Inequality involving periodic functions and Sobolev space.

Question

Set $n\in\mathbb{N}$. For $\psi=u+iv$ in $H^1_T(\mathbb{R}^n)=\lbrace \psi\in H^1_{loc}(\mathbb{R}^n,\mathbb{C}):\psi(x)=\psi(x_1+T,...,x_n+T)$, I wonder if $$ \int_Q |\nabla u|^2\geq C\int_Q (u-1)^3 $$ Where $Q$ is $[0,T]^n$ and $C>0$ is a constant. Is there some Gagliardo-Niremberg inequality for $H^1_T$-functions? Please let me know. Thanks in advance!