Existence and regularity of solutions to quasilinear systems of Maxwell type and Maxwell-Stokes type

Abstract

Solvability of nonlinear partial differential systems involving the operator curl depends on the nature of nonlinearity of the equations and the type of the boundary conditions, and very often depends also on the domain topology. For a quasilinear system of the Maxwell type where the right side is independent of the unknown vector field $\mathbf{{u}}(x)$, we have existence of classical solutions, and an example is the magneto-static problem of the Maxwell equations. If the right side does depend on the unknown vector field $\mathbf{{u}}(x)$, a scalar potential p(x) should be introduced to the equation and we are led to a quasilinear system of the Maxwell-Stokes type. The boundary condition on the potential p(x) should be determined according to the domain topology. We consider the Dirichlet boundary condition for the potential if the domain has no holes, and consider the Neumann boundary condition if the domain has holes. For each case existence of classical solutions $(\mathbf{{u}}(x), p(x))$ is proved and a priori estimates of the solutions are established.

Appendices

Appendix A: Spaces of vector fields

1.1 A.1: Notation

Subspaces of the Sobolev space $H^k(\Omega ,\mathbb {R}^3)$:

$$\begin{aligned} H^k_{t0}(\Omega ,\mathbb {R}^3)= & {} \{\mathbf{{u}}\in H^1(\Omega ,\mathbb {R}^3): \mathbf{{u}}_T=\mathbf{{0}}\},\\ H^k_t(\Omega ,\mathrm{div\,}0,\mathbf{{u}}^0_T)= & {} \{\mathbf{{u}}\in H^1(\Omega ,\mathbb {R}^3): \mathrm{div\,}\mathbf{{u}}=0,\; \mathbf{{u}}_T=\mathbf{{u}}^0_T\},\\ H^k_{t0}(\Omega ,\mathrm{div\,}0)= & {} \{\mathbf{{u}}\in H^1(\Omega ,\mathbb {R}^3): \mathrm{div\,}\mathbf{{u}}=0,\; \mathbf{{u}}_T=\mathbf{{0}}\},\\ H^k_{n0}(\Omega ,\mathrm{div\,}0)= & {} \{\mathbf{{u}}\in H^1(\Omega ,\mathbb {R}^3): \mathrm{div\,}\mathbf{{u}}=0,\; \nu \cdot \mathbf{{u}}=0\}. \end{aligned}$$

Subspaces of $L^2(\Omega ,\mathbb {R}^3)$:

$$\begin{aligned} \mathcal H(\Omega ,\mathrm{div\,})= & {} \{\mathbf{{u}}\in L^2(\Omega ,\mathbb {R}^3): \mathrm{div\,}\mathbf{{u}}\in L^2(\Omega )\},\\ \mathcal H(\Omega ,\mathrm{curl\,})= & {} \{\mathbf{{u}}\in L^2(\Omega ,\mathbb {R}^3): \mathrm{curl\,}\mathbf{{u}}\in L^2(\Omega ,\mathbb {R}^3)\},\\ \mathcal H(\Omega ,\mathrm{div\,}0)= & {} \{\mathbf{{u}}\in L^2(\Omega ,\mathbb {R}^3): \mathrm{div\,}\mathbf{{u}}=0\},\\ {\mathcal {H}}_0(\Omega ,\mathrm{div\,}0)= & {} \{\mathbf{{u}}\in L^2(\Omega ,\mathbb {R}^3): \mathrm{div\,}\mathbf{{u}}=0,\; \nu \cdot \mathbf{{u}}=0\},\\ \mathcal H(\Omega ,\mathrm{curl\,}0)= & {} \{\mathbf{{u}}\in L^2(\Omega ,\mathbb {R}^3): \mathrm{curl\,}\mathbf{{u}}=\mathbf{{0}}\}. \end{aligned}$$

Decompositions of $L^2(\Omega ,\mathbb {R}^3)$ (see [14, pp. 225–226]):

$$\begin{aligned} \begin{array}{ll} &{}L^2(\Omega ,\mathbb {R}^3)=\mathcal H(\Omega ,\mathrm{curl\,}0)\oplus _{L^2(\Omega )} \mathrm{curl\,}H^1_{t0}(\Omega ,\mathrm{div\,}0),\\ &{}L^2(\Omega ,\mathbb {R}^n)=\mathcal H(\Omega ,\mathrm{div\,}0)\oplus _{L^2(\Omega )} \mathrm{grad }H^1_0(\Omega ). \end{array} \end{aligned}$$

(A.1)

Image of the operator $\mathrm{curl\,}$ (see [14, p. 222, Proposition 3; p. 226, Remark 5]):

$$\begin{aligned}&\mathrm{curl\,}H^1(\Omega ,\mathbb {R}^3)=\mathrm{curl\,}H^1_{n0}(\Omega ,\mathrm{div\,}0)=\mathcal H^\Gamma (\Omega ,\mathrm{div\,}0),\nonumber \\&\mathrm{curl\,}H^1_{t0}(\Omega ,\mathrm{div\,}0)=\mathcal H_0^\Sigma (\Omega ,\mathrm{div\,}0), \end{aligned}$$

(A.2)

where (see $(O_1)$ and $(O_2)$ in Section 2 for the definition of $\Gamma _j$ and $\Sigma _i$)

$$\begin{aligned} \mathcal H^\Gamma (\Omega ,\mathrm{div\,}0)= & {} \{\mathbf{{u}}\in \mathcal H(\Omega ,\mathrm{div\,}0): \langle 1,\mathbf{{u}}\cdot \nu \rangle _{H^{1/2}(\Gamma _j),H^{-1/2}(\Gamma _j)}=0,\; j=1,\ldots , m+1\},\\ \mathcal H_0^\Sigma (\Omega ,\mathrm{div\,}0)= & {} \{\mathbf{{u}}\in L^2(\Omega ,\mathbb {R}^3): \mathrm{div\,}\mathbf{{u}}=0,\; \mathbf{{u}}\cdot \nu |_{\partial \Omega }=0,\\&\qquad \qquad \langle 1,\mathbf{{u}}\cdot \nu \rangle _{H^{1/2}(\Sigma _i),H^{-1/2}(\Sigma _i)}=0,\; i=1,\ldots , N\}. \end{aligned}$$

These yield the following decompositions of the kernel of the operators $\mathrm{curl\,}$ and $\mathrm{div\,}$:

$$\begin{aligned} \mathcal H(\Omega ,\mathrm{curl\,}0)=&\mathrm{grad }H^1(\Omega )\oplus _{L^2(\Omega )}\mathbb {H}_1(\Omega ),\nonumber \\ \mathcal H(\Omega ,\mathrm{div\,}0)=&{\mathcal {H}}^\Gamma (\Omega ,\mathrm{div\,}0)\oplus _{L^2(\Omega )}\mathbb {H}_2(\Omega ),\nonumber \\ \mathcal H_0(\Omega ,\mathrm{div\,}0)=&\mathcal H^\Sigma _0(\Omega ,\mathrm{div\,}0)\oplus _{L^2(\Omega )}\mathbb {H}_1(\Omega ). \end{aligned}$$

(A.3)

1.2 A.2 The div-curl-gradient inequalities

For various form of the div-curl-gradient inequalities we refer to [4, 9, 14, 25, 46] and the references therein. Here we list them in the form convenient to our use. In the following if $k=0$ then the $H^0$ norm stands for the $L^2$ norm.

