Let $f\geq0$ be a continuous function satisfying $\Delta f=0$ in $\{x\in U:f(x)>0\}$.
I was wondering if one could follow $\Delta f=0$ in $U$, especially in the cases $f\in C^2$ or $\Delta f=0$ in a weak sense.
So I've tried to show it for $\Delta u=0$ in weak sense, e.g. $$ \int_U\nabla f\nabla\phi=0 $$for all $\phi\in C_0^\infty(U)$. I've considered instead of just $\phi$ the function $\phi \eta_\delta(f)$ with a bounded function $\eta_\delta$ satisfying $\eta_\delta(x)=0$ if $x\leq \delta$ and $\eta_\delta(x)=1$ if $x\geq 2\delta$. Then we get for the left side $$ \int \eta_\delta(f)\nabla f\nabla\phi+\int\eta_\delta'(f)\nabla f\nabla f $$Now I've tried to focus on some other conditions for $\eta_\delta'$ but it didn't get me any further.
Any hints how you can show $\Delta f=0$ in $U$?
No. Let $U=\Bbb R^2$, $f(x,y)=|x|$.