Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 12 / 4

We prove the Interior second-derivative estimate for the Poisson equation using difference quotients, rather than differentiating the $L^2$ forcing. Fix a coordinate $e_k$ and write ${\delta }_{h}v(x)=(v(x+h{e}_k)-v(x))/h$. Choose a smooth cutoff function $\eta$ equal to one on ${\Omega }^{\prime }$ with support compactly contained in $\Omega$, leaving room for all translations with sufficiently small $|h|$. We can arrange $|D\eta |\le C(n)/d$, where $d=\mathrm{dist}({\Omega }^{\prime },\partial \Omega )$.

The standard difference quotient lemmas give discrete integration by parts, $\int a{\delta }_{-h}b=-\int ({\delta }_{h}a)b$, and $\Vert {\delta }_{h}v{\Vert }_2\le \Vert D_{k}v{\Vert }_2$ on the relevant enlarged set. In the weak formulation $\int Du\cdot D\phi =-\int f\phi$, use the admissible test function $\phi =-{\delta }_{-h}({\eta }^2{\delta }_{h}u)$. Moving a difference quotient onto $Du$ givesSet $X=\Vert \eta D{\delta }_{h}u{\Vert }_2$ and $Y=\Vert D\eta {\delta }_{h}u{\Vert }_2$. The cross term has absolute value at most $2XY$. Since $0\le \eta \le 1$, the difference quotient lemma on the right givesThe Young inequality now implies $X^2\le C(Y^2+\Vert f{\Vert }_2^2)$. Moreover $Y\le C(n)d^{-1}\Vert Du{\Vert }_{L^2(\Omega )}$, independently of $h$. Hence, for every coordinate $k$,By the Sobolev characterization by bounded difference quotients, applied to each first weak derivative on smaller interior sets, these uniform bounds give all second weak derivatives. Their bounds pass to the weak limit as $h\to 0$. Thus $u\in W_{\mathrm{loc}}^{2,2}(\Omega )$ andNo derivative of $f$ is used, and no boundary condition is needed for this interior elliptic regularity argument.

First perform homogenization of Dirichlet boundary data: put $v=u-\psi \in H_0^1(B_1^{+})$ and $g=f-\Delta \psi \in L^2(B_1^{+})$. Then $\Delta v=g$ weakly. Extend both functions oddly across the flat face:The zero Sobolev trace of $v$ makes this odd reflection an $H^1(B_1)$ function. This can also be seen by approximating $v$ by smooth compactly supported functions in $B_1^{+}$ and reflecting the approximants. Its $H^1$ norm is $\sqrt{2}$ times that of $v$, and $\Vert \tilde{g}{\Vert }_{L^2(B_1)}=\sqrt{2}\Vert g{\Vert }_{L^2(B_1^{+})}$.

For $\varphi \in C_c^{\infty }(B_1)$, the function $\varphi (x^{\prime },t)-\varphi (x^{\prime },-t)$ on the upper half-ball has zero trace on the flat face and vanishes near the curved face, hence is an admissible $H_0^1$ test. Splitting the integrals over the two half-balls and changing $t$ to $-t$ showsThus $\Delta \tilde{v}=\tilde{g}$ weakly throughout the full ball; there is no extra distribution on the reflecting face.

Apply part (i) to $B_{1/2}⋐B_1$, restrict to the upper half-ball, and then add $\psi$. Since $\Vert \Delta \psi {\Vert }_2\le C(n)\Vert \psi {\Vert }_{H^2}$, this gives $u\in W^{2,2}(B_{1/2}^{+})$ and the second-derivative estimate at a flat Dirichlet boundaryThe fixed radii leave distance $1/2$ for the interior estimate, so the constant depends only on dimension.