If $u$ solves Dirichlet problem $-\Delta u =f$, what is known about $fu$?

Question

Let $u \in H^1_0(\Omega)$ be the weak solution of $$-\Delta u = f$$ $$u|_{\partial\Omega} = 0$$

Is there anything known about the sign of $fu$ a.e?

Answer 1

As Lion said: the integral of $fu$ is nonnegative, but not because of any pointwise bound for $fu$. It's integration by parts: $\int_\Omega (-u\,\Delta u ) = \int_{\Omega}|\nabla u|^2\ge 0$.

To give a simple example: let $u (x)=x(x-1)(x-4) $ in one dimension, with $\Omega=(0,4)$. Then $f(x) = -u''(x) = 10-6x$. The product $fu$ changes sign at $1$ and at $5/3$: