbb{h}}_{2}}$ such that $\mathbb{g}\perp\mathbb{H}_{1}(\Omega)$, namely

[TABLE]

Recalling that $\mathcal{H}_{0}(\Omega,\text{\rm div\,}0)=\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)\oplus_{L^{2}(\Omega)}\mathbb{H}_{1}(\Omega),$ we see that for this choice of $\mathbb{h}_{1}$ we have $\mathbb{g}\in\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$. Then (7.24) with $\mathbb{h}_{1}=\mathbb{h}_{1,\mathbb{w},\hat{\mathbb{h}}_{2}}$ has a unique solution $\mathbb{z}\in H^{1}(\Omega,\text{\rm div\,}0)\cap\mathbb{H}_{2}(\Omega)^{\perp}_{L^{2}(\Omega)}=H_{1}(\Omega,\mathbb{R}^{3})\cap\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$.

To estimate $\mathbb{h}_{1,\mathbb{w},\hat{\mathbb{h}}_{2}}$, let $\{\mathbb{e}_{1},\cdots,\mathbb{e}_{N}\}$ be a orthonormal basis of $\mathbb{H}_{1}(\Omega)$. Then

[TABLE]

Using the above equalities and (2.1) we have

[TABLE]

where $C$ depends on $\Omega$, $\mathcal{H}(x,\mathbb{z})$ and $\mathbb{f}(x,\mathbb{z})$.

For the solution $\mathbb{z}$ of (7.24) obtained above we have

[TABLE]

Hence (7.22) has a unique solution $\mathbb{v}=\mathbb{z}+\mathbb{H}^{0}$ in $\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$. We denote this solution by $\mathbb{v}=\mathbb{V}_{\hat{\mathbb{h}}_{2}}[\mathbb{w}]$. Then we have (7.23).

(7.23) shows that the map $\mathbb{w}\mapsto\mathbb{V}_{\hat{\mathbb{h}}_{2}}[\mathbb{w}]$ is a bounded map from $\mathcal{H}(\Omega,\text{\rm div\,}0)$ into $H^{1}(\Omega,\mathbb{R}^{3})\cap\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$. Using $(H_{1}),(H_{2}),(H_{3})$ and $(f_{1})$ we see that the maps

[TABLE]

are continuous in the $L^{2}$ norm. Thus the map $\mathbb{w}\mapsto\mathbb{h}_{1,\mathbb{w},\hat{\mathbb{h}}_{2}}$ is continuous in the $L^{2}$ norm. By these facts and using the $L^{2}$-div-curl-gradient inequalities we see that the map $\mathbb{w}\mapsto\mathbb{V}_{\hat{\mathbb{h}}_{2}}[\mathbb{w}]$ is continuous from $\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)\to H^{1}(\Omega,\mathbb{R}^{3})\cap\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$. By the Sobolev imbedding theorem we know that $\mathbb{V}_{\hat{\mathbb{h}}_{2}}$ is compact from $\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ into itself.

By the assumption (7.12), then there exists $R>0$ such that $\mathbb{V}_{\hat{\mathbb{h}}_{2}}$ maps the closed ball $B_{R}$ with center $\mathbb{0}$ and radius $R$ in $\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$ into $B_{R}$ itself. By Schauder’s fixed point theorem we conclude that $\mathbb{V}_{\hat{\mathbb{h}}_{2}}$ has a fixed point $\mathbb{w}_{\hat{\mathbb{h}}_{2}}$ in $B_{R}$.

Step 3. Let

[TABLE]

Since $\mathbb{w}_{\hat{\mathbb{h}}_{2}}$ satisfies

[TABLE]

we see that $\mathbb{Y}_{\hat{\mathbb{h}}_{2}}$ is a solution of (7.20), and by Lemma 7.9 (iii) $\mathbb{Z}_{\hat{\mathbb{h}}_{2}}$ is a solution of (7.19).

Using the method in the proof of Proposition 4.7 (ii), we can show that, for the given $\mathbb{h}_{2}\in\mathbb{H}_{2}(\Omega)$, there exists $\hat{\mathbb{h}}_{2}\in\mathbb{H}_{2}(\Omega)$ such that

[TABLE]

Namely, there exists $\hat{\mathbb{h}}_{2}$ such that $\mathbb{Z}_{\hat{\mathbb{h}}_{2}}\perp_{L^{2}(\Omega)}\mathbb{H}_{2}(\Omega)$. Then $\mathbb{Z}_{\hat{\mathbb{h}}_{2}}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$. By Lemma 7.9 (ii) we know that there exists $(\mathbb{u},p)\in H_{n0}^{1}(\Omega,\text{\rm div\,}0)\times H^{1}(\Omega)$ which solves (7.15) for the given $\mathbb{h}_{2}$. ∎

