Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 13 / 6 / a

For smooth $u$, the fundamental theorem along the segment in direction $e_k$ gives, for either sign of $h$,The entire segment lies in $\Omega$ under the distance hypothesis. The Cauchy-Schwarz inequality on $[0,1]$, followed by integration and translation, yieldsFor general $u\in W^{1,2}(\Omega )$, extend $u$ and its weak gradient separately by zero as $L^2$ functions on ${\mathbb{R}}^n$, and mollify $u$ with radius $\epsilon$. At points farther than $\epsilon$ from the boundary the mollified gradient equals the convolution of the zero-extended gradient, because the convolution kernel is supported wholly inside $\Omega$. Choose $\epsilon <\mathrm{dist}({\Omega }^{\prime },\partial \Omega )-|h|$. All the segments then lie in this region, and the preceding calculation, followed by the $L^2$ contraction bound for convolution, gives the same right side $\Vert D_{k}u{\Vert }_{L^2(\Omega )}$. The mollifications converge locally in $L^2$, as do their fixed-$h$ translated differences, so passage to the limit preserves the estimate. HenceThe argument proves the sharper estimate with $D_{k}u$ on the right. It is the basic Sobolev difference quotient bound.