Inequality in Evans PDE section 5.7

Question

I'm stuck in the proof of the Compactness Theorem in Evans PDE 2nd edition book. On page 287, last line, how do you get the inequality $$ \epsilon \int_{B\left(0,1\right)}\eta\left(y\right)\int^{1}_{0}\int_{V}\left|Du_{m}\left(x-\epsilon ty\right)\right|dxdtdy \leq \epsilon\int_{V}\left|Du_{m}\left(z\right)\right|dz.$$ Any help will be appreciated.

Answer 1

The function $|Du_m|$ is compactly supported in $V$, and moreover it is assumed that its support is at distance more than $\epsilon$ from the boundary of $V$. Therefore, for every $t\in [0,1]$ and $y\in B(0,1)$ integration of $|Du_m(x-\epsilon ty)|$ over $V$ gives the same result as integration of $|Du_m(x)|$. (Translating the function by less than $\epsilon$ within the domain does not change the integral.) Thus,

$$\begin{split} \int_{B\left(0,1\right)}\eta\left(y\right)\int^{1}_{0}\int_{V}\left|Du_{m}\left(x-\epsilon ty\right)\right|dxdtdy &= \int_{B\left(0,1\right)}\eta\left(y\right)\int^{1}_{0}\int_{V}\left|Du_{m}\left(x \right)\right|dxdtdy \\ &= \int_{V}\left|Du_{m}\left(x \right)\right|dx \int_{B\left(0,1\right)}\eta\left(y\right)\,dy\int^{1}_{0}dt \\ &= \int_{V}\left|Du_{m}\left(x \right)\right|dx \\ \end{split} $$