# Extension of the Kelvin transform

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 $$B$$?