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Theorem: Let $u \in W^{1,p}(U)$ and let $V \subset \subset U$ (I.e. there is a compact set containing $V$ that is in $U$). Then for $1\le p <\infty$ there exists a constant $C$ such that for $0 < h < \frac{1}{2} \text{dist}(V,\partial U)$,

$$||D^h u||_{L^p(V)} \le C||Du||_{L^p(U)}$$

Where $D^h u$ is a vector such that $(D^h u)_i = \frac{u(x+he_i)-u(x)}{h}$.

Evans covers this proof, but I cannot see why we need $0<h < \frac{1}{2} \text{dist}(V,\partial U)$. It seems to me that $0<h < \text{dist}(V,\partial U)$ is enough.

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You are right. $0<h<\mathrm{dist}(V,\partial U)$ is enough, please refer to Gilbarg and Trudinger Lemma 7.23 about difference quotient.

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