Suppose I have:
$\begin{cases}-\Delta u= f, &\text{ on } \Omega\\ \nabla u \cdot n = g &\text{ on } \partial \Omega\\ \int_\Omega u = \operatorname{const}. \end{cases}$
I'm supposed to find what conditions $f$ and $g$ satisfy for existence of solutions. I have no idea where to use the last condition that the area of $u$ vanishes. Any help? Please do not tell me the answer as it's homework.
I tried looking at the weak formulation and coercivity and boundedness of the bilinear form are fine on $H_0^1$.