• (i) If $\Omega$ is a bounded domain in $\mathbb {R}^3$ with a $C^{k+2}$ boundary, $k\ge 0$, then we have the $L^2$-div-curl-gradient inequality (see [14, P. 212, Corollary 1])

  $$\begin{aligned} \Vert \mathbf{{u}}\Vert _{H^{k+1}(\Omega )}\le C\{\Vert \mathrm{curl\,}\mathbf{{u}}\Vert _{H^k(\Omega )}+\Vert \mathrm{div\,}\mathbf{{u}}\Vert _{H^k(\Omega )}+\Vert \mathbf{{u}}\Vert _{L^2(\Omega )}+\Vert \nu \cdot \mathbf{{u}}\Vert _{H^{k+1/2}(\partial \Omega )}\},\nonumber \\ \end{aligned}$$

  (A.4)
  $$\begin{aligned} \Vert \mathbf{{u}}\Vert _{H^{k+1}(\Omega )}\le C\{\Vert \mathrm{curl\,}\mathbf{{u}}\Vert _{H^k(\Omega )}+\Vert \mathrm{div\,}\mathbf{{u}}\Vert _{H^k(\Omega )}+\Vert \mathbf{{u}}\Vert _{L^2(\Omega )}+\Vert \mathbf{{u}}_T\Vert _{H^{k+1/2}(\partial \Omega )}\},\nonumber \\ \end{aligned}$$

  (A.5)

  where $C=C(\Omega ,k)$. If $\mathbf{{u}}\in \mathbb {H}_1(\Omega )^\perp _{L^2(\Omega )}$ (resp. $\mathbf{{u}}\in \mathbb {H}_2(\Omega )^\perp _{L^2(\Omega )}$) then the $L^2$ norm of $\mathbf{{u}}$ is the right side of (A.4) (resp. in the right side of (A.5)) can be removed (see [46]):

  $$\begin{aligned} \Vert \mathbf{{u}}\Vert _{H^{k+1}(\Omega )}\le & {} C\{\Vert \mathrm{curl\,}\mathbf{{u}}\Vert _{H^k(\Omega )}+\Vert \mathrm{div\,}\mathbf{{u}}\Vert _{H^k(\Omega )}+\Vert \nu \cdot \mathbf{{u}}\Vert _{H^{k+1/2}(\partial \Omega )}\},\nonumber \\&\forall \,\mathbf{{u}}\in H^{k+1}(\Omega ,\mathbb {R}^3)\cap \mathbb {H}_1(\Omega )^\perp _{L^2(\Omega )},\qquad \end{aligned}$$

  (A.6)
  $$\begin{aligned} \Vert \mathbf{{u}}\Vert _{H^{k+1}(\Omega )}\le & {} C\{\Vert \mathrm{curl\,}\mathbf{{u}}\Vert _{H^k(\Omega )}+\Vert \mathrm{div\,}\mathbf{{u}}\Vert _{H^k(\Omega )}+\Vert \mathbf{{u}}_T\Vert _{H^{k+1/2}(\partial \Omega )}\},\nonumber \\&\forall \,\mathbf{{u}}\in H^{k+1}(\Omega ,\mathbb {R}^3)\cap \mathbb {H}_2(\Omega )^\perp _{L^2(\Omega )}. \end{aligned}$$

  (A.7)

• (ii) If $\Omega$ is a bounded domain in $\mathbb {R}^3$ with a $C^{k+2+\alpha }$ boundary, $k\ge 0$ and $0<\alpha <1$, then we have the Hölder-div-curl-gradient inequality ([9]):

  $$\begin{aligned}&\Vert \mathbf{{u}}\Vert _{C^{k+1,\alpha }(\bar{\Omega })}\le C\{\Vert \mathrm{curl\,}\mathbf{{u}}\Vert _{C^{k,\alpha }(\bar{\Omega })}+\Vert \mathrm{div\,}\mathbf{{u}}\Vert _{C^{k,\alpha }(\bar{\Omega })} +\Vert \mathbf{{u}}\Vert _{L^2(\Omega )}+\Vert \nu \cdot \mathbf{{u}}\Vert _{C^{k+1,\alpha }(\partial \Omega )}\},\nonumber \\ \end{aligned}$$

  (A.8)
  $$\begin{aligned}&\Vert \mathbf{{u}}\Vert _{C^{k+1,\alpha }(\bar{\Omega })}\le C\{\Vert \mathrm{curl\,}\mathbf{{u}}\Vert _{C^{k,\alpha }(\bar{\Omega })}+\Vert \mathrm{div\,}\mathbf{{u}}\Vert _{C^{k,\alpha }(\bar{\Omega })}+\Vert \mathbf{{u}}\Vert _{L^2(\Omega )}+\Vert \mathbf{{u}}_T\Vert _{C^{k+1,\alpha }(\partial \Omega )}\},\nonumber \\ \end{aligned}$$

  (A.9)

  where $C=C(\Omega ,k,\alpha )$. If $\mathbf{{u}}\in \mathbb {H}_1(\Omega )^\perp _{L^2(\Omega )}$ (resp. $\mathbf{{u}}\in \mathbb {H}_2(\Omega )^\perp _{L^2(\Omega )}$), then the $L^2$ norm of $\mathbf{{u}}$ is the right side of (A.8) (resp. in the right side of (A.9)) can be removed:

  $$\begin{aligned} \Vert \mathbf{{u}}\Vert _{C^{k+1,\alpha }(\bar{\Omega })}\le & {} C\{\Vert \mathrm{curl\,}\mathbf{{u}}\Vert _{C^{k,\alpha }(\bar{\Omega })}+\Vert \mathrm{div\,}\mathbf{{u}}\Vert _{C^{k,\alpha }(\bar{\Omega })}+\Vert \nu \cdot \mathbf{{u}}\Vert _{C^{k+1,\alpha }(\partial \Omega )}\},\nonumber \\&\forall \,\mathbf{{u}}\in C^{k+1,\alpha }(\bar{\Omega },\mathbb {R}^3)\cap \mathbb {H}_1(\Omega )^\perp _{L^2(\Omega )}, \end{aligned}$$

