Let \(\Omega\subset \mathbb{R}^n\), \(u,v\in L^1_{\text{loc}}(\Omega)\) and \(\alpha\) is an multi-index . We call \(v\) the \(\alpha\)-th weak derivatve of \(u\) if
\begin{equation*} \int_{\Omega} v \varphi = (-1)^{\lvert \alpha\rvert}\int_{\Omega} u D^\alpha \varphi, \end{equation*}for every test function \(\varphi \in \mathcal{D}(\Omega)\).
Remarks
- The derivative is a.s. unique. Therefore we denote the \(\alpha\)-th weak derivative of \(u\) bu \(D^\alpha u\).
- Using distribution theory one can say that the derivative of \(u\) must be locally integrable .