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

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.

Put $\delta =\mathrm{dist}({\Omega }^{\prime },\partial \Omega )>0$. Choose a smooth cutoff $0\le \eta \le 1$, equal to one on ${\Omega }^{\prime }$, supported within its $\delta /2$ neighborhood, with $|D\eta |\le C_n/\delta$. Such a cutoff is obtained by smoothing a distance cutoff; its support is relatively compact in $\Omega$. Let $0<|h|<\delta /8$, fix a coordinate $k$, and put $z={\Delta }_k^{h}u$. It lies in $W^{1,2}$ on the cutoff neighborhood because both translated copies of $u$ do. Extend ${\eta }^{2}z$ by zero, and use the test functionThe weak equation is $\int Du\cdot D\varphi =-\int f\varphi$. The discrete integration-by-parts identity $\int a{\Delta }_k^{-h}b=-\int ({\Delta }_k^{h}a)b$, valid by translation with these supports, turns it intoWrite $X=\Vert \eta Dz{\Vert }_2$, $U=\Vert z{\Vert }_{L^2(\mathrm{supp}\eta )}$, $A=\Vert D\eta {\Vert }_{\infty }$ and $F=\Vert f{\Vert }_2$. Expanding the left side gives $X^2$ plus a cross term of absolute value at most $2AUX$. Applying part (a) on the whole space to the zero extension of ${\eta }^{2}z$ bounds the right side byConsequentlywhere the last step is the elementary Young inequality. Part (a), on a slightly larger cutoff neighborhood if needed, gives $U\le \Vert D_{k}u{\Vert }_{L^2(\Omega )}$. We conclude thatuniformly as $h\to 0$.

To obtain actual second weak derivatives, choose $h_j\to 0$. For each $i,k$, the bounded sequence ${\Delta }_k^{h_j}{D}_{i}u$ has a weakly convergent subsequence in $L^2({\Omega }^{\prime })$, say to $g_{ik}$. For every smooth compactly supported $\psi$ in ${\Omega }^{\prime }$,Thus $g_{ik}=D_k{D}_{i}u$ is the second weak derivative. weak lower semicontinuity gives the same uniform bound for these derivatives. Adding the already known zeroth and first derivative norms and summing over the finitely many coordinates provesAll derivative existence and estimates have been established by the difference-quotient argument; no second derivatives were assumed in choosing the test. This proves the Interior second-derivative estimate for the Poisson equation in the stated setting.