Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 12 / 3

Let $d=\mathrm{dist}({\Omega }^{\prime },\partial \Omega )>0$. Choose a smooth cutoff function $\eta \in C_c^{\infty }(\Omega )$ with $0\le \eta \le 1$, $\eta =1$ on ${\Omega }^{\prime }$ and $|D\eta |\le C(n)/d$. Such cutoffs can be constructed by mollifying the indicator of a fixed small neighbourhood of $\overline{{\Omega }^{\prime }}$. Write $M=\Vert A{\Vert }_{\infty ,\mathrm{op}}$, $B=\Vert b{\Vert }_{\infty }$ and $C_0=\Vert c{\Vert }_{\infty }$; these are controlled by the individual coefficient bounds and dimension.

The weak formulation, extended to compactly supported $H^1$ tests by density, isWe may use $\phi ={\eta }^{2}u$, since $u\in H_{\mathrm{loc}}^1$. Expanding its gradient and using uniform ellipticity givesApply Young inequality to the first two terms, allocating at most $\lambda /4$ of the gradient integral to each. Bound $|fu|\le (f^2+u^2)/2$ in the last term. Absorbing those gradient contributions yields the Caccioppoli inequality with bounded lower-order termsConsequently $\Vert Du{\Vert }_{L^2({\Omega }^{\prime })}\le C(\Vert u{\Vert }_{L^2(\Omega )}+\Vert f{\Vert }_{L^2(\Omega )})$. Adding the $L^2$ norm of $u$ proveswhere $C$ has precisely the stated dependence on dimension, ellipticity, coefficient bounds and $d$. The proof uses $|c|$, so no sign condition on the zeroth-order term is required.

First use weak compactness of bounded sets in the Hilbert space $L^2(\Omega )$ to pass to a subsequence with $u_k\rightharpoonup u$ globally in $L^2$. The weak lower semicontinuity of the Hilbert norm gives $\Vert u{\Vert }_{L^2(\Omega )}\le K$. This global step also applies if $\Omega$ is unbounded.

Choose a nested exhaustion $U_j⋐U_{j+1}⋐\Omega$ by bounded smooth open sets. Part (i) bounds $u_k$ in $H^1(U_j)$ for every $j$. By the Rellich-Kondrachov compactness theorem, $H^1(U_j)$ embeds compactly in $L^2(U_j)$. Weak compactness in $H^1(U_j)$ and a diagonal subsequence argument therefore give, for the same subsequence,The local limit agrees with the global weak $L^2$ limit. Using smooth exhaustion sets avoids imposing any unprinted boundary regularity on a general ${\Omega }^{\prime }$.

For a smooth compactly supported test function, the weak equation passes to the limit: the fixed bounded coefficients multiply fixed test functions, and local weak gradient convergence handles both the principal and drift terms. The zeroth-order term passes by local $L^2$ convergence. Thus $Lu=f$ weakly, $u\in H_{\mathrm{loc}}^1(\Omega )$, and the global bound already proved gives $u\in S_K$.

To improve convergence of the gradients, put $w_k=u_k-u$. Crucially the forcing is the same for all $k$, so $L{w}_k=0$. Given ${\Omega }^{\prime }⋐\Omega$, choose $j$ with $\overline{{\Omega }^{\prime }}\subset U_j$. Apply part (i) on $U_j$ to the homogeneous equation for $w_k$:This establishes strong local Sobolev compactness for a fixed elliptic equation and proves