A Characterization of Harmonic $L^r$-Vector Fields in Three Dimensional Exterior Domains

Abstract

Consider the space of harmonic vector fields u in $L^r(\Omega )$ for $1<r<\infty$ for three dimensional exterior domains $\Omega$ with smooth boundaries $\partial \Omega$ subject to the boundary conditions $u\cdot \nu =0$ or $u\times \nu =0$, where $\nu$ denotes the unit outward normal on $\partial \Omega$. Denoting these spaces by $X^r_{\tiny {\text{ har }}}(\Omega )$ and $V^r_{\tiny {\text{ har }}}(\Omega )$, it is shown that, in spite of the lack of compactness of $\Omega$, both of these spaces are finite dimensional and that $\dim V^r_{\tiny {\text{ har }}}(\Omega )$ equals L for $3/2<r<\infty$ and $L-1$ for $1<r\le 3/2$. Here L is a number representing topologically invariant quantities of $\partial \Omega$ and which in the case of bounded domains coincides with the first Betti number. In contrast to the situation of bounded domains, the dimension of $V^r_{\tiny {\text{ har }}}(\Omega )$ in exterior domains is depending on the Lebesgue exponent r. The critical value of this exponent for exterior domains is determined to be 3/2.

Appendix

Our argument for characterizing the dimension of $X^r_{{\mathrm{har}}}(\Omega )$ in Theorem 2.2 is based on the weak Neumann problem (5.7). In order to solve this problem, we need to guarantee the existence of a family $\{f_{l}^{(j)}\}_{1 \le l \le N(j)}^{1\le j \le L}$ of functions satisfying

$$\begin{aligned} f^{(j)}_{l} \in C^\infty \big (\dot{\Omega }\big ) \cap L^1_{loc}(\Omega ) \quad \text{ and } \quad \text{ supp } \,f^{(j)}_l\subset \Omega \cap B_R(0) \quad \text{ for } \text{ some } \, R>0, \end{aligned}$$

(6.1)

as well as the boundary and jump conditions

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \quad \frac{\partial f^{(j)}_{l}}{\partial \nu } &{} = \;\;0 \quad \text{ on } \;\;\partial \Omega , \\ \displaystyle \Big [\frac{\,\,\partial ^s\!f^{(j)}_{l}\,}{\partial \nu ^s_{k, m}}\Big ]_{\Sigma ^{(k)}_m} &{}= \;\;0, \quad s=1,2,\dots , \\ \displaystyle \quad f^{(j)}_{l}\Big |_{\Sigma ^{(k),+}_m} &{} = \;\;\delta ^j_k\delta ^{l}_{m}, \\ \displaystyle \quad f^{(j)}_{l}\Big |_{\Sigma ^{(k),-}_m} &{} = 0, &{} \\ \end{array} \right. \end{aligned}$$

(6.2)

for all $j,k \in \{1,\ldots ,L\}$, $l \in \{1,\ldots ,N(j)\}$ and $m \in \{1,\ldots ,N(k)\}$. In the following, we show that such a family of functions exists provided the underlying domain $\Omega$ fulfills the Assumption (A).

Main steps of the construction of a family $\{f_{l}^{(j)}\}_{1 \le l \le N(j)}^{1\le j \le L}$ of functions satisfying (6.1) and (6.2) can be summarized as follows:

• (i) We separate each smooth cut-surface $\Sigma ^{(j)}_l$ by taking an open set $U^{(j)}_l$ which is diffeomorphic to an open ball with $\overline{\Sigma ^{(j)}_l}\subset U^{(j)}_l$;

• (ii) Inside each open set $U^{(j)}_l$, we parametrize a part of the boundary $\Gamma _j\cup \Sigma ^{(j)}_l$ and introduce the Hanzawa transformation in a neighborhood of $\Gamma _j$ apart from the boundary $\partial \Sigma ^{(j)}_l$ where $\Sigma ^{(j)}_l$ intersects transversally $\Gamma _j$;

• (iii) We then construct the function $f^{(j)}_l$ which is smooth in $\Omega \cap U^{(j)}_l$ except for the cut-surface $\Sigma ^{(j)}_l$ with the compact support in $\overline{(\Omega \cap U^{(j)}_l)_+}$, and which takes the value 1 on a set near the side of $\Sigma ^{(j),+}_l$. (As for definition of the upper component $(\Omega \cap U^{(j)}_l)_+$, see the proof of Proposition 6.1 below.)

