Compactness of Sobolev Space in L infinity

I want to show that if $u_{m} \rightharpoonup u$ in $W^{1,\infty}(\Omega)$ then $u_{m} \rightarrow u$ in $L^{\infty}(\Omega)$.

I know that I can’t directly use the compactness of Rellich Kondrachov Theorem since I am taking $p = \infty$. From Morrey’s Inequality I have $||u||_{C^{0,\alpha}} \leq ||u||_{W^{1,p}}$ where $\alpha = 1 – \frac{n}{p}$. Is it possible to take this further and show that $W^{1,p}(\Omega) \Subset L^{\infty}(\Omega)$ where $\Omega \subset \mathbb{R}^{n}$ is $C^{1}$?

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Here is a direct proof for $L^\infty$, without Hölder spaces or Morrey inequality.

  1. By the Arzelà–Ascoli theorem, every infinite bounded subset of $W^{1,\infty}$ has a limit point in $L^\infty$.
  2. Therefore, the embedding of $W^{1,\infty}$ into $L^\infty$ is compact.
  3. A compact operator maps weakly convergent sequences into convergent sequences.

Old answer: Every $C^1$ domain, and more generally a Lipschitz domain, is a Sobolev extension domain, meaning that Sobolev functions on it can be extended to Sobolev functions on $\mathbb R^n$. In particular, all embedding theorems for Sobolev spaces hold on such domains.

When $p>n$, Morrey’s inequality gives a continuous embedding of $W^{1,p}$ into $C^\beta$ with $\beta=1-n/p$. In turn, $C^\beta$ compactly embeds into $C^\alpha$ for $0<\alpha<\beta$. See Is there a reference for compact imbedding of Hölder space? This topic was also discussed in Compact, continuous embeddings of $H^s := W^{s,2} \leftrightarrow C^{(\alpha)}$ (in one dimension).

Under your assumptions, you can get $u_m\to u$ in any space $C^\alpha$ with $\alpha\in (0,1)$. A fortiori, $u_m\to u$ in $L^\infty$.