7.3. The Co-normal-Neumann BVP

In this section we consider the natural BVP of the Maxwell-Stokes system

[TABLE]

In the following we assume (7.2) and $H_{n}^{0}\in H^{-1/2}(\partial\Omega).$

Definition 7.11**.**

We say that $(\mathbb{u},p)$ is a weak solution of (7.27) if $\mathbb{u}\in\mathcal{H}(\Omega,\text{\rm curl\,},\text{\rm div\,}0)$ and $p\in H^{1}(\Omega)$ such that

• (i)

$\mathcal{H}(x,\text{\rm curl\,}\mathbb{u}(x)+\mathbb{h}_{2}(x))\in L^{2,-1/2}_{\nu}(\Omega,\mathbb{R}^{3})$, $\mathbb{f}(x,\text{\rm curl\,}\mathbb{u}(x)+\mathbb{h}_{2}(x))\in L^{2,-1/2}_{\nu}(\Omega,\mathbb{R}^{3})$;

• (ii)

the equality $\nu\cdot\mathcal{H}(x,\text{\rm curl\,}\mathbb{u}(x)+\mathbb{h}_{2}(x))=H^{0}_{n}$ holds in the $H^{-1/2}(\partial\Omega)$ sense;

• (iii)

[TABLE]

[TABLE]

We shall show the relation between (7.27) with the $Z$-system

[TABLE]

and we shall show the equivalence of $Z$-system (7.28) and the $Y$-system

[TABLE]

Definition 7.12**.**

(i)* We say $\mathbb{Z}$ is a weak solution of (7.28) if $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ such that*

[TABLE]

(ii)* We say $\mathbb{Y}$ is a weak solution of (7.20) if $\mathbb{Y}\in L^{2,-1/2}_{\nu}(\Omega,\mathbb{R}^{3})$ such that*

[TABLE]

Similar to Lemma 7.3 we have

Lemma 7.13**.**

Assume (7.2) and $H^{0}_{n}\in H^{-1/2}(\partial\Omega)$.

• (i)

If $\mathbb{Y}\in L^{2,-1/2}_{\nu}(\Omega,\mathbb{R}^{3})$ is a weak solution of (7.29), then $\mathbb{Y}\in\mathcal{H}(\Omega,\text{\rm curl\,})$, and the first equality in (7.29) holds a.e. in $\Omega$.

• (ii)

If $(\mathbb{u},p)\in\mathcal{H}(\Omega,\text{\rm curl\,},\text{\rm div\,}0)\times H^{1}(\Omega)$ is a weak solution of (7.27) and letting $\mathbb{Z}=\text{\rm curl\,}\mathbb{u}$, then $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ and it is a weak solution of (7.28). On the other hand, if $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ is a weak solution of (7.28), then there exists $\mathbb{h}_{2}^{0}\in\mathbb{H}_{2}(\Omega)$ such that (7.27) with $\mathbb{h}_{2}$ replaced by $\mathbb{h}_{2}+\mathbb{h}_{2}^{0}$ has a weak solution $(\mathbb{u},p)\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)\times H^{1}(\Omega)$ with $\text{\rm curl\,}\mathbb{u}=\mathbb{Z}-\mathbb{h}_{2}^{0}$. If furthermore $\mathbb{Z}\perp_{L^{2}(\Omega)}\mathbb{H}_{2}(\Omega)$, then $\mathbb{h}^{0}_{2}=\mathbb{0}$.

• (iii)

If $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ is a weak solution of (7.28) and letting $\mathbb{Y}=\mathcal{H}(x,\mathbb{Z}(x)+\mathbb{h}_{2}(x)),$ then $\mathbb{Y}\in\mathcal{H}(\Omega,\text{\rm curl\,})$ and it is a weak solution of (7.29). On the other hand, if $\mathbb{Y}\in L^{2,-1/2}_{\nu}(\Omega,\mathbb{R}^{3})$ is a weak solution of (7.29) and letting $\mathbb{Z}=\mathcal{B}(x,\mathbb{Y}(x))-\mathbb{h}_{2}(x),$ then $\mathbb{Z}\in\mathcal{H}(\Omega,\text{\rm div\,}0)$ and it is a weak solution of (7.28).

