If the sequence $\{u_n\}\in C^\infty(\bar{\Omega})$ and $\|u_n\|_{L^2(\Omega)}\leq \frac{1}{n}$, what could I say about $\|u_n\|_{L^2(\partial\Omega)}$? $\partial \Omega$ is the boundary of $\Omega$.
Is that true $\|u_n\|_{L^2(\partial\Omega)}\rightarrow 0$?
No. The functions could have an extremely steep gradient near the boundary that the $L^2$ norm doesn't capture. For instance, take $\Omega$ to be the unit ball and $u_n(x) = |x|^{n^2}$. Then every $u_n$ is identically equal to $1$ on $\partial \Omega$, yet $\|u_n\|_{L^2(\Omega)} \le \frac Cn$ for some constant $C$ that is independent of $n$.