  (A.10)
  $$\begin{aligned} \Vert \mathbf{{u}}\Vert _{C^{k+1,\alpha }(\bar{\Omega })}\le & {} C\{\Vert \mathrm{curl\,}\mathbf{{u}}\Vert _{C^{k,\alpha }(\bar{\Omega })}+\Vert \mathrm{div\,}\mathbf{{u}}\Vert _{C^{k,\alpha }(\bar{\Omega })}+\Vert \mathbf{{u}}_T\Vert _{C^{k+1,\alpha }(\partial \Omega )}\},\nonumber \\&\forall \,\mathbf{{u}}\in C^{k+1,\alpha }(\bar{\Omega },\mathbb {R}^3)\cap \mathbb {H}_2(\Omega )^\perp _{L^2(\Omega )}, \end{aligned}$$

  (A.11)

• (iii) The $W^{k,p}$ versions of (A.4) and (A.5) has been obtained in [25], which yield the $W^{k,p}$ version of (A.6) and (A.7). For our use in section 4 we list the following:

  $$\begin{aligned} \Vert \mathbf{{u}}\Vert _{W^{1,p}(\Omega )}\le & {} C\{ \Vert \mathrm{curl\,}\mathbf{{u}}\Vert _{L^p(\Omega )} +\Vert \mathrm{div\,}\mathbf{{u}}\Vert _{L^p(\Omega )} +\Vert \nu \cdot \mathbf{{u}}\Vert _{W^{1-1/p,p}(\partial \Omega )}\},\nonumber \\&\forall \,\mathbf{{u}}\in W^{1,p}(\Omega ,\mathbb {R}^3)\cap \mathbb {H}_1(\Omega )^\perp _{L^2(\Omega )},\; 1<p\le 6, \end{aligned}$$

  (A.12)

where $C=C(\Omega ,p)$.

Appendix B: Linear curl-Stokes system

We collect here some existence and regularity results for the linear curl-Stokes system

$$\begin{aligned} \left\{ \begin{array}{lll} &{}\mathrm{curl\,}[{\mathcal {A}}(x)\mathrm{curl\,}\mathbf{{u}}]=\mathbf{{J}}+\nabla p,\qquad \mathrm{div\,}\mathbf{{u}}=0 &{}\text {in }\Omega ,\\ &{}\mathbf{{u}}_T=\mathbf{{u}}^0_T,\quad p=0\quad &{}\text {on }\partial \Omega , \end{array} \right. \end{aligned}$$

(B.1)

and the linear Maxwell type system

$$\begin{aligned} \left\{ \begin{array}{lll} &{}\mathrm{curl\,}[{\mathcal {A}}(x)\mathrm{curl\,}\mathbf{{u}}]=\mathbf{{J}},\qquad \mathrm{div\,}\mathbf{{u}}=0 &{}\text {in }\Omega ,\\ &{}\mathbf{{u}}_T=\mathbf{{u}}^0_T&{}\text {on }\partial \Omega . \end{array} \right. \end{aligned}$$

(B.2)

If $\mathbf{{J}}\in L^2(\Omega ,\mathbb {R}^3)$, existence of weak solutions of (B.1) follows from Proposition 3.1. Here we state a result of existence for slightly more general $\mathbf{{J}}$:

Lemma B.1

Assume that $\Omega$ is a bounded domain in $\mathbb {R}^3$ with a $C^2$ boundary, ${\mathcal {A}}$ satisfies $(A_0)$, $\mathbf{{J}}\in H^{1,*}_{t0}(\Omega ,\mathbb {R}^3)$ and $\mathbf{{u}}^0_T\in T\!H^{1/2}(\partial \Omega ,\mathbb {R}^3)$.

• (i) Eq.(B.1) has weak solutions if and only if

  $$\begin{aligned} \langle \mathbf{{J}},\mathbf{{h}}\rangle _{H^{1,*}_{t0}(\Omega ),H^1_{t0}(\Omega )} =0,\quad \forall \,\mathbf{{h}}\in \mathbb {H}_2(\Omega ). \end{aligned}$$

  (B.3)

• (ii) If (B.3) is satisfied, then all the weak solutions of (B.1) are given by $\mathbf{{u}}=\mathbf{{u}}_J+\mathbf{{h}}$ and $p=\phi _J$ for an arbitrary $\mathbf{{h}}\in \mathbb {H}_2(\Omega )$, where $\phi _J$ is given in (2.10) and $(\mathbf{{u}}_J,\phi _J)$ is a particular solution of (B.1).

• (iii) If $\mathbf{{J}}\in L^2(\Omega ,\mathbb {R}^3)$, then (B.3) is equivalent to $\mathbf{{J}}\in \mathbb {H}_2(\Omega )^\perp _{L^2(\Omega )}$. Therefore (B.1) has weak solutions if and only if $\mathbf{{J}}\in \mathbb {H}_2(\Omega )^\perp _{L^2(\Omega )}$; (B.2) has weak solutions if and only if $\mathbf{{J}}\in {\mathcal {H}}^\Gamma (\Omega ,\mathrm{div\,}0)$.

Proof

Assume $\mathbf{{J}}$ satisfies (B.3). Let $\mathcal U\in H^1_{t0}(\Omega ,\mathrm{div\,}0)$ be a divergence-free and tangential component-preserving extension of $\mathbf{{u}}^0_T$ satisfying (2.11) with $k=1$, and set $\mathbf{{J}}_1=\mathbf{{J}}-\mathrm{curl\,}({\mathcal {A}}\mathrm{curl\,}\mathcal U)$. For any $\mathbf{{h}}\in \mathbb {H}_2(\Omega )$, from (B.3) we have

$$\begin{aligned} \langle \mathbf{{J}}_1,\mathbf{{h}}\rangle _{H^{1,*}_{t0}(\Omega ),H^1_{t0}(\Omega )}=\langle \mathbf{{J}},\mathbf{{h}}\rangle _{H^{1,*}_{t0}(\Omega ),H^1_{t0}(\Omega )} -\int _\Omega {\mathcal {A}}\mathrm{curl\,}\mathcal U,\mathrm{curl\,}\mathbf{{h}}dx=0. \end{aligned}$$

