---- Let $\Omega$ be an open subset of $\mathbb{R}^{n}$. > [!definition] Definition. ([[Sobolev space]]) > For $k \geq 0$ and $1 \leq p \leq \infty$, the **Sobolev space** $W^{k,p}(\Omega)$ is the [[vector space|space]] of functions of $f \in L^{p}(\Omega)$ whose [[weak derivative|weak partial derivatives]] up to and including order $k$ exist and belong also to $L^{p}(\Omega)$. > > The [[norm]][^1] $\|u\|_{k,p}:= \left( \sum_{|\alpha| \leq k} \|\partial^{\alpha}u\|_{p}^{p} \right)^{1/p}$ > makes $W^{k,p}(\Omega)$ into a [[Banach space]]. When $p=2$, $W^{k,2}(\Omega)$ is in fact a [[Hilbert space]] with respect to the [[inner product]] $\langle u, v \rangle_{k}:= \sum_{|\alpha| \leq k} \langle \partial^{\alpha}u, \partial^{\alpha} v \rangle , $ > where $\langle -,- \rangle$ denotes the [[inner product]] [[nontrivial Lp spaces are Hilbert iff p=2|making]] $L^{2}(\Omega)$ into a [[Hilbert space]]. In this case, the notation $W^{k,2}(\Omega)=:H^{k}(\Omega)$ is also used. > [[Lp density of compactly supported functions|In contrast to]] [[Lp-norm|the]] $L^{p}$-case, usually $\overline{C_{c}^{\infty}(\Omega)} \subsetneq W^{k,p}(\Omega)$. We denote $W_{0}^{k,p}(\Omega):=\overline{C_{c}^{\infty}(\Omega)}$ the [[closure]] of $C_{c}^{\infty}(\Omega)$ in $W^{k,p}(\Omega)$. ($H_{0}^{k}(\Omega)$ when $p=2$.) It is true that $W_{0}^{k,p}(\mathbb{R}^{n})=W^{k,p}(\mathbb{R}^{n})$, ([[Global Sobolev approximation by functions smooth up to the boundary|see]]), though we omitted the proof in our course. > [!justification] > ^justification > [!basicproperties] > - For $k \geq 0$ and $1< p < \infty$, $W^{k,p}(\Omega)$ is [[reflexive space|reflexive]], though we will not prove it here. ^properties $C_{c}^{\infty}(\mathbb{R}^{n})$ a subset of $C^{\infty}(\overline{\Omega}) \cap W^{k,p}(\Omega)$ ---- #### [^1]: Here the summation ranges over all multi-indices $\alpha=(\alpha_{1},\dots,\alpha_{k})$ satisfying $|\alpha|=\alpha_{1}+\dots+\alpha_{k} \leq k$. Since this includes $\alpha=(0,\dots,0)$, for which the corresponding term is $\|u\|_{p}^{p}$, the Sobolev norm can be though about quantifying both the size ($\|u\|_{p}^{p}$) and regularity $(\|\partial ^{\alpha}u\|_{p}^{p}, \alpha \neq 0)$ of $u$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```