----
- $\Omega$ is an open subset of $\mathbb{R}^{n}$.
- $f \in L^{1}_{\text{loc}}(\Omega)$ is [[locally integrable function|locally integrable]].
- $\alpha=(\alpha_{1},\dots,\alpha_{n})$ is a multi-index.
> [!definition] Definition. ([[weak derivative]])
> A function $g \in L^{1}_{\text{Loc}}(\Omega)$ is said to be a **weak $\alpha$-derivative** of $f$ if $\int _{\Omega} f\partial_{}^{\alpha} \varphi \, dx= (-1)^{|\alpha|} \int _{\Omega} g \varphi \, dx \text{ for all } \varphi \in C_{c}^{\infty}(\Omega). $
In this case we say **$g=\partial_{}^{\alpha} f$ in the weak sense**. Sometimes we'll just write $g=\partial_{}^{\alpha}f$ and 'in the weak sense' will have to be ascertained from context. The functions $\varphi \in C_{c}^{\infty}(\Omega)$ are called **[[bump function|test functions]]**.
^definition
> [!specialization]
> Since the classical $\alpha$-[[derivative]] $g$ of $f$, should it exist, of course satisfies the required integration by parts formula, we see that weak differentiation is indeed a generalization.
^specialization
- [ ] Sharpening in 1D: If $f \in W^{1,1}_{\text{loc}}(I)$, then $f$ has uniformly (in fact absolutely) continuous representative (I think we use this lots in FEM reasoning, in fact I think proved it going as far back as 395)
> [!basicproperties]
> - It follows from the [[fundamental lemma of the calculus of variations]] that weak derivatives are unique[^1] when they exist.
> - Sequential property of weak derivatives (Remark 2.2.3 in notes): If $\partial_{}^{\alpha}u_{m} \xrightarrow{\text{ in }L^{p}} v$ in $L^{p}$ for some $(u_{m}) \xrightarrow{\text{in }L^{p}} u$ in $L^{p}$, then $v$ is the $\alpha$-weak derivative of $u$. For $p<\infty$, the conclusion continues to hold upon relaxing to [[weak topology|weak convergence]].
>
>
> > [!proof]- Proof.
> > We'll prove for $k=1$. (It is also implicitly done in the proof that $W^{k,p}$ is complete.)
> >
> > Let $\varphi \in C_{c}^{\infty}$. $\int_{}u_{m} \partial_{i} \varphi= -\int_{}\partial_{i} u_{m} \, \varphi$
> > and want to show $\int_{} u \partial_{i} \varphi = - \int_{}v \varphi.$
> > By [[Hölder's inequality|Holder]], $\int_{}(u_{m} - u) \partial_{i} \varphi \leq \|u_{m} - u\|_{L^{p}} \|\partial_{i} \varphi\|_{L^{p'}}\xrightarrow{m \to \infty} 0$
> > and $-\int (\partial_{i} u _{m} - v) \varphi \leq \|\partial_{i} u_{m}- v\|_{L^{p} } \|\varphi\|_{L^{p'}} \xrightarrow{m \to \infty}0.$
> > The result follows.
> >
>
[^1]: Unique as elements of the [[quotient module|quotient]] $L^{1}_{\text{loc}}$, that is — unique up to a set of [[Lebesgue measure]] zero. Indeed, if $g,h$ both make the IBP identity hold, then $\int _{\Omega} g \varphi =\int _{\Omega} \, h \varphi$ for all test functions $\varphi$ hence $\int _{\Omega} (g-h) \varphi \, =0$ for all test functions $\varphi$ and the [[fundamental lemma of the calculus of variations|FLOTCOV]] says $g-h=0$ up to a null set.
> [!intuition]
>
>
>
> First let $f \in C^{1}_{c}(\Omega)$. Recall the [[integration by parts]] formula $(i \in [n])$ $\int _{\Omega} f \partial_{i} g =- \int_{\Omega} g \partial_{i} f \, \text{ for all } g \in C^{1}_{c}(\Omega) .$
> If we relax to allow $f \in L^{1}_{\text{loc}}$, then the LHS still makes sense but the RHS need not; $\partial_{i}f$ might not be defined. If $h \in L^{1}_{\text{loc}}$ is such that $\int _{\Omega}f \partial_{i}g \, = - \int _{\Omega} g h \text{ for all } g \in C ^{1}_{c}(\Omega),$ then $h$ 'behaves like it were the derivative of $f