Let $\Omega \subset \mathbb{R}^n$ be an open and bounded set and $u^\varepsilon : \mathbb{R}^n \rightarrow \mathbb{R}$ an $\varepsilon$-periodic function, i.e. $$ u(x + \tau) = u(x) \quad \text{for all } \tau \in \varepsilon \mathbb{Z}^n. $$ Let $Q(0, \varepsilon)$ be the cube centered at $0$ with side length $2\varepsilon$. Assume $$ \frac{1}{\varepsilon^n} \int_{Q(0, \varepsilon)} |u^\varepsilon|^2 \, dx \to I$$ as $\varepsilon \to 0$. Is it true that $$ \int_{\Omega} |u^\varepsilon|^2 \, dx \to \frac{|\Omega|}{2^n} I? $$
Clearly, the number of non-overlapping cubes of side length $2\varepsilon$ contained in $\Omega$ is asymptotically equal to $|\Omega|/(2\varepsilon)^n$ so that $$ \int_{\Omega} |u^\varepsilon|^2 \, dx \approx \frac{|\Omega|}{(2\varepsilon)^n} \int_{Q(0, \varepsilon)} |u^\varepsilon|^2 \, dx $$ However, I don't think this is enough to prove the claim since $|u^\varepsilon|^2$ might have a big integral on the small subset of $\Omega$ not covered by the cubes. I would appreciate any help.
Edit: The claim is made in the paper "A Strange Term Coming from Nowhere" which can be found here (page 56). There a function $w^\varepsilon$ is defined on $Q(0, \varepsilon)$ as follows: For $x$ in the annulus $\{x \in \mathbb{R}^n : a^\varepsilon < x < \varepsilon \}$ set $$ w^\varepsilon(x) = \frac{(a^\varepsilon)^{2 - n} - |x|^{2 - n}}{(a^\varepsilon)^{2 - n} - \varepsilon^{2 - n}}, $$ where $0 < a^\varepsilon < \varepsilon $. Otherwise set $w^\varepsilon(x) = 0$. Extend $w^\varepsilon$ periodically to $\mathbb{R}^n$. Take $u^\varepsilon = \nabla w^\varepsilon$.
Note that $u^\varepsilon$ is rotationally symmetric on $Q(0, \varepsilon)$ and $$ \int_{Q(0, \varepsilon)} |u^\varepsilon|^2 \, dx = \frac{\omega_n(n - 2)}{(a^\varepsilon)^{2 - n} - \varepsilon^{2 - n}}$$ where $\omega_n$ is the surface area of the $(n - 1)$-dimensional unit sphere. Finally, take $a^\varepsilon = C_0 \varepsilon^{n/(n -2)}$ for some constant $C_0$ and observe that $$ \frac{1}{\varepsilon^n} \int_{Q(0, \varepsilon)} |u^\varepsilon|^2 \, dx \to \omega_n(n - 2)C_0^{n - 2}.$$
Edit: Corrected the formula for $w^\varepsilon$.