Let $U \subset \mathbb{R}^2$ be a bounded open set with Lipschitz boundary. Suppose $f^n \rightharpoonup f$ in $W^{1,p}(U)$. Then $f_n \to f$ in $L^{\infty}(U)$ for $2<p<\infty$.
Attempt: Since $p>n=2$, I know we have to use Morrey's inequality. As $f_n$ is weakly convergent, it is uniformly bounded in $W^{1,p}$. Thus by Morrey's $\{f_n\}$ is precompact in $C(\overline{U})$.
Edit: I think I figured it out. Firstly wlog $f_n$ and $f$ are Holder continuous. Now, as $f_n$ convegres to $f$ weakly in $L^p$ and $1<p<\infty$, by reflexivity $f_n$ converges to $f$ strongly in $L^p$.
By Morrey's $W^{1,p}$ is compactly embedded in $C(\overline{U})$, thus there exists a constant $C$ such that $||f_n-f||_{\infty} \leq C||f_n-f||_p$
Does this look correct?