If $u_k$ converges uniformly on $\partial \Omega$, does it converge uniformly on $\Omega$?
Solution 1:
Careful: This is not true for any $\Omega.$ For example let $u_k(x,y) = (-1)^ky$ in the upper half plane $\Omega $ of $\mathbb {R}^2.$ Then all $u_k$ vanish on $\partial \Omega,$ hence converge uniformly there, but the $u_k$ converge nowhere in $\Omega.$
This will be true for all bounded $\Omega$ however. There it's just the maximum principle:
$$\sup_{\overline {\Omega}} |u_k-u_j|\le \sup_{ \partial \Omega}|u_k-u_j|.$$