[proofplan] We prove the compact embedding $W^{1,p}(U) \hookrightarrow\hookrightarrow C(\overline{U})$ for $p > n$ by showing that every bounded sequence in $W^{1,p}(U)$ has a uniformly convergent subsequence. The strategy has two parts: first, use [Morrey's Inequality](/theorems/62) (via the Extension Theorem) to establish that the sequence is uniformly bounded and [equicontinuous](/page/Equicontinuity) in $C^{0,\gamma}(\overline{U})$ with $\gamma = 1 - n/p$; then apply the Arzela-Ascoli Theorem on the [compact](/page/Compact%20Space) set $\overline{U}$ to extract a uniformly convergent subsequence. [/proofplan] [step:Extend the bounded sequence and apply Morrey's Inequality to obtain uniform Holder bounds] Let $(u_k)_{k \in \mathbb{N}}$ be a bounded sequence in $W^{1,p}(U)$ with $\|u_k\|_{W^{1,p}(U)} \le M$ for all $k \in \mathbb{N}$. Since $\partial U$ is $C^1$, the Extension Theorem provides a bounded linear operator $E: W^{1,p}(U) \to W^{1,p}(\mathbb{R}^n)$ with $\|Eu_k\|_{W^{1,p}(\mathbb{R}^n)} \le C_E M$. Since $p > n$, [Morrey's Inequality](/theorems/62) gives the continuous embedding $W^{1,p}(\mathbb{R}^n) \hookrightarrow C^{0,\gamma}(\mathbb{R}^n)$ with $\gamma = 1 - n/p \in (0, 1)$. Restricting to $\overline{U}$: \begin{align*} \|u_k\|_{C^{0,\gamma}(\overline{U})} \le \|Eu_k\|_{C^{0,\gamma}(\mathbb{R}^n)} \le C_{\text{Morrey}} \|Eu_k\|_{W^{1,p}(\mathbb{R}^n)} \le C_{\text{Morrey}} C_E M. \end{align*} Setting $K = C_{\text{Morrey}} C_E M$, we have $\|u_k\|_{C^{0,\gamma}(\overline{U})} \le K$ for all $k$. [guided] The Extension Theorem is needed because [Morrey's Inequality](/theorems/62) is stated for functions on all of $\mathbb{R}^n$, while our sequence $(u_k)$ lives on the bounded domain $U$. Since $\partial U$ is $C^1$, the Extension Theorem provides a bounded linear operator $E: W^{1,p}(U) \to W^{1,p}(\mathbb{R}^n)$ satisfying $Eu_k|_U = u_k$ and $\|Eu_k\|_{W^{1,p}(\mathbb{R}^n)} \le C_E \|u_k\|_{W^{1,p}(U)} \le C_E M$. Since $p > n$, [Morrey's Inequality](/theorems/62) applies to $Eu_k \in W^{1,p}(\mathbb{R}^n)$ and gives \begin{align*} \|Eu_k\|_{C^{0,\gamma}(\mathbb{R}^n)} \le C_{\text{Morrey}} \|Eu_k\|_{W^{1,p}(\mathbb{R}^n)} \end{align*} with Holder exponent $\gamma = 1 - n/p \in (0,1)$. Restricting from $\mathbb{R}^n$ to $\overline{U}$ can only decrease norms, so \begin{align*} \|u_k\|_{C^{0,\gamma}(\overline{U})} \le \|Eu_k\|_{C^{0,\gamma}(\mathbb{R}^n)} \le C_{\text{Morrey}} C_E M =: K. \end{align*} The $C^{0,\gamma}$ norm controls two quantities simultaneously, both uniform in $k$: \begin{align*} \sup_{x \in \overline{U}} |u_k(x)| \le K \quad \text{and} \quad \sup_{\substack{x, y \in \overline{U} \\ x \neq y}} \frac{|u_k(x) - u_k(y)|}{|x - y|^\gamma} \le K. \end{align*} The first bound gives uniform pointwise boundedness of the sequence. The second gives [equicontinuity](/page/Equicontinuity): for any $\epsilon > 0$, choosing $\delta = (\epsilon/K)^{1/\gamma}$ ensures $|u_k(x) - u_k(y)| \le K|x-y|^\gamma < \epsilon$ whenever $|x - y| < \delta$, uniformly in $k$. These are precisely the two hypotheses of the Arzela-Ascoli Theorem. [/guided] [/step] [step:Verify the hypotheses of the Arzela-Ascoli Theorem] We check the two conditions of the Arzela-Ascoli Theorem for the family $\{u_k\}_{k \in \mathbb{N}} \subset C(\overline{U})$: **Uniform pointwise boundedness.