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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 test functions|In contrast to]] [[Lp-norm|the]] $L^{p}$-case $L^{p}=W^{0, p}$, for $k>0$ usually $\overline{C_{c}^{\infty}(\Omega)}^{W^{k,p}(\Omega)} \subsetneq W^{k,p}(\Omega)$. We denote $W_{0}^{k,p}(\Omega):=\overline{C_{c}^{\infty}(\Omega)}^{W^{k,p}(\Omega)}$ 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.
>
We interpret $W_{0}^{k,p}(\Omega)$ as the subspace of $W^{k,p}(\Omega)$ such that "$\partial_{}^{\alpha}u=0$ on $\partial_{}\Omega
quot; for $|\alpha| \leq k-1$. The sense in which this property understood is made precise via [[the trace theorem|trace theory]].
> [!note] Remark.
> Some definitions use an [[norm|equivalent norm]]. Most commonly this is ($k=1$ for simplicity) $\|u\|_{k,p}:=\|u\|_{p}+\|Du\|_{p}, \, (\text{alternative definition})$
where [[Lp-norm|we recall]] $\|Du\|_{L^{p}}^{p} = \int |Du(x)|_{2}^{p} \, dx$. (Have equality when $p=2$.)
^note
**Completeness.** Observe that $(u_{m}) \subset W^{k,p}$ Cauchy implies $(\partial^{\alpha}u_{m}) \subset L^{p}$ [[Cauchy sequence|Cauchy]] for all $|\alpha| \leq k$, and since $L^{p}$ is [[complete]], have $(\partial^{\alpha}u_{m}) \to v_{\alpha}$ for some $v_{\alpha} \in L^{p}$. In particular for $\alpha=0$ we get $u_{m} \to v_{0}=:u$ in $L^{p}$. The claim is then that in fact $u \in W^{k,p}$ with weak derivatives equalling precisely the $v_{\alpha}$.
For each $m$, $\partial^{\alpha}u_{m}$ is the $\alpha$-weak derivative of $u_{m}$ and so $\int_{\Omega}u_{m} \, \partial^{\alpha} \varphi= (-1)^{|\alpha|} \int_{\Omega} \partial^{\alpha}u_{m}\, \varphi$ for all $\varphi \in C_{c}^{\infty}$. If we can pass $m \to \infty$ on both sides we obtain $\int_{\Omega} u \partial^{\alpha} \varphi = \int_{\Omega} v_{\alpha} \varphi$
for all $\varphi \in C_{c}^{\infty}$ which is precisely the statement that $v_{\alpha}$ is the $\alpha$-weak derivative of $u$. We can indeed do this via [[Hölder's inequality|Holder]], see the relevant property and proof in [[weak derivative]] (short). Hence $u \in W^{k,p}$, and $\|u_{m}- u\|_{W^{k,p}} = \sum_{|\alpha| \leq k}\|\partial^{\alpha} u_{m}- \partial^{\alpha} u\|_{L^{p}} =\sum_{|\alpha| \leq k}\|\partial^{\alpha} u_{m} - v_{\alpha}\|_{L^{p}} \to 0.$
> [!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 $C^{\infty}(\overline{\Omega}) \cap W^{k,p}(\Omega)$
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####
[^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 $u$.
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#### 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
> ```