Consider the elliptic equation $$ \begin{aligned} -\nabla \cdot A \nabla u=f, & \text { in } \Omega, \\ u=0, & \text { on } \partial \Omega. \end{aligned} $$ where $\Omega$ is a bounded domain. Suppose there is the regularity assumptions $u\in H^2$. In the paper I am reading, it mentions that even if the coefficient is smooth, say at frequency $\varepsilon^{-1}$ for some small parameter $\varepsilon$, $\nabla^2 u$ may oscillate at the same scale, a fact that is hidden in the constant $\left\|\nabla^2 u\right\|_{L^2(\Omega)} \approx \varepsilon^{-1}$.
Q: how to prove $\left\|\nabla^2 u\right\|_{L^2(\Omega)} \approx \varepsilon^{-1}$?
This is the paper : enter link description here