I'm studing weak and distributional derivatives and solutions and I have a few questions about it.
- From my understanding, one defines a weak derivative of $u \in L^{1}_{loc}(\Omega)$ such that
$$ \int_{\Omega} D^{\alpha} u \varphi dx = (-1)^{|a|}\int_{\Omega} v D^{\alpha}\varphi dx$$
for all test functions $\varphi \in C^{\infty}_{c}$ and $v \in L^{1}_{loc}(\Omega)$. So $v$ is a weak derivative of u.
My question is: what is the difference between this definition and the definition of distributional derivatives? From what I understood, the integral of ($\int_{\Omega} u \varphi dx$) defines a distribution so the distributional derivative is defined as seen in equation above.
I understand that while working with weak derivatives, I'm differentiating functions and when one is working with distributional derivatives it's obviously differentiating distributions. But what is the difference between these two definitions?
- Is it necessary, for the definition of a weak derivative, that the function $u \in L^{1}_{loc}$?
Thanks.