Thus $\mathbf{{J}}_1$ also satisfies (B.3). Let $\phi _J$ and $\phi _{J_1}$ be the functions in (2.10) determined by $\mathbf{{J}}$ and $\mathbf{{J}}_1$ respectively. Since $\mathrm{div\,}\mathbf{{J}}_1=\mathrm{div\,}\mathbf{{J}}$, we see that $\phi _{J_1}=\phi _J$. Let $\mathbf{{u}}={\mathbf{v}}+\mathcal U$. Then (B.1) can be written as

$$\begin{aligned} \left\{ \begin{array}{lll} &{}\mathrm{curl\,}({\mathcal {A}}\,\mathrm{curl\,}{\mathbf{v}})=\mathbf{{J}}_1+\nabla p,\quad \mathrm{div\,}{\mathbf{v}}=0\quad &{}\text {in }\Omega ,\\ &{}{\mathbf{v}}_T=\mathbf{{0}},\quad p=0\quad &{}\text {on }\partial \Omega . \end{array} \right. \end{aligned}$$

(B.4)

Using (A.7), we can apply the Lax-Milgram theorem on $H^1_{t0}(\Omega ,\mathrm{div\,}0)\cap \mathbb {H}_2(\Omega )^\perp$, and conclude that there exists ${\mathbf{v}}_0\in H^1_{t0}(\Omega ,\mathrm{div\,}0)\cap \mathbb {H}_2(\Omega )^\perp$ such that the following equality holds for all ${\mathbf{w}}\in H^1_{t0}(\Omega ,\mathrm{div\,}0)\cap \mathbb {H}_2(\Omega )^\perp$:

$$\begin{aligned} \int _\Omega \langle {\mathcal {A}}\mathrm{curl\,}\mathbf{{u}},\mathrm{curl\,}{\mathbf{v}_0}\rangle dx=\langle \mathbf{{J}}_1,{\mathbf{v}}\rangle _{H^{1,*}_{t0}(\Omega ),H^1_{t0}(\Omega )}. \end{aligned}$$

(B.5)

Since $\mathbf{{J}}_1$ satisfies (B.3), we see that (B.5) holds for all $\mathbf{{w}}\in \mathbb {H}_2(\Omega )$, hence it holds for all $\mathbf{{w}}\in H^1_{t0}(\Omega ,\mathrm{div\,}0)$. Then by Lemma 2.2 we conclude that there exists $p\in L^{2,-1/2}(\Omega )$ such that $({\mathbf{v}}_0,p)$ is a weak solution of (B.4). From Corollary 2.6 we see that $p=\phi _{J_1}=\phi _J$. $\square$

Now we derive regularity results for the linear curl-Stokes system.

Lemma B.2

Assume that $\Omega$ is a bounded domain in $\mathbb {R}^3$ with a $C^3$ boundary, ${\mathcal {A}}\in C^1(\bar{\Omega },S^+(3))$ satisfies $(A_0)$, $\mathbf{{J}}\in L^2(\Omega ,\mathbb {R}^3)$ and $\mathbf{{u}}^0_T\in T\!H^{3/2}(\partial \Omega ,\mathbb {R}^3)$. Let $(\mathbf{{u}},p)\in H^1(\Omega ,\mathbb {R}^3)\times L^{2,-1/2}(\Omega )$ be a weak solution of (B.1). Then $\mathbf{{u}}\in H^2(\Omega ,\mathbb {R}^3)$, $p\in H^1_0(\Omega )$,

$$\begin{aligned}&\Vert \mathbf{{u}}\Vert _{H^2(\Omega )}\le C_1\left\{ \Vert \mathbf{{J}}\Vert _{L^2(\Omega )}+\Vert \mathbf{{u}}\Vert _{L^2(\Omega )}+\Vert \mathbf{{u}}^0_T\Vert _{H^{3/2}(\partial \Omega )}\right\} ,\nonumber \\&\Vert \nabla p\Vert _{L^2(\Omega )}\le C(\Omega )\Vert \mathbf{{J}}\Vert _{L^2(\Omega )}. \end{aligned}$$

(B.6)

If furthermore $\mathbf{{u}}$ satisfies (2.13), then we have

$$\begin{aligned} \Vert \mathbf{{u}}\Vert _{H^2(\Omega )}\le C_1\left\{ \Vert \mathbf{{J}}\Vert _{L^2(\Omega )}+\Vert \mathbf{{u}}^0_T\Vert _{H^{3/2}(\partial \Omega )}\right\} . \end{aligned}$$

(B.7)

The constant $C_1$ depends on $\Omega$, $\lambda$, $\Vert {\mathcal {A}}\Vert _{C^1(\bar{\Omega })}$ and the constants in $(A_0)$.

Lemma B.3

Assume that $\Omega$ is a bounded domain in $\mathbb {R}^3$ with a $C^{3,\alpha }$ boundary, $0<\alpha <1$, ${\mathcal {A}}\in C^{1,\alpha }(\bar{\Omega }, S^+(3))$ satisfies $(A_0)$, $\mathbf{{J}}\in C^\alpha (\bar{\Omega },\mathbb {R}^3)$ and $\mathbf{{u}}^0_T\in T\!C^{2,\alpha }(\partial \Omega ,\mathbb {R}^3)$. Let $(\mathbf{{u}},p)\in H^1(\Omega ,\mathbb {R}^3)\times L^{2,-1/2}(\Omega )$ be a weak solution of (B.1). Then $\mathbf{{u}}\in C^{2,\alpha }(\bar{\Omega },\mathbb {R}^3)$, $p\in C^{1,\alpha }(\bar{\Omega })$,

$$\begin{aligned}&\Vert \mathbf{{u}}\Vert _{C^{2,\alpha }(\bar{\Omega })}\le C_2\left\{ \Vert \mathbf{{J}}\Vert _{C^\alpha (\bar{\Omega })}+\Vert \mathbf{{u}}\Vert _{L^2(\Omega )}+\Vert \mathbf{{u}}^0_T\Vert _{C^{2,\alpha }(\partial \Omega )}\right\} ,\nonumber \\&\Vert p\Vert _{C^{1,\alpha }(\bar{\Omega })}\le C(\Omega ,\alpha )\Vert \mathbf{{J}}\Vert _{C^\alpha (\bar{\Omega })}. \end{aligned}$$

(B.8)

If furthermore $\mathbf{{u}}$ satisfies (2.13), then we have

$$\begin{aligned} \Vert \mathbf{{u}}\Vert _{C^{2,\alpha }(\bar{\Omega })}\le C_2\left\{ \Vert \mathbf{{J}}\Vert _{C^\alpha (\bar{\Omega })}+\Vert \mathbf{{u}}^0_T\Vert _{C^{2,\alpha }(\partial \Omega )}\right\} . \end{aligned}$$

(B.9)

The constant $C_2$ depends on $\Omega$, $\lambda$, $\Vert {\mathcal {A}}\Vert _{C^{1,\alpha }(\bar{\Omega })}$ and the constants in $(A_0)$.

