[proofplan] We construct the extension operator $E: W^{1,p}(U) \to W^{1,p}(\mathbb{R}^n)$ in three stages. First, we flatten the $C^1$ [boundary](/page/Boundary) using finitely many charts and a subordinate partition of unity, reducing to the model case of the upper half-cube $Q_+$. Second, we define a higher-order reflection across $\{y_n = 0\}$ that extends functions from $Q_+$ to all of $Q$ while preserving $W^{1,p}$ regularity, and we verify the explicit norm estimates by changes of variable. Third, we synthesise the local extensions into a global operator, prove the norm bound $\|Eu\|_{W^{1,p}(\mathbb{R}^n)} \le C\|u\|_{W^{1,p}(U)}$ for $u \in C^1(\overline{U})$, and extend $E$ to all of $W^{1,p}(U)$ by density of $C^1(\overline{U})$ in $W^{1,p}(U)$ (which requires the $C^1$ boundary via the [Meyers--Serrin Theorem](/theorems/58)). [/proofplan] [step:Flatten the boundary using charts and a partition of unity] Since $\partial U$ is compact and of class $C^1$, there exists a finite open cover $\{W_i\}_{i=1}^N$ of $\partial U$ and, for each $i$, a $C^1$ diffeomorphism \begin{align*} \Phi_i: W_i &\to Q := (-1,1)^n \end{align*} mapping $W_i \cap U$ onto $Q_+ := \{y \in Q : y_n > 0\}$ and $W_i \cap \partial U$ onto $Q_0 := \{y \in Q : y_n = 0\}$. Let $W_0 \subset\subset U$ be an open set such that $U \subseteq W_0 \cup \bigcup_{i=1}^N W_i$. By the [Existence of Smooth Partitions of Unity](/theorems/57), there exists a smooth partition of unity $\{\zeta_i\}_{i=0}^N$ subordinate to $\{W_i\}_{i=0}^N$, so that $u = \sum_{i=0}^N \zeta_i u$ on $U$. [/step] [step:Define the higher-order reflection extending $C^1(\overline{Q}_+)$ functions to $Q$] For $v \in C^1(\overline{Q}_+)$, define the reflection operator \begin{align*} E_0 v: Q &\to \mathbb{R}, \\ (y', y_n) &\mapsto \begin{cases} v(y', y_n) & \text{if } y_n \ge 0, \\ -3\,v(y', -y_n) + 4\,v(y', -y_n/2) & \text{if } y_n < 0. \end{cases} \end{align*} The coefficients $-3$ and $4$ are chosen so that $E_0 v$ is continuous across $\{y_n = 0\}$ (since $-3 + 4 = 1$) and the normal derivative matches: $\partial_{y_n}(E_0 v)\big|_{y_n = 0^-} = 3\,\partial_{y_n} v - 2\,\partial_{y_n} v = \partial_{y_n} v\big|_{y_n = 0^+}$. This matching of values and first derivatives ensures $E_0 v \in W^{1,p}(Q)$. [/step] [step:Verify the $W^{1,p}$ norm estimate for the reflection operator $E_0$] We bound $\|E_0 v\|_{W^{1,p}(Q)}$ in terms of $\|v\|_{W^{1,p}(Q_+)}$. For the $L^p$ norm on $Q_- := \{y \in Q : y_n < 0\}$, apply the convexity inequality $|a + b|^p \le 2^{p-1}(|a|^p + |b|^p)$ and change variables $z_n = -y_n$ (Jacobian determinant $1$), $w_n = -y_n/2$ (Jacobian determinant $2$): \begin{align*} \int_{Q_-} |E_0 v|^p \, d\mathcal{L}^n(y) &\le 2^{p-1}\Bigl(3^p \int_{Q_-} |v(y', -y_n)|^p \, d\mathcal{L}^n(y) + 4^p \int_{Q_-} |v(y', -y_n/2)|^p \, d\mathcal{L}^n(y)\Bigr) \\ &= 2^{p-1}\Bigl(3^p \int_{Q_+} |v(y', z_n)|^p \, d\mathcal{L}^n(z) + 4^p \cdot 2 \int_{Q_+ \cap \{w_n < 1/2\}} |v(y', w_n)|^p \, d\mathcal{L}^n(w)\Bigr) \\ &\le C_0 \|v\|_{L^p(Q_+)}^p. \end{align*} For tangential derivatives $j < n$: $\partial_{y_j}(E_0 v) = -3\,(\partial_{y_j} v)(y', -y_n) + 4\,(\partial_{y_j} v)(y', -y_n/2)$ on $Q_-$, which has the same structure, so $\|\partial_{y_j}(E_0 v)\|_{L^p(Q_-)} \le C\|\partial_{y_j} v\|_{L^p(Q_+)}$. For the normal derivative $j = n$, the chain rule gives: \begin{align*} \partial_{y_n}(E_0 v)(y', y_n) &= -3\,(\partial_{y_n} v)(y', -y_n) \cdot (-1) + 4\,(\partial_{y_n} v)(y', -y_n/2) \cdot (-1/2) \\ &= 3\,(\partial_{y_n} v)(y', -y_n) - 2\,(\partial_{y_n} v)(y', -y_n/2) \quad \text{for } y_n < 0. \end{align*} The same change-of-variables argument yields $\|\partial_{y_n}(E_0 v)\|_{L^p(Q_-)}^p \le C\|\partial_{y_n} v\|_{L^p(Q_+)}^p$. Summing all contributions: \begin{align*} \|E_0 v\|_{W^{1,p}(Q)} &\le C_{\mathrm{refl}} \|v\|_{W^{1,p}(Q_+)}. \end{align*} [guided] We need to verify that the reflection $E_0$ is bounded as an operator from $W^{1,p}(Q_+)$ to $W^{1,p}(Q)$. The strategy is to estimate the $L^p$ norms of $E_0 v$ and its derivatives on the lower half-cube $Q_-$ in terms of the corresponding norms of $v$ on $Q_+$, using explicit changes of variable. For the $L^p$ norm of $E_0 v$ on $Q_-$, where $E_0 v(y', y_n) = -3\,v(y', -y_n) + 4\,v(y', -y_n/2)$, apply the convexity inequality $|a + b|^p \le 2^{p-1}(|a|^p + |b|^p)$: \begin{align*} \int_{Q_-} |E_0 v(y', y_n)|^p \, d\mathcal{L}^n(y) &\le 2^{p-1}\Bigl(3^p \int_{Q_-} |v(y', -y_n)|^p \, d\mathcal{L}^n(y) + 4^p \int_{Q_-} |v(y', -y_n/2)|^p \, d\mathcal{L}^n(y)\Bigr). \end{align*} For the first integral, substitute $z_n = -y_n$: as $y_n$ ranges over $(-1, 0)$, $z_n$ ranges over $(0, 1)$, with $d\mathcal{L}^1(z_n) = d\mathcal{L}^1(y_n)$ (Jacobian determinant is $1$). This gives $\int_{Q_-} |v(y', -y_n)|^p \, d\mathcal{L}^n(y) = \int_{Q_+} |v(y', z_n)|^p \, d\mathcal{L}^n(z) = \|v\|_{L^p(Q_+)}^p$. For the second integral, substitute $w_n = -y_n/2$: as $y_n$ ranges over $(-1, 0)$, $w_n$ ranges over $(0, 1/2)$, with $d\mathcal{L}^1(y_n) = 2\,d\mathcal{L}^1(w_n)$. This gives $\int_{Q_-} |v(y', -y_n/2)|^p \, d\mathcal{L}^n(y) = 2\int_{Q_+ \cap \{w_n < 1/2\}} |v(y', w_n)|^p \, d\mathcal{L}^n(w) \le 2\|v\|_{L^p(Q_+)}^p$. Combining: $\|E_0 v\|_{L^p(Q_-)}^p \le 2^{p-1}(3^p + 2 \cdot 4^p)\|v\|_{L^p(Q_+)}^p =: C_0\|v\|_{L^p(Q_+)}^p$. For tangential derivatives ($j < n$), we differentiate: $\partial_{y_j}(E_0 v) = -3\,(\partial_{y_j} v)(y', -y_n) + 4\,(\partial_{y_j} v)(y', -y_n/2)$. This has the identical algebraic structure as the function values, so the same change-of-variables argument gives $\|\partial_{y_j}(E_0 v)\|_{L^p(Q_-)} \le C\|\partial_{y_j} v\|_{L^p(Q_+)}$. For the normal derivative ($j = n$), the chain rule introduces factors from differentiating the arguments $-y_n$ and $-y_n/2$: \begin{align*} \partial_{y_n}(E_0 v)(y', y_n) &= 3\,(\partial_{y_n} v)(y', -y_n) - 2\,(\partial_{y_n} v)(y', -y_n/2) \quad \text{for } y_n < 0. \end{align*} The same substitutions $z_n = -y_n$ and $w_n = -y_n/2$ yield $\|\partial_{y_n}(E_0 v)\|_{L^p(Q_-)}^p \le C\|\partial_{y_n} v\|_{L^p(Q_+)}^p$. Adding $Q_+$ and $Q_-$ contributions: \begin{align*} \|E_0 v\|_{W^{1,p}(Q)} &\le C_{\mathrm{refl}} \|v\|_{W^{1,p}(Q_+)}. \end{align*} [/guided] [/step] [step:Synthesise the local extensions into a global operator and establish the norm bound] Let $u \in C^1(\overline{U})$. Decompose $u = \sum_{i=0}^N \zeta_i u$ using the partition of unity. **Interior term ($i = 0$).** The function $\zeta_0 u$ has $\operatorname{supp}(\zeta_0 u) \subseteq W_0 \subset\subset U$. Extend by zero to $\mathbb{R}^n$. By the Leibniz rule for products of smooth functions: \begin{align*} \|\zeta_0 u\|_{W^{1,p}(\mathbb{R}^n)} &= \|\zeta_0 u\|_{W^{1,p}(U)} \le C_\zeta \|u\|_{W^{1,p}(U)}. \end{align*} **Boundary terms ($i \ge 1$).** For each boundary patch $W_i$, set $u_i := \zeta_i u$. Map to the model domain via $\Phi_i$, reflect, and map back: \begin{align*} Eu_i &:= (E_0(u_i \circ \Phi_i^{-1})) \circ \Phi_i. \end{align*} Since $\Phi_i$ and $\Phi_i^{-1}$ are $C^1$ diffeomorphisms on compact domains, their Jacobians are uniformly bounded. Let $K_1, K_2 > 0$ bound the Sobolev-norm distortion under the change-of-variables formula for $\Phi_i$ and $\Phi_i^{-1}$ respectively. Then: \begin{align*} \|Eu_i\|_{W^{1,p}(W_i)} &\le K_1 \|E_0(u_i \circ \Phi_i^{-1})\|_{W^{1,p}(Q)} \\ &\le K_1 C_{\mathrm{refl}} \|u_i \circ \Phi_i^{-1}\|_{W^{1,p}(Q_+)} \\ &\le K_1 C_{\mathrm{refl}} K_2 \|u_i\|_{W^{1,p}(W_i \cap U)} \\ &\le C_i \|u\|_{W^{1,p}(U)}. \end{align*} Define the total extension: \begin{align*} Eu &:= (\zeta_0 u)_{\text{ext}} + \sum_{i=1}^N Eu_i. \end{align*} By the triangle inequality: \begin{align*} \|Eu\|_{W^{1,p}(\mathbb{R}^n)} &\le \sum_{i=0}^N C_i \|u\|_{W^{1,p}(U)} = C \|u\|_{W^{1,p}(U)}. \end{align*} Multiplying by a cutoff $\chi \in C_c^\infty(V)$ with $\chi = 1$ on $\overline{U}$ ensures $\operatorname{supp}(Eu) \subset V$ without affecting the restriction to $U$ or the norm bound (up to a modified constant). [/step] [step:Extend $E$ from $C^1(\overline{U})$ to all of $W^{1,p}(U)$ by density] Since $\partial U$ is $C^1$ and $U$ is bounded, the [Meyers--Serrin Theorem](/theorems/58) guarantees that $C^\infty(\Omega) \cap W^{1,p}(U)$ is dense in $W^{1,p}(U)$, and in fact $C^1(\overline{U})$ is dense in $W^{1,p}(U)$ (the $C^1$ boundary allows approximation up to the boundary). The operator $E: C^1(\overline{U}) \to W^{1,p}(\mathbb{R}^n)$ is linear and satisfies $\|Eu\|_{W^{1,p}(\mathbb{R}^n)} \le C\|u\|_{W^{1,p}(U)}$, so it is bounded. By the Bounded Linear Transformation theorem (a bounded linear operator defined on a dense subspace of a [Banach space](/page/Banach%20Space) extends uniquely to a bounded linear operator on the whole space), $E$ extends uniquely to a bounded linear operator $E: W^{1,p}(U) \to W^{1,p}(\mathbb{R}^n)$ with the same norm bound. The restriction, compact support, and norm stability properties hold for the extended operator by continuity. [/step]