Let $B(0,r)$ denote the ball of radius $r$ centered at the origin in $\mathbb{R}^d$, $d\geq 2$.
Let $0<r_0<1$. Suppose I have a real-valued function $v$ that is harmonic in the annulus $\{r_0<|x|<1\}$ that vanishes on $\partial B(0,1)$. Now let $\tilde{v}$ be an extension of $v$ to $\{r_0<|x|<1/r_0\}$ defined by $\tilde{v}(x) = -|x|^{2-d}v(|x|^{-2}x)$ for $1<|x|<1/r_0$. It can be shown that $\tilde{v}$ is $C^1$ on $\{r_0<|x|<1/r_0\}$.
In a paper that I am reading (Lemma 1), the author seems to makes the claim that from the above, it follows that $\tilde{v}$ is harmonic on $\{r_0<|x|<1/r_0\}$. My question is, what is the general fact that is being used here? Is it true that a $C^1$ extension of a harmonic function is harmonic? My sense here is that this has something to do with the fact that (real) harmonic functions are (real) analytic, but I can't seem to find any reference on this.
$\tilde v$ is the Kelvin transform of $v$, not just some regular extension. The Kelvin transform of a harmonic function (in this case with respect to the unit ball) is always harmonic in the complementary domain (interior domains become exterior domains etc. Just find the Laplacian of $\tilde v$ and use the fact that $v$ is harmonic to convince yourself.