Proof of Lemmas B.2 and B.3

Step 1. Let $\mathbf{{J}}\in L^2(\Omega ,\mathbb {R}^3)$, and let $\mathbf{{J}}_0$ and $\phi _J$ be given in the decomposition of $\mathbf{{J}}$ in (2.9). Then $\phi _J\in H^1(\Omega )$ and we have

$$\begin{aligned}&\Vert \phi _J\Vert _{H^1(\Omega )}\le C(\Omega )\Vert \mathbf{{J}}\Vert _{L^2(\Omega )},\\&\Vert \mathbf{{J}}_0\Vert _{L^2(\Omega )}\le \Vert \mathbf{{J}}\Vert _{L^2(\Omega )}+\Vert \nabla \phi _J\Vert _{L^2(\Omega )}\le C(\Omega )\Vert \mathbf{{J}}\Vert _{L^2(\Omega )}. \end{aligned}$$

Let $(\mathbf{{u}},p)$ be a weak solution of (B.1). From Lemma 2.5 we know that $p=\phi _J$, and $\mathbf{{u}}$ is a weak solution of (B.2) with $\mathbf{{J}}$ replaced by $\mathbf{{J}}_0$. So we can apply the difference-quotient method to (B.2) with $\mathbf{{J}}$ replaced by $\mathbf{{J}}_0$ (see [7, proof of Theorem 4.1]) and show that $\mathbf{{u}}\in H^2(\Omega ,\mathbb {R}^3)$ and the first inequality of (B.6) with $\mathbf{{J}}$ replaced by $\mathbf{{J}}_0$ holds. If furthermore $\mathbf{{u}}$ satisfies (2.13), then the estimate (B.7) with $\mathbf{{J}}$ replaced by $\mathbf{{J}}_0$ holds.

Now assume $\Omega$ has a $C^{3,\alpha }$ boundary and $\mathbf{{J}}\in C^\alpha (\bar{\Omega },\mathbb {R}^3)$. Then $\phi _J\in C^{1,\alpha }(\bar{\Omega })$ and we have (see [17, Theorem 8.33])

$$\begin{aligned}&\Vert \phi _J\Vert _{C^{1,\alpha }(\bar{\Omega })}\le C(\Omega ,\alpha )\Vert \mathbf{{J}}\Vert _{C^\alpha (\bar{\Omega })},\nonumber \\&\Vert \mathbf{{J}}_0\Vert _{C^\alpha (\bar{\Omega })}\le \Vert \mathbf{{J}}\Vert _{C^\alpha (\bar{\Omega })}+\Vert \nabla \phi _J\Vert _{C^\alpha (\bar{\Omega })}\le C(\Omega )\Vert \mathbf{{J}}\Vert _{C^\alpha (\bar{\Omega })}. \end{aligned}$$

(B.10)

We shall prove that $\mathbf{{u}}\in C^{2,\alpha }(\bar{\Omega },\mathbb {R}^3)$ and

$$\begin{aligned} \Vert \mathbf{{u}}\Vert _{C^{2,\alpha }(\bar{\Omega })}\le C\{\Vert \mathbf{{J}}_0\Vert _{C^\alpha (\bar{\Omega })}+\Vert \mathbf{{u}}\Vert _{L^2(\Omega )}+\Vert \mathbf{{u}}^0_T\Vert _{C^{2,\alpha }(\partial \Omega )}\}, \end{aligned}$$

(B.11)

where C depends on $\Omega ,\lambda ,\alpha$, $\Vert {\mathcal {A}}\Vert _{C^{1,\alpha }(\bar{\Omega })}$ and $(A_0)$; If $\mathbf{{u}}$ satisfies (2.13), then the term $\Vert \mathbf{{u}}\Vert _{L^2(\Omega )}$ in the right side of (B.11) is not necessary. Using (B.10) and (B.11) we get the first inequality in (B.8), and (B.9).

Step 2. Now we prove (B.11), where $\mathbf{{u}}\in H^1(\Omega ,\mathbb {R}^3)$ is the solution of (B.2) with $\mathbf{{J}}$ replaced by $\mathbf{{J}}_0$. We may assume $\mathbf{{u}}^0_T=\mathbf{{0}}$, otherwise we consider $\mathbf{{w}}=\mathbf{{u}}-\mathcal U$, where $\mathcal U$ is a divergence-free and tangential component-preserving extension of $\mathbf{{u}}^0_T$ satisfying (2.12) with $k=2$. Using (A.5) we get

$$\begin{aligned}&\Vert \mathbf{{u}}\Vert _{H^1(\Omega )}\le C(\Omega )\{\lambda ^{-1}\Vert \mathbf{{J}}_0\Vert _{H^{1,*}_{t0}(\Omega )}+\Vert \mathbf{{u}}\Vert _{L^2(\Omega )}\}. \end{aligned}$$

(B.12)

If furthermore $\mathbf{{u}}\in \mathbb {H}_2(\Omega )^\perp _{L^2(\Omega )}$, then we use (A.7) instead of (A.5) to get

$$\begin{aligned}&\Vert \mathbf{{u}}\Vert _{H^1(\Omega )}\le C(\Omega )\lambda ^{-1}\Vert \mathbf{{J}}_0\Vert _{H^{1,*}_{t0}(\Omega )}. \end{aligned}$$

(B.13)

Using (B.2) and arguing as in step 3 of the proof of Theorem 4.1 (see (4.11)) we find a vector field ${\mathbf{j}}\in H^1(\Omega ,\mathbb {R}^3)\cap \mathbb {H}_1(\Omega )^\perp$ such that $\mathrm{curl\,}{\mathbf{j}}=\mathbf{{J}}_0$ and $\mathrm{div\,}{\mathbf{j}}=0$ in $\Omega$, and $\nu \cdot {\mathbf{j}}=0$ on $\partial \Omega$. Then

$$\begin{aligned} \Vert {\mathbf{j}}\Vert _{H^1(\Omega )}\le C(\Omega )\Vert \mathbf{{J}}_0\Vert _{L^2(\Omega )},\qquad \Vert {\mathbf{j}}\Vert _{C^{1,\alpha }(\bar{\Omega })}\le C(\Omega ,\alpha )\Vert \mathbf{{J}}_0\Vert _{C^\alpha (\bar{\Omega })}. \end{aligned}$$

(B.14)

From (B.2) and the first equality of (A.3), there exist $\phi \in \dot{H}^1(\Omega )$ and $\mathbf{{h}}\in \mathbb {H}_1(\Omega )$, such that

$$\begin{aligned} {\mathcal {A}}\mathrm{curl\,}\mathbf{{u}}-{\mathbf{j}}=\nabla \phi +\mathbf{{h}}. \end{aligned}$$

(B.15)

