Check if the function $v(x)=\log(-\log|x|^2)$ belongs to $H^1(\Omega)$ if $\Omega=\{x\in\mathbb{R}^2:|x|<\frac{1}{2}\}$. Are functions in $H^1(\Omega)$ necessarily bounded and continuous?
So we have to show that $\int_\Omega|v(x)|^2dx$ and $\int_\Omega|v'(x)|^2dx$ converges. Now $$\int_\Omega|v(x)|^2dx=\int_\Omega|\log(-\log|x|^2)|^2dx$$
and $$\int_\Omega|v'(x)|^2dx=\int_\Omega\Bigg|\frac{1}{x\log x}\Bigg|^2dx$$
I don't know what to do now. Also for the second part I think the functions for $H^1(\Omega)$ are not necessarily bounded and continuous, but I couldn't construct any example to support that.
Any help would be great. Thanks in advance.