Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 2

Freeze the lower-order coefficients on a ball and apply the interior Schauder estimate for the Poisson equation to a cutoff of $u$. The terms $b_i{D}_{i}u+cu$ are controlled by the Hölder interpolation inequalityChoosing $\eta$ in terms of the prescribed $\delta$ yields

The standard iteration lemma for nested balls absorbs the first term. The remaining $C^2$ norm is bounded by the interior derivative estimate and interpolation, producingEquivalently, a contradiction-and-rescaling proof would produce a globally Hölder harmonic limit forbidden by the Polynomial-growth Liouville theorem for harmonic functions.

Fix $B_r⋐B_R⋐B$. Local uniform convergence bounds $|u_k{|}_{0;B_R}$ uniformly. Interior Schauder estimates therefore bound $u_k$ uniformly in $C^{2,\alpha }(B_r)$. The Arzela-Ascoli theorem gives a subsequence converging in $C^2(B_r)$. Its limit must be the locally uniform limit $v$, and passing to the limit in $L{u}_k=0$ gives $Lv=0$. A diagonal argument over $r\uparrow 1$ proves

Yes. The local $L^1$-to-$L^{\infty }$ estimate for solutions of a uniformly elliptic equation with bounded Hölder coefficients gives, whenever $B_r⋐B_R$,Thus $(u_k)$ is locally uniformly Cauchy and converges locally uniformly to a continuous representative of the $L^1$ limit $v$. Part (b) then applies, so $v\in C^2(B)$ and $Lv=0$.

No. Let ${\lambda }_1>0$ be the First Dirichlet eigenvalue of $-\Delta$ on $B$, choose a corresponding nonzero eigenfunction $w$, and takeThen $L{u}_k=0$ and every $u_k$ vanishes on $\partial B$, so the boundary values converge uniformly to those of $v=0$. But $|u_k{|}_{0;B}\to \infty$, and hence no subsequence converges uniformly on $B$.