Using (B.15) and since $\mathbf{{h}}\in \mathbb {H}_1(\Omega )$ and ${\mathbf{j}}\in \mathbb {H}_1(\Omega )^\perp$, we have

$$\begin{aligned} \int _\Omega |\mathbf{{h}}|^2dx=\int _\Omega \langle \mathbf{{h}},{\mathcal {A}}\mathrm{curl\,}\mathbf{{u}}\rangle dx. \end{aligned}$$

From this and (A.8) we get

$$\begin{aligned} \Vert \mathbf{{h}}\Vert _{C^{1,\alpha }(\bar{\Omega })}\le C(\Omega ,\alpha )\Vert \mathbf{{h}}\Vert _{L^2(\Omega )}\le C(\Omega ,\alpha )\Vert {\mathcal {A}}\Vert _{C^0(\bar{\Omega })}\Vert \mathbf{{u}}\Vert _{H^1(\Omega )}. \end{aligned}$$

(B.16)

From (B.15) we can write

$$\begin{aligned} \mathrm{curl\,}\mathbf{{u}}=M(\nabla \phi +{\mathbf{j}}+\mathbf{{h}}), \end{aligned}$$

(B.17)

where $M={\mathcal {A}}^{-1}$. Since ${\mathcal {A}}\in C^{1,\alpha }(\bar{\Omega }, S_+(3,\mathbb {R}))$, so $M\in C^{1,\alpha }(\bar{\Omega },S_+(3,\mathbb {R}))$. Since $\mathbf{{u}}_T=\mathbf{{u}}_T^0=\mathbf{{0}}$, we have $\nu \cdot \mathrm{curl\,}\mathbf{{u}}=\nu \cdot \mathrm{curl\,}\mathbf{{u}}_T=0$ on $\partial \Omega$, and we find that $\phi$ satisfies the following

$$\begin{aligned} \left\{ \begin{array}{ll} &{}\mathrm{div\,}[M(\nabla \phi +{\mathbf{j}}+\mathbf{{h}})]=0\quad \text {in }\Omega ,\\ &{}\nu \cdot M(\nabla \phi +{\mathbf{j}}+\mathbf{{h}})=0\quad \text {on }\partial \Omega . \end{array} \right. \end{aligned}$$

(B.18)

By $(A_0)$ we have $\langle M\nu ,\nu \rangle \ge \gamma$ for some $\gamma >0$. So we can apply the Calderón-Zygmund estimate to (B.18) and find $\nabla \phi \in L^r(\Omega ,\mathbb {R}^3)$ and

$$\begin{aligned} \Vert \nabla \phi \Vert _{L^r(\Omega )}\le C_1\Vert {\mathbf{j}}+\mathbf{{h}}\Vert _{L^r(\Omega )} \end{aligned}$$

for any $1<r<\infty$, where $C_1$ depends on $\Omega$, $\lambda (M)$, $\Lambda (M)$, r, and the modula of continuity of M. Then from the Sobolev embedding theorem, for $0<\delta <1/2$ we have

$$\begin{aligned} \Vert \phi \Vert _{C^\delta (\bar{\Omega })}\le&C_2 \Vert {\mathbf{j}}+\mathbf{{h}}\Vert _{L^6(\Omega )}\le C_2\Vert {\mathbf{j}}+\mathbf{{h}}\Vert _{H^1(\Omega )}. \end{aligned}$$

(B.19)

$C_2$ depends on $\Omega ,\lambda (M),\Lambda (M),\delta$, and the modula of continuity of M. Then we apply the Schauder estimate to (B.18) to find $\phi \in C^{2,\alpha }(\bar{\Omega })$, and using (B.19) to conclude that

$$\begin{aligned} \Vert \phi \Vert _{C^{2,\alpha }(\bar{\Omega })}\le C_3\{\Vert {\mathbf{j}}+\mathbf{{h}}\Vert _{C^{1,\alpha }(\bar{\Omega })}+\Vert \phi \Vert _{C^0(\bar{\Omega })}\}\le C_4\Vert {\mathbf{j}}+\mathbf{{h}}\Vert _{C^{1,\alpha }(\bar{\Omega })}, \end{aligned}$$

(B.20)

where $C_3$ and $C_4$ depend on $\Omega$, $\alpha$, $\lambda (M)$, $\Lambda (M)$, $\Vert M\Vert _{C^1(\bar{\Omega })}$. Going back to (B.17) and using (A.9) we find $\mathbf{{u}}\in C^{2,\alpha }(\bar{\Omega },\mathbb {R}^3)$, and using (A.9), (B.14), (B.16) and (B.20) we find

$$\begin{aligned} \Vert \mathbf{{u}}\Vert _{C^{2,\alpha }(\bar{\Omega })}\le & {} C(\Omega ,\alpha )\{\Vert M\Vert _{C^{1,\alpha }(\bar{\Omega })}\Vert {\mathbf{j}}+\mathbf{{h}}+\nabla \phi \Vert _{C^{1,\alpha }(\bar{\Omega })}+\Vert \mathbf{{u}}\Vert _{L^2(\Omega )}\}\\ {}\le & {} C_5\{\Vert \mathbf{{J}}_0\Vert _{C^\alpha (\bar{\Omega })}+\Vert \mathbf{{u}}\Vert _{H^1(\Omega )}\}, \end{aligned}$$

where $C_5$ depends on $\Omega$, $\alpha$, $\lambda (M)$, $\Lambda (M)$, $\Vert M\Vert _{C^{1,\alpha }(\bar{\Omega })}$, $\Vert {\mathcal {A}}\Vert _{C^0(\bar{\Omega })}$. From this and (B.12) we get (B.11) with the constant C depending on $\Omega$, $\alpha$, $\lambda (M)$, $\Lambda (M)$, $\Vert M\Vert _{C^{1,\alpha }(\bar{\Omega })}$, $\Vert {\mathcal {A}}\Vert _{C^0(\bar{\Omega })}$.