In above procedure, the Hanzawa transformation plays an essential role for ensuring that $f^{(j)}_l$ satisfies the first boundary condition in (6.2) with keeping its smoothness in $(\Omega \cap U^{(j)}_l)_+$.

Proposition 6.1

Let $\Omega \subset \mathbb {R}^3$ be an exterior domain with smooth boundary $\partial \Omega$ satisfying Assumption (A). Then, for every $j \in \{1,\ldots ,L\}$, there exist N(j) functions $f_{1}^{(j)},\ldots , f_{N(j)}^{(j)}$ satisfying (6.1) and (6.2).

Proof

First, we consider the case where $L=1$ and $N(1) =1$. In this case, in order to simply the notation, we write $\Gamma := \Gamma _1$ and $\Sigma := \Sigma ^1_{N(1)}$. By taking the smooth cut-surface $\Sigma$ appropriately and by rotating the coordinates axis of $\mathbb {R}^3$, we may choose an open set U diffeomorphic to an open ball with $\Sigma \subset U$ in such a way that there exist open disks $D_0, D \subset \mathbb {R}^2$ with $\overline{D_0}\subset D$ and a function $h \in C^{0, 1}(D)\cap C^{\infty }(D\setminus \partial D_0)$ with the following properties;

$$\begin{aligned} \big (\Gamma \cup \Sigma \big ) \cap U&= \{x=(x',x_3) \in \mathbb {R}^3; x_3 = h(x'), x':=(x_1, x_2) \in D\}, \\ \Sigma&= \{x = (x', x_3)\in \mathbb {R}^3; x_3 = h(x'), x' \in D_0\}. \end{aligned}$$

Let us define $\big (\Omega \cap U \big )_+$ by

$$\begin{aligned} \big (\Omega \cap U \big )_+ := \{x=(x',x_3); x_3 > h(x'), x'\in D\}\cap (\Omega \cap U). \end{aligned}$$

Here we regard the cut $\Sigma$ as a surface having the upper side $\Sigma ^+$ and the lower side $\Sigma ^-$. Then the open set $\Omega \cap U$ is devided into two components by the surface $\Gamma \cup \Sigma$ so that one component includes $\Sigma ^+$ as a part of the boundary and so that the other component includes $\Sigma ^-$ as a part of the boundary. Notice that set $\big (\Omega \cap U \big )_+$ defined as above coincides with its upper component including $\Sigma ^+$ as a part of the boundary.

We denote by $\nu '$ the unit normal to $\Sigma$ with the direction from $\Sigma ^-$ to $\Sigma ^+$ and define $\tilde{\nu }$ by

$$\begin{aligned} \tilde{\nu }= \left\{ \begin{array}{ll} \nu \;\;&{}\text{ on } \;\;\Gamma , \\ \nu ' \;\;&{}\text{ on } \;\;\Sigma . \end{array} \right. \end{aligned}$$

Since $\overline{D_0} \subset D$, we may choose further a disk $D_1 \subset \mathbb {R}^2$ in such a way that $\overline{D_0} \subset D_1$ and $\overline{D_1}\subset D$. Since $h \in C^{\infty }(D\setminus \overline{D_1})$, the normal $\tilde{\nu }$ on the surface $\{(x', x_3)\in \mathbb {R}^3; x_3 = h(x'), x'\in D\setminus \overline{D_1}\}$ can be expressed as

$$\begin{aligned} \tilde{\nu }(x') = \frac{1}{\sqrt{|\nabla ' h|^2 + 1}}\; \big (-\nabla ' h, 1\big ), \;\;x' \in D\setminus \overline{D_1}, \end{aligned}$$

where $\nabla ' h:= \Big (\frac{\partial h}{\partial x_1},\frac{\partial h}{\partial x_2}\Big )$. Following the strategy known as the Hanzawa transformation(see, e.g.,[6]), we consider the mapping F defined by

$$\begin{aligned} F(\omega ', \lambda ):= \big (\omega ',h(\omega ')\big ) + \lambda \; \tilde{\nu }(\omega '), \quad \omega ':=(\omega _1,\omega _2) \in D\setminus \overline{D_1}, \,\, \lambda \ge 0. \end{aligned}$$

