I have recently been trying to understand the theory regarding harmonic extensions in $\mathbb R^n$. I have, however, had some difficulties to find the kind of results I am looking for. For that reason I was hoping to get an answer to the below question here.

Let $g:B(x, 2r) \to \mathbb R$ be real analytic and suppose $u$ is the harmonic function in $B(x, r)$ such that $u=g$ on $\partial B(x, r)$.

Now my question is whether it is possible to extend $u$ to a harmonic function $\tilde u$ in $B(x, (1+\delta)r)$ for some $\delta >0$. In other words, is it possible to find a function $\tilde u$ such that $\tilde u = u$ in $\overline{B(x,r)}$ and $\tilde u$ is harmonic in $B(x, (1+\delta)r)$?