Suppose $B=B(y,r)$ is ball in $\mathbb{R}^m$ ($m\geq2$), and $u$ a superharmonic function on a neighborhood of the closure $\overline{B}$ of $B$. We know that the Kelvin transform of $u$ with respect to the sphere $S(y,r)$, given by $$u^*(x^*)=\Big(\frac{r}{|x^*-y|}\Big)^{m-2}u(x)$$ and $$x^*-y=\Big(\frac{r}{|x^*-y|}\Big)^{2}(x-y),$$ is superharmonic on $\mathbb{R}^m\setminus \overline{B}.$ Is it possible to extend $u^*$ to a function $\overline{u}$ that is superharmonic everywhere coinciding with $u^*$ outside $B$?

Suppose $B=B(y,r)$ is ball in $\mathbb{R}^m$ ($m\geq2$), and $u$ a superharmonic function on a neighborhood of the closure $\overline{B}$ of $B$. We know that the Kelvin transform of $u$ with respect to the sphere $S(y,r)$, given by $$u^*(x^*)=\Big(\frac{r}{|x^*-y|}\Big)^{m-2}u(x)$$ and $$x^*-y=\Big(\frac{r}{|x-y|}\Big)^{2}(x-y),$$ is superharmonic on $\mathbb{R}^m\setminus \overline{B}.$ Is it possible to extend $u^*$ to a function $\overline{u}$ that is superharmonic everywhere coinciding with $u^*$ outside $B$?