It is known that in 3D exterior domains Ω with the compact smooth boundary $\partial \Omega$, two spaces $X^{r}_{har}\left ( \Omega \right )$ and $V^{r}_{har}\left ( \Omega \right )$ of $L^{r}$-harmonic vector fields $h$ with $h\cdot v\mid _{\partial \Omega }= 0$ and $h\times v\mid _{\partial \Omega }= 0$ are both of finite dimensions, where $v$ denotes the unit outward normal to $\partial \Omega$. We prove that for every $L^{r}$-vector field $u$, there exist $h\in X^{r}_{har}\left ( \Omega \right )$, $w\in H^{1,r}\left ( \Omega \right )^{3}$ with div $w= 0$ and $p\in H^{1,r}\left ( \Omega \right )$ such that $u$ is uniquely decomposed as $u= h$ + rot $w$ + $\bigtriangledown p$.
On the other hand, if for the given $L^{r}$-vector field $u$ we choose its harmonic part $h$ from $V^{r}_{har}\left ( \Omega \right )$, then we have a similar decomposition to above, while the unique expression of $u$ holds only for $1< r< 3$. Furthermore, the choice of $p$ in $H^{1,r}\left ( \Omega \right )$ is determined in accordance with the threshold $r= 3/2$.
Our result is based on the joint work with Matthias Hieber, Anton Seyferd (TU Darmstadt), Senjo Shimizu (Kyoto Univ.) and Taku Yanagisawa (Nara Women Univ.).