** For all $x \in \overline{U}$ and all $k \in \mathbb{N}$: \begin{align*} |u_k(x)| \le \sup_{z \in \overline{U}} |u_k(z)| \le \|u_k\|_{C^{0,\gamma}(\overline{U})} \le K. \end{align*} **Equicontinuity.** For all $x, y \in \overline{U}$ and all $k \in \mathbb{N}$, the Holder bound gives: \begin{align*} |u_k(x) - u_k(y)| \le K |x - y|^\gamma. \end{align*} Given $\epsilon > 0$, choosing $\delta = (\epsilon / K)^{1/\gamma}$ ensures $|x - y| < \delta$ implies $|u_k(x) - u_k(y)| < \epsilon$ uniformly in $k$. [/step] [step:Apply the Arzela-Ascoli Theorem to extract a uniformly convergent subsequence] The domain $\overline{U}$ is [compact](/page/Compact%20Space) (since $U$ is bounded and $\overline{U}$ is closed in $\mathbb{R}^n$). The sequence $(u_k)$ is uniformly bounded and [equicontinuous](/page/Equicontinuity) on $\overline{U}$. By the Arzela-Ascoli Theorem, there exists a subsequence $(u_{k_j})_{j \in \mathbb{N}}$ and a function $u \in C(\overline{U})$ such that \begin{align*} \lim_{j \to \infty} \|u_{k_j} - u\|_{C(\overline{U})} = \lim_{j \to \infty} \sup_{x \in \overline{U}} |u_{k_j}(x) - u(x)| = 0. \end{align*} Since every bounded sequence in $W^{1,p}(U)$ admits a subsequence converging in $C(\overline{U})$, the embedding $W^{1,p}(U) \hookrightarrow\hookrightarrow C(\overline{U})$ is compact. [guided] The Arzela-Ascoli Theorem states: if $X$ is a [compact](/page/Compact%20Space) metric space and $\mathcal{F} \subset C(X)$ is pointwise bounded and [equicontinuous](/page/Equicontinuity), then every sequence in $\mathcal{F}$ has a uniformly convergent subsequence. We apply it with $X = \overline{U}$ and $\mathcal{F} = \{u_k : k \in \mathbb{N}\}$. The compactness of $\overline{U}$ follows from the Heine-Borel theorem: $U \subset \mathbb{R}^n$ is bounded, so $\overline{U}$ is closed and bounded in $\mathbb{R}^n$, hence compact. The two hypotheses were verified in the previous step. Pointwise boundedness: \begin{align*} |u_k(x)| \le \|u_k\|_{C^{0,\gamma}(\overline{U})} \le K \quad \text{for all } x \in \overline{U}, \; k \in \mathbb{N}. \end{align*} Equicontinuity with Holder modulus: for any $\epsilon > 0$, setting $\delta = (\epsilon / K)^{1/\gamma}$ gives \begin{align*} |x - y| < \delta \implies |u_k(x) - u_k(y)| \le K |x - y|^\gamma < K \cdot \frac{\epsilon}{K} = \epsilon \end{align*} uniformly in $k$. The Arzela-Ascoli conclusion is that a subsequence $(u_{k_j})$ converges uniformly: \begin{align*} \|u_{k_j} - u\|_{C(\overline{U})} = \sup_{x \in \overline{U}} |u_{k_j}(x) - u(x)| \to 0 \quad \text{as } j \to \infty. \end{align*} Note the contrast with the [Rellich-Kondrachov Theorem](/theorems/64) for $p < n$: there, compactness required mollification, diagonal extraction, and $L^p$ interpolation because the embedding into continuous functions is unavailable. Here, $p > n$ gives Holder continuity via [Morrey's Inequality](/theorems/62), and Arzela-Ascoli directly provides the compact embedding into $C(\overline{U})$ without any mollification or interpolation argument. [/guided] [/step]