A function $u$ on the square $\Omega=[0,L]\times[0,L]$ is said to satisfy periodic boundary conditions (PBCs) if $u(x,0)=u(x,L)$ and $u(0,y)=u(L,y)$ for all $x,y\in[0,L]$. Now consider the Poisson equation $$\Delta u=f \;\;\text{ on }\;\;\Omega,$$ where $f\in L^2(\Omega)$. Suppose that we seek a solution $u$ such that $u$, $\partial_xu$ and $\partial_yu$ satisfy periodic boundary conditions.
Question. Which theorem or method of solving should we use to tackle such a problem? If there is a solution, is it unique?
For instance, in variational form we have $$\langle \nabla u,\nabla v\rangle=\langle f,v\rangle\;\;\;\;\text{ for all }\;\;\;\;\;v\in H:=\{\phi\in H^1(\Omega):Trace(\phi) \text{ satisfies PBCs}\}$$ (the boundary integral will vanish after applying Green's formula). So it almost seems we can apply Lax-Milgram on $H$ provided we know that $a(u,v)=\langle \nabla u,\nabla v\rangle$ is coercive on $H$. However that does not seem to be the case!
I further noticed that the divergence theorem implies that $$\int_\Omega f=\int_\Omega \Delta u=\int_{\partial\Omega} \nabla u\cdot\vec{n}=0, $$ where the last equality holds because of the periodicity of $\partial_xu$ and $\partial_yu$.