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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.

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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,

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

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