I am studying PDE's and we have defined the following notion of weak derivative:
Given a domain $\Omega\subset\mathbb{R}^{n}$ a function $f\in L^1_{loc}(\Omega)$ is wealy differentiable with respect to the $i-th$ variable if there exist a function $f_{x_i}\in L^1_{loc}$ such that for all test functions $\phi\in C^{\infty}_c(\Omega)$ we have $$\int f_{x_i}\phi =-\int f\phi_{x_i}$$
Now my questions are:
When the weak and classical and weak derivative do or do not coincide?
I can think that they coincide $f\in C^1$, while they cannot coincide when f is piecewise $C^1$ and shows jumps (since then we get a Dirac $\delta$ in the distributional derivative).
If the classical derivative presents a singularity whcich is not a jump we have to test its action on the generic $\phi$ (this boils down to showing the singularity is summable).
Is the definition of weak derivative the most safe and convenient way to compute it ?
Thanks for help!
Best Answer
In $\mathbb R$:
If $f$ is derivable everywhere, then weak derivative and derivative coincides.
In somehow, weak derivative is rather an equivalence class than a function.
If $f$ is absolutely continuous, then weak derivative and strong derivative coincides (although strong derivative is defined only a.e.)
If $f$ is a.e. derivable (but not absolutely continuous), then the weak derivative may not exist or be different (example : Cantor function $F$ is derivable a.e. it's derivative if $0$, but if the weak derivative exist they can't coincides because otherwise $$\forall \varphi \in \mathcal C_0^\infty [0,1],\int F\varphi '=0\implies \exists C>0: F=C\text{ a.e.}$$ which doesn't hold.)