> [!definition] > > Let $U \subset \real^n$ be an [[Open Set|open set]], and $f: U \to \real$ be a function, then $f \in W^{k, p}(U)$ is in the **Sobolev space** if > 1. $f \in \loci$ is [[Locally Integrable|locally integrable]]. > 2. The [[Distributional Derivative|weak derivative]] $D^{\alpha}f$ exists for each [[Multi-Index|multi-index]] $\alpha$ with $\abs{\alpha} \le k$. > 3. $D^\alpha f$ is in [$L^p$](Lp%20Space) for each $\alpha$ with $\abs{\alpha} \le k$. > > which comes with a [[Normed Vector Space|norm]] > $ > \norm{f}_{W^{k, p}(U)} = \begin{cases} > \braks{\sum_{\abs{\alpha} \le k}\norm{D^\alpha f}_p^p}^{1/p} & 1 \le p < \infty \\ > \sum_{\abs{\alpha} \le k}\norm{D^\alpha f}_\infty &p = \infty > \end{cases} > $ > If $k = 2$, then $W^{k, p}(U) = H^k(U)$ is given a [[Hilbert Space|Hilbert space]] structure. The space $W_{0}^{k, p}(U)$ is the [[Topological Closure|closure]] of $C_c^\infty(U)$. > > Let $V \subset\subset U$ be a [[Compactness|compactly]] contained open set, then $\norm{f}_{W^{k, p}(V)} = \norm{f|_V}_{W^{k, p}}(V)$ is a [[Seminorm|seminorm]]. The set of functions in $W^{k, p}(U)$ with topology given by this family of seminorms is denoted $W^{k, p}_{\text{loc}}(U)$. # The Rundown The Sobolev space forms a Banach space with its norm. Functions in $W^{k, p}(U)$ where $p \in [1, \infty)$ can be [[Approximation of Sobolev Space by Smooth Functions|approximated by smooth functions]] with respect to $\norm{\cdot }_{W^{k, p}}$ to various degrees: - If $U$ is any open set, then functions in $W^{k, p}(U)$ can be approximated by smooth functions when restricted to compactly contained open sets. - If $U$ is bounded, then elements can be approximated by smooth functions on $U$. - If $\partial U$ is $C^1$, then elements in $W^{k, p}(U)$ can be approximated by smooth [[Space of Uniformly Continuously Differentiable Functions|UC]] functions on $U$. Given a bounded open set $U$ with $\partial U$ being [[Differentiable Boundary|of class]] $C^1$, functions in $W^{1, p}(U)$ or $W^{2, p}(U)$ can be [[Extension Theorem for Sobolev Spaces|extended to]] $W^{k, p}(\real^d)$ via a [[Higher Order Reflection|reflection]], with the extension operator being bounded. # Completeness > [!theorem] > > Let $\seq{u_n} \subset W^{k, p}$ such that $u_n \to u$ in $L^p$. Let $\alpha$ be a multi-index such that $\abs{\alpha} \le k$, then there exists $u_\alpha \in L^p$ such that $D^\alpha u_n \to u_\alpha$ in $L^p$. Moreover, $D^\alpha u = u_\alpha$. Therefore $W^{k, p}$ is a [[Banach Space|Banach space]]. > > *Proof*. By completeness of $L^p$, limits for the function and each partial derivative exists in $L^p$. Let $\phi \in C_c^\infty$, then since convergence in $L^p$ implies weak convergence as linear functionals of $L^q$, > $ > \angles{u_n, D^\alpha \phi} \to \angles{u, D^\alpha \phi} \quad \angles{D^\alpha u_n, \phi} \to \angles{u_\alpha, \phi} > $ # Multiplication by Smooth Functions > [!theorem] > > Let $U \subset \real^d$ be an open set, $W^{k, p}(U) = W^{k, p}$, $\zeta \in C_c^\infty(U)$, then there exists $C_{\zeta, k, d} \ge 0$ such that > $ > \norm{\zeta u}_{W^{k, p}} \le C_{\zeta, k, d}\norm{u}_{W^{k, p}} \quad \forall u \in W^{k, p} > $ > *Proof*. Since $\zeta \in C_c^\infty$, the weak derivative coincides with the strong derivative. In addition, $\norm{D^\alpha \zeta}_u < \infty$ for all multi-indices. For a crude estimate, let $C_\zeta = \sum_{\abs{\alpha} \le k}\norm{D^\alpha \zeta}_u$. > > By the product rule, > $ > \begin{align*} > D^\alpha(\zeta u) &= \sum_{{\beta} \le \alpha} {\alpha \choose \beta}D^\beta \zeta \cdot D^{\alpha - \beta}u \\ > \abs{D^\alpha (\zeta u)}^p &\le C_\zeta^p \sum_{{\beta} \le \alpha}{\alpha \choose \beta}^p\abs{D^{\alpha - \beta}u} ^p > \end{align*} > $ > Now let $C_\alpha = \max_{\beta \le \alpha}{\alpha \choose \beta} \le n!$, then > $ > \begin{align*} > \abs{D^\alpha(\zeta u)}^p &\le C_\zeta^p C_\alpha^p\sum_{\beta \le \alpha} \abs{D^{\alpha - \beta}u}^p \\ > \norm{D^\alpha(\zeta u)}_p^p &\le C_\zeta^p C_\alpha^p \sum_{\beta \le \alpha}\norm{D^{\alpha -\beta}u}_p^p \le C_\zeta^pC_\alpha^p \norm{u}_{W^{k, p}}^p > \end{align*} > $ > Lastly, let $C_k = \max_{\abs{\alpha} \le k}C_\alpha \le k!$, then > $ > \norm{\zeta u}_{W^{k, p}}^p =\sum_{\abs{\alpha} \le k}\norm{D^\alpha(\zeta u)}_p^p \le \sum_{\abs{\alpha} \le k}C_\zeta^p C_{k}^p \norm{u}_{W^{k, p}} \le d^kC_\zeta^p C_{k}^p \norm{u}_{W^{k, p}} > $ > so > $ > \norm{\zeta u}_{W^{k, p}} \le d^{k}C_\zeta C_k \norm{u}_{W^{k, p}} > $