Let $\Sigma$ be a compact surface and denote by $\nu$ the unit conormal for $\partial \Sigma$. Let
$$E = \left\{ \phi \in C^{2,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\} $$
and
$$F = \left\{ \phi \in C^{0,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\},$$
for fixed $0 < \alpha < 1$.
I would like to know why, given $(f,g) \in F \times C^{1,\alpha}(\partial \Sigma)$, the problem
$$\begin{cases} -\Delta_{\Sigma} u = f + \frac{1}{\vert \Sigma \vert} \int_{\partial \Sigma} g \, \mathrm{d}L, \quad \text{on } \Sigma \\ \frac{\partial u}{\partial \nu} = -g, \quad \text{on } \partial \Sigma \end{cases}$$
has a solution in $E$. This came up for me in a paper. I am sorry if the question is silly, but I am not very good at PDE's.