(6.3)

Choosing $\Lambda >0$ sufficiently small, we see that $F: (D\setminus \overline{D_1}) \times [0, \Lambda ] \rightarrow \overline{(\Omega \cap U)_+}$ is a $C^{\infty }$-mapping. Since the Jacobian $\frac{\partial F(\omega ',\lambda )}{\partial (\omega ',\lambda )}$ at $\omega ' \in D\setminus \overline{D_1}$ and $\lambda = 0$ satisfies

$$\begin{aligned} \Big .\frac{\partial F(\omega ', \lambda )}{\partial (\omega ', \lambda )}\Big |_{\omega ' \in D\setminus \overline{D_1},\lambda =0}= \sqrt{|\nabla ' h|^2 + 1} \ne 0, \quad \omega ' \in D\setminus \overline{D_1}, \end{aligned}$$

there exist $\lambda _0 \in (0,\Lambda )$ and a $C^{\infty }$-mapping $\Psi : V(\lambda _0) \rightarrow (D\setminus \overline{D_1})\times [0, \lambda _0]$ such that

$$\begin{aligned} F\circ \Psi = \text{ id}_{V(\lambda _0)}, \quad \Psi \circ F = \text{ id}_{(D\setminus \overline{D_1})\times [0, \lambda _0]} \end{aligned}$$

with $V(\lambda _0)\subset \overline{(\Omega \cap U)_+}$, where

$$\begin{aligned} V(\lambda _*) := \{x = (\omega ', h(\omega ')) + \lambda \tilde{\nu }(\omega ') \in \mathbb {R}^3: \omega ' \in D\setminus \overline{D_1}, \lambda \in [0, \lambda _*]\}, \quad \lambda _*>0. \end{aligned}$$

Notice that, in view of (6.3), the function $\Psi = F^{-1}$ may be expressed as

$$\begin{aligned} \Psi (x) = \big (\omega '(x), \lambda (x)\big ), \quad x \in V(\lambda _0), \end{aligned}$$

and that every $f \in C^1(\overline{\Omega })$ satisfies

$$\begin{aligned} \Big .\frac{\partial }{\partial \lambda }f\big (\Psi ^{-1}(\omega ', \lambda )\big )\Big |_{\lambda =0} = \frac{\partial f}{\partial \tilde{\nu }}\; \big (x', h(x')\big ), \quad \big (x',h(x')\big )\in \Gamma \text{ with } x'\in D\setminus \overline{D_1}. \end{aligned}$$

(6.4)

We choose a $C^\infty$-extension $\tilde{F}(\omega ',\lambda _0 )$ on $\omega ' \in D$ of $F(\omega ',\lambda _0)$ on $\omega ' \in D\setminus \overline{D_1}$, and define a $C^\infty$-surface $S_1$ by

$$\begin{aligned} S_1 := \{\tilde{F}(\omega ', \lambda _0)\in \mathbb {R}^3; \omega ' \in \overline{D_1} \}. \end{aligned}$$

It is easy to see that such an extension $\tilde{F}(\omega ',\lambda _0 )$ can be taken to satisfy that $S_1\cap (\Gamma \cup \Sigma ) = \emptyset$ and $\overline{S_1}\subset (\Omega \cap U)_+$. In addition, we define two surfaces $S_2$ and $S_3$ by

$$\begin{aligned}&S_2 := \{x = F(\omega ', \lambda )\in \mathbb {R}^3; \omega ' \in \partial D_1, \lambda \in [0, \lambda _0]\}, \\&S_3 := \{ x =(x',h(x'))\in \mathbb {R}^3; x'\in \overline{D_1}\}. \end{aligned}$$

Denote by W a closed region surrounded by $S_1$, $S_2$ and $S_3$. Since $\overline{S_1} \subset (\Omega \cap U)_+$, we see that $W \subset \overline{(\Omega \cap U)_+}$.

