> [!theorem] > > Let $U$ be a bounded [[Open Set|open]] set with [[Differentiable Boundary|differentiable boundary]] $\partial U \in C^1$, and $p \in [1, \infty)$. Let $V \in \cn^o(U)$ be a [[Neighbourhood|neighbourhood]] of $U$ such that $U \subset \subset V$. Then there exists a [[Bounded Linear Map|bounded linear map]] > $ > E: W^{1, p}(U) \to W^{1, p}(\real^n) > $ > that extends elements of the [[Sobolev Space|Sobolev space]], such that for each $u \in W^{1, p}(U)$, > 1. $Eu|_U = u$ [[Almost Everywhere|almost everywhere]]. > 2. $\supp{Eu} \subset V$. > 3. The [[Space of Bounded Linear Maps|operator norm]] $\norm{E}$ depends only on $p$, $U$, and $V$. > > *Proof*. Let $u \in W^{1, p}(U) \cap UC^1(U)$ be a [[Space of Uniformly Continuously Differentiable Functions|UC]] function. By the [[UC Extension Theorem]], there exists $V_0 \in \cn^o(\ol{U})$ and a linear operator > $ > E_0: UC^1(U) \to C_c^1(\real^d) \quad u \mapsto \ol{u} > $ > such that $\supp{\ol{u}} \subset V_0$ for all $u \in UC^1(U)$, and $\norm{\ol{u}}_{W^{1, p}(\real^d)} \le C_0\norm{u}_{W^{1, p}(U)}$. > > Since [[Approximation of Sobolev Space by Smooth Functions|UC functions are dense]], by the [[Linear Extension Theorem]], $E$ admits a unique extension $\ol{E}_0: W^{1, p}(U) \to W^{1, p}(\real^d)$ with $||{\ol{E}_0}|| = \norm{E_0}$. > > Let $\zeta \in C_c^\infty(\real^d)$ be a [[Smooth Bump Function|smooth bump function]] such that $\zeta|_U = 1$ and $\zeta|_{V^c} = 0$, then this induces a linear operator > $ > R: W^{1, p}(\real^d) \to W^{1, p}(\real^d) \quad u \mapsto \zeta u > $ > where there exists $C_\zeta$ such that $\norm{\zeta u}_{W^{1, p}} \le C_\zeta \norm{U}_{W^{1, p}}$. From here, define $E = R \circ \ol{E}_0$, then $E$ is the desired operator.