Poisson functional on bounded domain I was wondering if it is actually clear that on bounded domains the Poisson integral is bounded from below:
$$I[u]=\int_{\Omega} \left( \frac{1}{2}\lvert \nabla u \rvert^2 - u\rho \right)\, dx,$$
I tried to show it with Poisson's inequality, but I did not get it. So is this even true and does anybody know how to show it?
If anything is unclear, please let me know.

I will assume that $\rho \in L^2(\Omega)$ and that $u \in H_0^1(\Omega)$. By Holder's inequality,
$$
\int_\Omega \rho u \leq \|\rho\|_2 \|u\|_2.
$$
Therefore, recalling that $H_0^1(\Omega) \subset L^2(\Omega)$ with continuous embedding,
$$
I(u) \geq S \|u\|_2^2 - \|\rho\|_2 \|u\|_2 ,
$$
and you should get the result from this rather easily.
