Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 9 / 4

We use a blow-up compactness proof of an interior Schauder estimate, with Taylor normalization in elliptic blow-up arguments. Fix $n,\mu ,\delta$ and suppose no suitable constant exists. There would be solutions $u_j$ with forcing $f_j$ for whichChoose distinct $x_j,y_j\in B_{1/2}$ such that, for $r_j=|y_j-x_j|$,Then $S_j/A_j<2/\delta$, $N_j/A_j<2/j$, andIn particular $r_j\to 0$. Subtract the quadratic Taylor polynomial of $u_j$ at $x_j$ and rescale:Its domain $\{z:x_j+r_{j}z\in B_1\}$ exhausts ${\mathbb{R}}^n$. The normalization givesTaylor theorem, or integration along line segments from zero, consequently bounds $D^2{v}_j,\nabla v_j,v_j$ on every fixed ball by constants times $|z{|}^{\mu },|z{|}^{1+\mu },|z{|}^{2+\mu }$, respectively. The Hessian matrices are equicontinuous. The Arzela-Ascoli theorem and a diagonal subsequence therefore give $v_j\to v$ in $C^2$ on compact subsets of ${\mathbb{R}}^n$.

The rescaled equation ison each compact set. Thus $v$ is harmonic and is smooth. Every entry of its Hessian matrix is harmonic and has a global Hölder seminorm at most $2/\delta$. The allowed Liouville lemma for globally Hölder harmonic functions makes each entry constant. Since $D^{2}v(0)=0$, all these constants are zero.

On the other hand, $e_j=(y_j-x_j)/r_j$ is a unit vector and $|D^2{v}_j(e_j)|=1$ by construction. A further subsequence has $e_j\to e$; the local $C^2$ convergence implies $|D^{2}v(e)|=1$, a contradiction. Hence the desired constant exists, andThe global Hölder seminorm on the limiting Hessian matrix, rather than a bound on its magnitude throughout space, is what makes this Liouville argument work.

We prove compactness of zero-boundary Poisson solutions. Here the printed $\partial \Omega$ must mean $\partial B_1$; otherwise the boundary condition refers to an unspecified set. Use this intended boundary condition throughout. Let $f=\Delta u$. A maximum-principle barrier gives a uniform bound on $u$: with $\psi (x)=(1-|x{|}^2)/(2n)$, we have $\Delta \psi =-1$ and $\psi =0$ on $\partial B_1$. Since $|f|\le 1$,The weak maximum principle gives $-\psi \le u\le \psi$, so $\Vert u{\Vert }_{\infty }\le 1/(2n)$.

Standard elliptic regularity on the smooth ball first gives $u\in C^{2,\mu }(\overline{B_1})$. The global Schauder estimate then givesThus the functions and all their derivatives through order two are uniformly bounded and equicontinuous on the closed ball. The Arzela-Ascoli theorem gives a subsequence converging uniformly along with all these derivatives. Integration along line segments identifies the limiting derivatives, so the convergence is in $C^2(\overline{B_1})$.

It remains to keep the limit in the set, rather than merely proving relative compactness. If $u_j\to u$ in $C^2$, then the zero boundary values pass to the limit and $f_j=\Delta u_j\to f=\Delta u$ uniformly. For each pair of distinct points,Taking suprema gives $[f{]}_{\mu }\le {\mathrm{liminf}}_j[f_j{]}_{\mu }$, while $\Vert f_j{\Vert }_{\infty }\to \Vert f{\Vert }_{\infty }$. Hence $\Vert f{\Vert }_{C^{0,\mu }}\le 1$. The limit belongs to the set, proving sequential compactness in $C^2(\overline{B_1})$. The fixed positive Hölder exponent supplies the compactness; a bound on the supremum norm of $\Delta u$ alone would not supply this argument.