Theorem 7.14**.**

Assume (7.2), $(H_{3N})$, $(H^{0})$ and (7.12). Then for any given $\mathbb{h}_{1},\hat{\mathbb{h}}_{1}\in\mathbb{H}_{1}(\Omega)$ and $\mathbb{h}_{2}=\mathbb{h}_{2}(\hat{\mathbb{h}}_{1})\in\mathbb{H}_{2}(\Omega)$ which depends on $\hat{\mathbb{h}}_{1}$ such that (7.27) has a weak solution $(\mathbb{u},p)\in H^{1}(\Omega,\mathbb{R}^{3})\times H^{1}(\Omega)$ for the given $\mathbb{h}_{1}$ and for $\mathbb{h}_{2}=\mathbb{h}_{2}(\hat{\mathbb{h}}_{1})$.

Proof.

Step 1. By Lemma 7.13, we first solve (7.29). Using (2.6), we write

[TABLE]

From the last two equalities in (7.29) and using $(H^{0})$ we have

[TABLE]

Using $(H_{3})$ and $(H_{3N})$ we see that this BVP can be written as

[TABLE]

Hence $(\mathbb{w},\hat{\mathbb{h}}_{1},\nabla\psi)$ satisfies (7.30).

By the monotone operator methods we can show that, for any given $\mathbb{w}\in\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$ and $\hat{\mathbb{h}}_{1}\in\mathbb{H}_{1}(\Omega)$, (7.30) has a unique solution $\psi=\psi_{\mathbb{w},\hat{\mathbb{h}}_{1}}\in\dot{H}^{1}(\Omega)$.

Step 2. Denote

[TABLE]

We look for $\mathbb{v}\in\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$ such that

[TABLE]

By the definition of $\mathcal{P}_{\nu}$ we see that the right side of the first equation of (7.31) lies in $\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$. So (7.31) has a unique solution $\mathbb{v}=\mathbb{V}_{\hat{\mathbb{h}}_{1}}[\mathbb{w}]\in H^{1}_{n0}(\Omega,\text{\rm div\,}0)\cap\mathbb{H}_{1}(\Omega)^{\perp}_{L^{2}(\Omega)}=H_{1}(\Omega,\mathbb{R}^{3})\cap\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$. Similar to Lemma 7.5 we can show that

[TABLE]

where $C$ depends on $\Omega,\mathcal{H}(x,\mathbb{z})$ and $\mathbb{f}(x,\mathbb{z})$. Using the Sobolev imbedding theorem we know that $\mathbb{V}_{\hat{\mathbb{h}}_{1}}$ is compact from $\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$ into itself.

Since $\mathbb{f}(x,\mathbb{z})$ satisfies (7.12), then there exists $R>0$ such that $\mathbb{V}_{\hat{\mathbb{h}}_{1}}$ maps the closed ball $B_{R}$ with center $\mathbb{0}$ and radius $R$ in $\mathcal{H}^{\Sigma}_{0}(\Omega,\text{\rm div\,}0)$ into $B_{R}$ itself. By Schauder’s fixed point theorem we conclude that $\mathbb{V}_{\hat{\mathbb{h}}_{1}}$ has a fixed point $\mathbb{w}_{\hat{\mathbb{h}}_{1}}$ in $B_{R}$.

Step 3. Let

[TABLE]

$\mathbb{w}_{\hat{\mathbb{h}}_{1}}$ satisfies

[TABLE]

Then $\mathbb{Y}_{\hat{\mathbb{h}}_{1}}$ is a solution of (7.29), and by Lemma 7.13 (iii) we know that $\mathbb{Z}_{\hat{\mathbb{h}}_{1},\mathbb{h}_{2}}$ is a solution of (7.28).

Now we choose $\mathbb{h}_{2}=\mathbb{h}_{2}(\hat{\mathbb{h}}_{1})$ such that

[TABLE]

Denote $\mathbb{Z}_{\hat{\mathbb{h}}_{1}}=\mathbb{Z}_{\hat{\mathbb{h}}_{1},\mathbb{h}_{2}(\hat{\mathbb{h}}_{1})}$. Then $\mathbb{Z}_{\hat{\mathbb{h}}_{1}}\in\mathcal{H}^{\Gamma}(\Omega,\text{\rm div\,}0)$. By Lemma 7.13 (ii), there exists $(\mathbb{u},p)\in H_{n0}^{1}(\Omega,\text{\rm div\,}0)\times H^{1}(\Omega)$ which solves (7.27) for the given $\mathbb{h}_{1}$ and for $\mathbb{h}_{2}=\mathbb{h}_{2}(\hat{\mathbb{h}}_{1})$.

∎

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China grants no. 11671143 and no. 11431005.