Extending a harmonic function

Question

Suppose that $u$ is a harmonic function on some open set $U$ (assume that $\overline{U}$ is compact). It is well known than in this case $u$ is smooth. Is it true that we can extend $u$ to the whole plane in such a way that the extended function is still smooth and has compact support?
EDIT: Apparently in the argument which inspired me to post this question in fact the following weaker assetion would be enough for my purposes: is it true that if I have a compact set $K$ and a function $u$ which is harmonic in some neighbourhood of $K$ then $u$ can be approximated uniformly on $K$ by functions $(u_n)_n$ harmonic in the neighbourhood of $K$, such that each $u_n$ is smooth on whole $\mathbb{R}^2$ and having compact supports?

Answer 1

No. For example, the principal branch of $\arg(x+iy)$ is harmonic on the complement of the nonpositive real axis in $\mathbb R^2$, but can't be extended to be continuous at any point of that axis.