Assume we have established the following version of Caccioppoli inequality
$$\int |\nabla u|^2 \psi^2 dA\leq C \int u^2 |\nabla \psi| ^2 dA$$
for $C^2(\mathbb C)$- smooth functions $u\geq 0$ with $\Delta u\geq 0$, and $\psi\in C_c^\infty (\mathbb C)$ (compactly supported, smooth) test functions.
Is there a way to upgrade this inequality, so that it holds for $\psi \in C_c(\mathbb C)$ (continuous, compactly supported), such that $\nabla \psi$ exists almost everywhere, and it is bounded, and supported on a finite measure set?
The reason is that I want to use a bump function $\psi$ such that $\psi=1$ on a disk $D(0,a)$, $\psi=0$ outside $D(0,b)$ ($b>a$), but $\nabla \psi$ does not exist on $|z|=a,|z|=b$.
Best Answer
As written: not enough. On the right, $|\nabla \psi|^2$ must be the weak derivative of $\psi$; nothing short of it can control the oscillation of $\psi$. Bounded pointwise a.e. derivative need not be weak.
But, I see that the function $\psi$ you want to use is Lipschitz. Then everything is fine. As a matter of fact, $\psi\in W^{1,2}_0(\mathbb C)$ is enough. Just note that $C^\infty_c$ is norm-dense in $W^{1,2}_0(\mathbb C)$, and both sides of the Cacciopoli inequality depend continuously on $\psi$ with respect to the $W^{1,2}$ norm.