Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 / 3

The Campanato space ${\mathcal{L}}^{p,\mu }(\Omega )$ consists of $f\in L^p(\Omega )$ for whichwhere $f_E=|E{|}^{-1}{\int }_{E}f$. On a smooth bounded domain,with equivalent norms for $0<\alpha \le 1$.

Fix $B_R(x_0)\subset B_1$, put $A_0=A(x_0)$, and let $v$ have the same boundary values as $u$ while solving $\mathrm{div}(A_0\nabla v)=0$. Then $w=u-v\in W_0^{1,2}(B_R)$ satisfiesSince $u\in C^1$ gives $|Du|\le L$ locally, the supplied energy estimate and Hölder continuity of $A$ yieldThe constant-coefficient decay estimate givesThus, for $\Phi (r)={\int }_{B_r}|\nabla u-(\nabla u{)}_{B_r}{|}^2$,Because $n+2>n+2\alpha$, the Campanato iteration lemma gives $\Phi (\rho )\le C{\rho }^{n+2\alpha }$ on balls in $B_{1/2}$. Hence $\nabla u\in {\mathcal{L}}^{2,n+2\alpha }=C^{0,\alpha }$ and $u\in C^{1,\alpha }(B_{1/2})$.