Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 107 / 3

The De Giorgi-Nash-Moser theorem gives numbers $\alpha =\alpha (n,\lambda ,\Lambda )\in (0,1)$ and $C=C(n,\lambda ,\Lambda ,\theta )$ such that every weak solution satisfiesfor each fixed $0<\theta <1$. In particular, every such weak solution has a locally Hölder-continuous representative despite the coefficients being only bounded and measurable.

The first variation of the graph area functional isThus the minimal surface equation for a graph isor, in nondivergence form,Differentiate the divergence equation with respect to $x_k$. Then $w=D_{k}u$ satisfieswhere

If $|Du|\le L$, the eigenvalues of $a^{ij}(Du)$ lie between $(1+L^2{)}^{-3/2}$ and $1$, so the differentiated equations are uniformly elliptic with bounded measurable coefficients depending only on $L$. Apply the De Giorgi-Nash-Moser theorem to each $D_{k}u$. Since $|D_{k}u|\le L$, for every $\theta <1$,for some $\alpha =\alpha (n,L)\in (0,1)$. The mapis smooth with bounded derivative on $|p|\le L$. Composition therefore gives

The function $u$ solves the linear nondivergence equationThe coefficient matrix is uniformly elliptic, and part (ii) makes it $C^{0,\alpha }$ on every smaller ball. Solve the corresponding linear Dirichlet problem on $B_r⋐B_1$ with boundary value $u$. Existence gives a $C^{2,\alpha }$ solution, and the weak maximum principle for elliptic operators gives uniqueness, so that solution is $u$. Hence $u\in C_{\mathrm{loc}}^{2,\alpha }(B_1)$.

The local boundedness estimate for uniformly elliptic equations first controls $\Vert u{\Vert }_{C^0(B_r)}$ by $\Vert u{\Vert }_{L^2(B_1)}$. Applying the Interior Schauder estimate and covering $B_{\theta }$ by finitely many balls gives

SetPart (b.iii), applied on each compactly contained ball, gives uniform $C^{2,\alpha }$ bounds for $v_k$. A diagonal use of the Arzela-Ascoli theorem therefore gives a subsequence converging in $C^2(K)$ on every compact $K⋐B_1$ to a $C^2$ function $w$.

The same estimate applied to $u_k$ gives $D{u}_k\to 0$ locally uniformly because $\Vert u_k{\Vert }_2\to 0$. Dividing the minimal-surface equation by $\Vert u_k{\Vert }_2$ shows thatThe coefficients converge locally uniformly to ${\delta }_{ij}$. Passing to the $C^2$ limit yieldsThus $w$ is harmonic and the required normalized subsequence converges to it locally in $C^2$.