If $\mathbf{{u}}\in \mathbb {H}_2(\Omega )^\perp _{L^2(\Omega )}$, since we have assumed $\mathbf{{u}}_T=\mathbf{{u}}_T^0=0$, using (B.13) instead of (B.12) we find

$$\begin{aligned} \Vert \mathbf{{u}}\Vert _{C^{2,\alpha }(\bar{\Omega })}\le C\Vert \mathbf{{J}}_0\Vert _{C^\alpha (\bar{\Omega })}. \end{aligned}$$

$\square$

Appendix C: The projections ${\mathcal {P}}_0$ and ${\mathcal {P}}_1$

For $\mathbf{{J}}\in L^2(\Omega ,\mathbb {R}^3)$, let

$$\begin{aligned} {\mathcal {P}}_0(\mathbf{{J}})=\mathbf{{J}}+\nabla p, \end{aligned}$$

where $p\in H^1_0(\Omega )$ is a solution of (1.9). ${\mathcal {P}}_0(\mathbf{{J}})$ and $\nabla p$ are orthogonal in $L^2(\Omega ,\mathbb {R}^3)$, hence ${\mathcal {P}}_0: L^2(\Omega ,\mathbb {R}^3)\rightarrow \mathcal H(\Omega ,\mathrm{div\,}0)$ is an orthonormal projection (see the second equality in (A.1)). By the classical elliptic estimates (see for instance [17, 19]) we have the following

Lemma C.1

Let $\Omega$ be a bounded domain in $\mathbb {R}^3$ with a Lipschitzian boundary. Then ${\mathcal {P}}_0: L^2(\Omega ,\mathbb {R}^3)\rightarrow \mathcal H(\Omega ,\mathrm{div\,}0)$ is an orthogonal project and it is surjective.

• (i) If $\partial \Omega$ is of $C^{k+1}$ with $k\ge 1$, then ${\mathcal {P}}_0: H^{k}(\Omega ,\mathbb {R}^3)\rightarrow H^{k}(\Omega ,\mathrm{div\,}0)$ is surjective and

  $$\begin{aligned} \Vert {\mathcal {P}}_0(\mathbf{{w}})\Vert _{H^{k}(\Omega )}\le C(\Omega ,k)\{\Vert \mathrm{curl\,}\mathbf{{w}}\Vert _{H^{k-1}(\Omega )}+\Vert {\mathcal {P}}_0(\mathbf{{w}})\Vert _{L^2(\Omega )}+\Vert \mathbf{{w}}_T\Vert _{H^{k-1/2}(\partial \Omega )}\}.\nonumber \\ \end{aligned}$$

  (C.1)

• (ii) If $\partial \Omega$ is of $C^{k+1,\alpha }$ with $k\ge 1$ and $0<\alpha <1$, then ${\mathcal {P}}_0: C^{k,\alpha }(\bar{\Omega },\mathbb {R}^3)\rightarrow C^{k,\alpha }(\Omega ,\mathrm{div\,}0)$ is surjective, and

  $$\begin{aligned} \Vert {\mathcal {P}}_0(\mathbf{{w}})\Vert _{C^{k,\alpha }(\bar{\Omega })}\le C(\Omega ,k,\alpha )\{\Vert \mathrm{curl\,}\mathbf{{w}}\Vert _{C^{k-1,\alpha }(\Omega )}+\Vert {\mathcal {P}}_0(\mathbf{{w}})\Vert _{L^2(\Omega )}+\Vert \mathbf{{w}}_T\Vert _{C^{k,\alpha }(\partial \Omega )}\}.\nonumber \\ \end{aligned}$$

  (C.2)

The term $\Vert {\mathcal {P}}_0(\mathbf{{w}})\Vert _{L^2(\Omega )}$ in the right side of (C.1) and (C.2) can be replaced by $\Vert \mathbf{{w}}\Vert _{L^2(\Omega )}$, and if $\Omega$ has no holes then this term can be removed.

Now we consider the projection ${\mathcal {P}}_1$. Let $\Omega$ satisfy $(O_1)$. For $\mathbf{{J}}\in {\mathcal {H}}(\Omega ,\mathrm{div\,})$, let $c(\mathbf{{J}})$ be the piece-wise constant function on $\partial \Omega$ defined by (1.11). Denote

$$\begin{aligned}&\Vert \mathbf{{J}}\Vert _{{\mathcal {H}}(\Omega ,\mathrm{div\,})}=\left( \Vert \mathrm{div\,}\mathbf{{J}}\Vert _{L^2(\Omega )}^2+\Vert \mathbf{{J}}\Vert _{L^2(\Omega )}^2\right) ^{1/2},\\&\Vert c(\mathbf{{J}})\Vert _1=\sum _{i=1}^{m+1}|c_i(\mathbf{{J}})||\Gamma _i|,\qquad \Vert c(\mathbf{{J}})\Vert _\infty =\mathrm{max}_{1\le i\le m+1}|c_i(\mathbf{{J}})|. \end{aligned}$$

We have

$$\begin{aligned} \Vert c(\mathbf{{J}})\Vert _q\le C(\Omega )\Vert \nu \cdot \mathbf{{J}}\Vert _{H^{-1/2}(\partial \Omega )}\le C(\Omega )\Vert \mathbf{{J}}\Vert _{{\mathcal {H}}(\Omega ,\mathrm{div\,})} \end{aligned}$$

for $q=1,\infty$.

Lemma C.2

Assume $\Omega$ is a bounded domain in $\mathbb {R}^3$ satisfying $(O_1)$ with $r\ge 2$. Given $\mathbf{{J}}\in {\mathcal {H}}(\Omega ,\mathrm{div\,})$, (1.10) has a unique solution p, and

$$\begin{aligned}&\Vert p\Vert _{H^2(\Omega )}\le C(\Omega )\Vert \mathbf{{J}}\Vert _{{\mathcal {H}}(\Omega ,\mathrm{div\,})},\nonumber \\&\Vert p\Vert _{H^k(\Omega )} \le C(\Omega ,k)\left\{ \Vert \mathrm{div\,}\mathbf{{J}}\Vert _{H^{k-2}(\Omega )}+\Vert \mathbf{{J}}\Vert _{{\mathcal {H}}(\Omega ,\mathrm{div\,})}\right\} ,\qquad \quad 3\le k\le r,\nonumber \\&\Vert p\Vert _{C^{1,\alpha }(\bar{\Omega })}\le C(\Omega ,\alpha )\left\{ \Vert \mathbf{{J}}\Vert _{C^\alpha (\bar{\Omega })}+\Vert \mathbf{{J}}\Vert _{{\mathcal {H}}(\Omega ,\mathrm{div\,})}\right\} ,\nonumber \\&\Vert p\Vert _{C^{k,\alpha }(\bar{\Omega })}\le C(\Omega ,k,\alpha )\left\{ \Vert \mathrm{div\,}\mathbf{{J}}\Vert _{C^{k-2+\alpha }(\bar{\Omega })}+\Vert \mathbf{{J}}\Vert _{{\mathcal {H}}(\Omega ,\mathrm{div\,})}\right\} ,\quad 2\le k\le r. \end{aligned}$$

(C.3)

If $\mathrm{div\,}\mathbf{{J}}=0$, then for $1\le k\le r$ and $0<\alpha <1$ we have

$$\begin{aligned} \Vert p\Vert _{H^k(\Omega )}\le C(\Omega ,k)\Vert c(\mathbf{{J}})\Vert _\infty ,\qquad \Vert p\Vert _{C^{k,\alpha }(\bar{\Omega })}\le C(\Omega ,\alpha )\Vert c(\mathbf{{J}})\Vert _\infty . \end{aligned}$$

