This question is in reference to the earlier question asked here.
Let $u(x)=\log\log \frac{2}{\sqrt{x_1^2+x_2^2}}$ where $x=(x_1,x_2)$ and $\Omega=B(0;2)$. It is easy to see that $u$ is in $H^1(\Omega)$.
How do I extend $u$ to $\mathbb{R}^2\,$? I have seen many standardized texts, but I don't seem to find anything relevant to this.
Does anyone know about any source for this or an extension operator that would work here?
For your very special function $u$ defined in a very special domain $B(0,2)$, one can extend $u$ to $Eu \in H^1(\mathbb R^2)$ by
$$ Eu (x) = \log\log \phi(|x|),$$
where $\phi: (0,\infty) \to \mathbb R^+$ is smooth, so that $\phi(y) = \frac{2}{y}$ for $y\le 2$ and $\phi(y) = 0$ for $y>2+\delta$ for some $\delta >0$.
In general the extension operator is defined for all Lipschitz domain, see Theorem 7.25 in Gilbarg-Trudinger.