I would like some reference about Periodic Sobolev Spaces. I found some references, like the book of R. Teman, but all these references deal with Periodic Sobolev Spaces that have the period defined in a cube like $Q=[0,L]^n$ of $\mathbb{R}^n$, i.e. the functions are periodic in all variables. I need Periodic Sobolev Spaces such that the function are periodic only for some variable.
In order to be more precise, I need to define spaces $L^2$ and $H^2$ for functions that satisfy:
$$u(x+Le_1)=u(x+Le_1), \ \ \forall x\in \mathbb{R}^n, \ \ \forall t>0,$$ tha is, they are periodic only for one direction.
Edited: Actually, I need these spaces because I am concerned with the following problem:
$$\begin{cases} -u_{xx}-au_{yy}+b_1u_{x}+b_2u_{y}+cu=f, & (x,y)\in \mathbb{R}\times (0,1)\\ u(x+L,y)=u(x,y), & (x,y)\in \mathbb{R}\times (0,1)\ \ \text{(periodic condition)}\\ u(x,0)=u_0(x), \ \ u(x,1)=u_1(x) & x\in \mathbb{R}, \end{cases}$$ where $a,b,c,f \in C^1(\mathbb{R}\times [0,1])$, periodic with respect to $x$ and $u_0,u_1$ are regular enough periodic functions.