Proof

(1.10) has a solution p because

$$\begin{aligned} \int _{\partial \Omega }c(\mathbf{{J}})(x)dS=\int _{\Omega }\mathrm{div\,}\mathbf{{J}}dx, \end{aligned}$$

and the solution is unique. Multiplying the equation (1.10) by p and integrating we have

$$\begin{aligned} \int _\Omega |\nabla p|^2dx= & {} -\int _\Omega \mathbf{{J}}\cdot \nabla p dx+\int _{\partial \Omega }(\nu \cdot \mathbf{{J}}-c(\mathbf{{J}}))p\;dS\\\le & {} \Vert \mathbf{{J}}\Vert _{L^2(\Omega )}\Vert \nabla p\Vert _{L^2(\Omega )}+\Vert \nu \cdot \mathbf{{J}}-c(\mathbf{{J}})\Vert _{H^{-1/2}(\partial \Omega )}\Vert p\Vert _{H^{1/2}(\partial \Omega )}. \end{aligned}$$

Since the integral of p vanishes, we have $\Vert p\Vert _{H^{1/2}(\partial \Omega )}\le C(\Omega )\Vert \nabla p\Vert _{L^2(\Omega )}$. So we find

$$\begin{aligned} \Vert \nabla p\Vert _{L^2(\Omega )}\le \Vert \mathbf{{J}}\Vert _{L^2(\Omega )}+C(\Omega )\Vert \nu \cdot \mathbf{{J}}-c(\mathbf{{J}})\Vert _{H^{-1/2}(\partial \Omega )} \le C(\Omega )\Vert \mathbf{{J}}\Vert _{{\mathcal {H}}(\Omega ,\mathrm{div\,})}. \end{aligned}$$

(C.4)

From the $H^k$ estimates of Neumann problem (see for instance [19, Ch.2], [28, Ch.5], also see [18, Theorem I.1.10]) and using (C.4) we see that, for $2\le k\le r$,

$$\begin{aligned} \Vert p\Vert _{H^k(\Omega )}\le & {} C(\Omega ,k)\{\Vert \mathrm{div\,}\mathbf{{J}}\Vert _{H^{k-2}(\Omega )}+\Vert p\Vert _{L^2(\Omega )}+\Vert c(\mathbf{{J}})\Vert _{H^{k-1/2}(\partial \Omega )}\}\\\le & {} C(\Omega ,k)\{\Vert \mathrm{div\,}\mathbf{{J}}\Vert _{H^{k-2}(\Omega )}+\Vert \mathbf{{J}}\Vert _{{\mathcal {H}}(\Omega ,\mathrm{div\,})}+\Vert c(\mathbf{{J}})\Vert _\infty \}. \end{aligned}$$

The proof of the other inequalities is standard. $\square$

For $\mathbf{{J}}\in {\mathcal {H}}(\Omega ,\mathrm{div\,})$ we define

$$\begin{aligned} {\mathcal {P}}_1(\mathbf{{J}})=\mathbf{{J}}+\nabla p, \end{aligned}$$

where p is the solution of (1.10) associated with $\mathbf{{J}}$. Then ${\mathcal {P}}_1$ is a projection from ${\mathcal {H}}(\Omega ,\mathrm{div\,})$ onto ${\mathcal {H}}^\Gamma (\Omega ,\mathrm{div\,}0)$. ${\mathcal {P}}_1$ is not an orthogonal projection as ${\mathcal {P}}_1(\mathbf{{J}})$ is not orthogonal to $\nabla p$ with respect to the $L^2$ inner product. From Lemma C.2 we get the following estimates of ${\mathcal {P}}_1(\mathbf{{J}})$.

Lemma C.3

Let $\Omega$ be a bounded domain in $\mathbb {R}^3$ satisfying $(O_1)$ with $r\ge 2$. Then ${\mathcal {P}}_1$ is a projection from ${\mathcal {H}}(\Omega ,\mathrm{div\,})$ onto ${\mathcal {H}}^\Gamma (\Omega ,\mathrm{div\,}0)$, and

$$\begin{aligned} \Vert {\mathcal {P}}_1(\mathbf{{J}})\Vert _{L^2(\Omega )}\le C(\Omega )\Vert \mathbf{{J}}\Vert _{{\mathcal {H}}(\Omega ,\mathrm{div\,})}. \end{aligned}$$

If $1\le k\le r-1$ and $\mathbf{{J}}\in H^k(\Omega ,\mathbb {R}^3)$, then ${\mathcal {P}}_1(\mathbf{{J}})\in H^k(\Omega ,\mathbb {R}^3)$ and

$$\begin{aligned} \Vert {\mathcal {P}}_1(\mathbf{{J}})\Vert _{H^k(\Omega )}\le C(\Omega ,k)\Vert \mathbf{{J}}\Vert _{H^k(\Omega )}. \end{aligned}$$

If $0\le k\le r-1$, $0<\alpha <1$ and $\mathbf{{J}}\in C^{k,\alpha }(\bar{\Omega },\mathbb {R}^3)$, then ${\mathcal {P}}_1(\mathbf{{J}})\in C^{k,\alpha }(\bar{\Omega },\mathbb {R}^3)$ and

$$\begin{aligned}&\Vert {\mathcal {P}}_1(\mathbf{{J}})\Vert _{C^\alpha (\bar{\Omega })}\le C(\Omega ,\alpha )\left( \Vert \mathbf{{J}}\Vert _{C^\alpha (\bar{\Omega })}+\Vert \mathrm{div\,}\mathbf{{J}}\Vert _{L^2(\Omega )}\right) ,\\&\Vert {\mathcal {P}}_1(\mathbf{{J}})\Vert _{C^{k,\alpha }(\bar{\Omega })}\le C(\Omega ,k,\alpha )\Vert \mathbf{{J}}\Vert _{C^{k,\alpha }(\bar{\Omega })},\quad 1\le k\le r-1. \end{aligned}$$

Appendix D: List of assumptions

$(A_0)$: Section 3.

$(B_2)$: Subsection 2.3.

(F): Section 3.

$(f_1)$, $(f_2)$: Subsection 4.2. $(f_2')$: Subsection 6.1. $(f_3)$, $(f_4)$: Subsection 4.3.

$(H_1)$, $(H_2)$, $(H_3)$: Subsection 2.3. $(H_4)$: Subsection 2.5. $(H_5)$: Subsection 5.2.

$(O_1)$, $(O_2)$: Subsection 2.1.

(P): Section 3.

$(U_1)$, $(U_2)$: Subsection 4.3.