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This question is about the proof of Theorem 1 in Chapter 8 of Evans' PDE book (p. 468 in the 2nd edition). Let $u,u_k\in\mathrm{W}^{1,q}(U)$ for all $k\in\mathbb{N}$, $U\subset\mathbb{R}^n$ be open, bounded, $L$ smooth, $1

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    What are $u_k$?2012-09-15
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    Ooops, forgot the most important part :D.2012-09-15

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It's a consequence of mean value inequality, as $u_k(x)$ and $u(x)$ is uniformly bounded (says by $C$). Then \begin{align} \small\left|\int_{G_\varepsilon}L(Du,u_k,x)dx-\int_{G_\varepsilon}L(Du,u,x)dx\right|&\leqslant \int_{G_\varepsilon}\left|\int_{u_k(x)}^{u(x)}\partial_2L(Du,t,x)dt\right|dx\\ &\leqslant |G_{\varepsilon}|\sup_{-R\leqslant t_1,t_2,t_3\leqslant R}|\partial_2L(t_1,t_2,t_3)|\cdot \sup_{x\in G_\varepsilon}|u_k(x)-u(x)|. \end{align} As $G_\varepsilon$ is bounded it has a finite measure and the supremum is finite (as $L$ is smooth hence $\partial_2L$ is continuous on the compact set $[-R,R]^3$).

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    Is there a term $|u-u_k|$ missing in the last step?2012-09-15
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    Yes, I will add it readily.2012-09-15
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    Thanks for solving, should have thought of that myself :).2012-09-15