Definition of the Sobolev space $H^1_{periodic}(\mathbb{R}^2)$

Question

please what's the definition of the space $H^1_{periodic}(\mathbb{R}^2)$ and what is the inclusion between $H^1(\mathbb{R}^2)$ and $H^1_{periodic}(\mathbb{R}^2)$?

Thank you in advance for the help.

Answer 1

I don't think there is a standard definition. A possible one is: Given $f \in H^1_{\text{loc}}(\mathbb{R^2})$ you say that $f\in H^1_{\text{periodic}}(\mathbb{R^2})$ if $f(x+e_i)=f(x)$ for all $x\in \mathbb{R}^2$ and for $i=1,2$, where $e_1=(1,0)$ and $e_2=(0,1)$. There is no relation with $H^1(\mathbb{R^2})$, since if you decompose $\mathbb{R^2}$ into countably many disjoint cubes, then by periodicity $$\int_\mathbb{R^2}|f|^2dx=\sum_n\int_{Q_n}|f|^2dx=\sum_n\int_{Q}|f|^2dx=\infty.$$