Now we construct a function f such that $f \in C^\infty (\dot{\Omega }) \cap L^1_{loc}(\Omega )$ with the supp $f\subset B_R(0)$ for some $R>0$ satisfying the boundary and jump conditions

$$\begin{aligned} \displaystyle \left\{ \begin{array}{lll} \frac{\partial f}{\partial \nu } &{} = \;\;0 \quad \text{ on } \;\;\partial \Omega , \\ \Big [\frac{\partial ^s f}{\partial \nu '^s}\Big ]_{\Sigma } &{}= \;\;0, \quad s=1,2,\dots ,\\ f|_{\Sigma ^+} &{}= \;\;1,\\ f|_{\Sigma ^-} &{}= \;\;0.\\ \end{array} \right. \end{aligned}$$

(6.5)

Since $D_1$ and D are open disks in $\mathbb {R}^2$ with $\overline{D_1} \subset D$, we see that $d_1:=\text{ dist }(\partial D_1, \partial D) > 0$. Setting $\varphi \in C^\infty (D\setminus \overline{D_1})$ by

$$\begin{aligned} \varphi (\omega ') := \left\{ \begin{array}{ll} 1, \;\;&{}\omega '\in (D\setminus \overline{D_1}) \cap S_{\frac{d_1}{2}}(\partial D_1), \\ 0, \;\;&{}\omega '\in (D\setminus \overline{D_1}) \cap S_{\frac{2d_1}{3}}(\partial D_1)^c, \end{array} \right. \end{aligned}$$

where $S_d(\partial D_1):= \{\omega ' \in \mathbb {R}^2: \text{ dist }(\omega ', \partial D_1) < d\}$, we first define $f_1\in C^{\infty }\big (W\cup V(\lambda _0)\big )$ by

$$\begin{aligned} f_1(x) := \left\{ \begin{array}{ll} 1, \;\;&{}x \in W, \\ \varphi \big (\omega '(x)\big ), \;\;&{}x \in V(\lambda _0). \end{array} \right. \end{aligned}$$

(6.6)

Since $\overline{\Sigma } \subset S_3 \subset W$ and since $f_1 \equiv 1$ on W, we see that

$$\begin{aligned} f_1|_{\Sigma ^+} =1\,\,\, \text{ and }\,\,\, \frac{\partial ^s f_1}{\partial \tilde{\nu }^s}{\big |_{S_3}} = 0, \,\,s=1,2,\dots . \end{aligned}$$

(6.7)

Since $f_1(x)=f_1(\Psi ^{-1}(\omega '(x), \lambda (x)))=\varphi (\omega '(x))$ is constant along $\lambda (x)$ for $x \in V(\lambda _0)$, it follows from (6.4) that

$$\begin{aligned} \frac{\partial ^s f_1}{\partial {\tilde{\nu }}^s} \big |_{\tilde{S}_3} =0,\quad s=1,2,\dots , \end{aligned}$$

where $\tilde{S}_3 =\{ x =(x',h(x'))\in \mathbb {R}^3; x'\in D\setminus \overline{D_1}\}$. Since $(\Gamma \cup \Sigma )\cap U = S_3 \cup \tilde{S}_3$, we obtain

$$\begin{aligned} \Big .\frac{\partial ^s f_1}{\partial {\tilde{\nu }}^s}\Big |_{(\Gamma \cup \Sigma )\cap U} =0,\quad s=1,2,\dots . \end{aligned}$$

(6.8)

By (6.6) and the definition of $\varphi (\omega ')$, we also have

$$\begin{aligned} f_1(x)=0, \,\,\text{ for } \,\,x = F(\omega ', \lambda ) \text{ with } \omega ' \in \partial D, \lambda \in [0, \lambda _0]. \end{aligned}$$

(6.9)

Furthermore, we see the fact that $f_1$ is of class $C^\infty$ on the $C^\infty$-surface $S_1 \cup \tilde{S}_1$ with $\tilde{S}_1=\{\,x=F(\omega ', \lambda _0)\in \mathbb {R}^3\,|\, \omega '\in D\setminus \overline{D_1}\,\}$. Since

$$\begin{aligned} \partial (W\cup V(\lambda _0))=(S_1 \cup \tilde{S}_1) \cup \{x=F(\omega ',\lambda )\in \mathbb {R}^3 \,|\, \omega '\in \partial D, \, \lambda \in [0,\lambda _0]\,\}\cup (S_3 \cup \tilde{S}_3 ), \end{aligned}$$

we see by (6.9) that the function $f_1$ defined on $W\cup V(\lambda _0)$ has a $C^\infty$-extension $f_2$ on $\overline{(\Omega \cap U)_+}$ with $\text{ supp } f_2\subset \overline{(\Omega \cap U)_+}$. We next define the extension f on $\Omega$ of $f_2$ with its value 0 outside of $\overline{(\Omega \cap U)_+}$. Clearly, such an f satisfies that $f|_{\Sigma ^-}=0$, $f\in C^\infty (\dot{\Omega })\cap L^1_{loc}(\Omega )$ and that $\text{ supp } f\subset \overline{(\Omega \cap U)_+}\subset \Omega \cap B_R(0)$ for some $R>0$. In addition, by (6.7) and (6.8), we conclude that f fulfills all conditions of (6.5). Hence this f is a desired function in the case of $L=1$ and $N(1)=1$.

Finally, we turn our attention to the general case of $L>1$. By Assumption (A), for each fixed $j=1,\dots , L$ and $l=1,\dots , N(j)$, we can choose an open set $U^{(j)}_l \subset \mathbb {R}^3$ diffeomorphic to an open ball in such a way that $\overline{\Sigma ^{(j)}_l}\subset U^{(j)}_l$, and in such a way that the other $\Sigma ^{(k)}_m$ for all $1\le m \le N(k)$ with $m\ne l$ when $k=j$, and for all $1\le m\le N(k)$ when $k\ne j$, lie outside of $U^{(j)}_l$. In addition, without loss of generality we may assume that $U^{(j)}_l$ satisfies that $\partial U^{(j)}_l \cap \Gamma _k = \emptyset$ for all $k\ne j$ . By the similar way to the case of $L=1$, we may find open disks $D_0, \,D\subset \mathbb {R}^2$ with $\overline{D_0}\subset D$, and a function $h^{(j)}_l \in C^{0,1}(D)\cap C^\infty (D\setminus \partial D_0)$ such that

$$\begin{aligned} \big (\Gamma _j \cup \Sigma ^{(j)}_l \big ) \cap U^{(j)}_l&= \{x=(x',x_3) \in \mathbb {R}^3; x_3 =h^{(j)}_l (x'), x':=(x_1, x_2) \in D\}, \\ \Sigma ^{(j)}_l&= \{x = (x', x_3)\in \mathbb {R}^3; x_3 = h^{(j)}_l (x'), x' \in D_0\}. \end{aligned}$$

Let us define $\big (\Omega \cap U^{(j)}_l \big )_+$ by

$$\begin{aligned} \big (\Omega \cap U^{(j)}_l \big )_+:= \{x=(x',x_3); x_3 > h^{(j)}_l (x'), x'\in D\}\cap (\Omega \cap U^{(j)}_l ). \end{aligned}$$

Notice that $\big (\Omega \cap U^{(j)}_l \big )_+$ is the upper component of the set $\Omega \cap U^{(j)}_l$ devided by $\Gamma _j\cup \Sigma ^{(j)}_l$ including $\Sigma ^{(j),+}_l$ as a part of the boundary. Then, we can proceed in each $U^{(j)}_l$ in the same way as the proof in the case when $L=1$ to obtain the desired function $f^{(j)}_l$. The proof of Proposition 6.1 is now complete.