# Poisson equation on square with periodic boundary conditions up to first derivatives

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 $$\partial_yu$$.

$$H^1(\Omega)$$ is not the correct space to analyze this problem. The correct space to analyze is to use $$H^1(\mathbb{T}^2)$$, where $$\mathbb{T}^2$$ is the 2D torus. The $$L^2$$-norm can be most easily understood using Parseval's identity for Fourier series: let $$k\in \mathbb{Z}^2$$ and $$\hat{v}_k := \int_{[0,L]^2} e^{-2\pi i k\cdot x} v(x)\, dx \quad \text{ and } \quad \|v\|_{L^2(\mathbb{T}^2)}^2 = c\sum_{k\in \mathbb{Z}^2} |k|^2,$$ where $$c$$ is a constant determined by the Fourier basis you choose but fixed once the basis are chosen. $$H^1$$ follows naturally. $$H^1(\mathbb{T}^2) := \{v\in L^2(\mathbb{T}^2): \|v\|_{H^1(\mathbb{T}^2)}<\infty\},$$ where $$\|v\|_{H^1(\mathbb{T}^2)}^2 = \sum_{k \in\mathbb{Z}^2}(1 +|k|^2) |\hat{v}_k|^2.$$ Poincare inequality is naturally true for this space (which is needed to prove Lax-Milgram), and you are good to go.