Let $Q=[0,1]^{n}$. Define a $Q$-periodic function as follows:
A function $f:\mathbb{R}^{n}\to\mathbb{C}$ is said to be $Q$-periodic if for any $x\in\mathbb{R}^{n}$ and any $z\in\mathbb{Z}^{n}$, it holds that
$$f(x+z)=f(x)$$
Consider the (Hilbert) space $H^{1}_{\text{per}}(Q)$ of $Q$-periodic functions that are $H^{1}_{\text{loc}}(\mathbb{R}^{n})$ (i.e. $H^{1}$ on any compact), endowed with the following hermitian product:
$$(u,v)\mapsto (u,v)_{1,Q}:= \int_{Q}\nabla u\cdot\nabla \overline{v}+\int_{Q}u\overline{v}$$
We denote $\Vert u\Vert_{H^{1}_{\text{per}}(Q)}:=(u,u)_{1,Q}^{1/2}$. Similarly, we consider the (Hilbert) space $L^{2}_{\text{per}}(Q)$ of $Q$-periodic functions that are $L^{2}_{\text{loc}}(\mathbb{R^{n}})$, endowed with the following hermitian product:
$$(u,v)\mapsto (u,v)_{0,Q}:=\int_{Q}u\overline{v}\hspace{1em}\text{ and }\hspace{1em}\Vert u\Vert_{L^{2}_{\text{per}}(Q)}:=(u,u)_{0,Q}^{1/2}$$
I want to bound the following quantity:
$$\min_{W_{m}\subset H^{1}_{\text{per}}(Q)}\max_{\substack{u\in W_{m}\\ \Vert u\Vert_{L^{2}_{\text{per}}(Q)}=1}}\int_{Q}\nabla u\cdot\nabla\bar{u}$$
where $W_{m}$ denotes any $m$-dimensional subspace of $H^{1}_{\text{per}}(Q)$.
I thought of the laplacian eigenvalues, but the functions are not in $H^{1}_{0}(Q)$. How can we exploit the periodicity here?
Any help is appreciated (hints are good of course).