I have trouble understanding how can a sequence $\{u_k\} \in C^1{(\mathbb{U})}$, with $\mathbb{U} \subset \mathbb{R}^2$ bounded and $\|u_k\|_{H_1(\mathbb{U})} < \infty$, can be Cauchy with respect to the $H^1$ norm but not be essentially bounded.
Doesn't essentially unbounded mean that it diverges in non-zero measure set? Could you provide me any ideas/hints to construct such a sequence in the 2D unit disk?