According to the Wiki article, if $u$ and $v$ are locally integrable functions on some open subset of $\mathbb{R}^n$, then $v$ is the weak derivative of $u$ if, for any infinitely differentiable function $\varphi$ on $U$ with compact support, we have $$\int_U u D^{\alpha}\varphi = (-1)^{|\alpha|}\int v\varphi,$$ where $$D^{\alpha}\varphi = \frac{\partial^{|\alpha|}\varphi}{\partial x_1^{\alpha_1}...\partial x_n^{\alpha_n}}.$$

They then go on to say that the weak derivative is often notated $D^{\alpha}u$. Replacing this in the above definition, we get

$$\int_U u D^{\alpha}\varphi = (-1)^{|\alpha|}\int D^{\alpha}u \varphi.$$

Just checking my understanding here: $D^{\alpha}$ is used to signify two different things here, right? The one of $\varphi$ is a big partial derivative, while the one on $u$ denotes the weak derivative (i.e. a locally integrable function that satisfies that satisfies that identity). If so, is there no better notation that we can use? This looks terribly confusing.

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    $\begingroup$ It rarely happens that you need to distinguish between the various kinds of derivatives (strong, weak), so it would be much more confusing to have different symbols for them. $\endgroup$ – Giuseppe Negro Jan 17 '17 at 15:21

Yes $D^{\alpha}u$ is a notation for the weak derivative that satisfies the "partial integration-equation" and $D^{\alpha}\varphi$ denotes the classical derivative which is of course a weak derivative too, because it satisfies

$\int_U\phi D^{\alpha}\psi=(-1)^{|\alpha|}\int_U D^{\alpha}\phi \psi$

for all smooth $\psi$ with compact support. So the notation is now understood as the weak derivative but is still consistent with that of a classical partial derivative. Even if it is confusing at the beginning it´s better to get used to it because it is used very often (cf. Evans Partial differential equations and many others).


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