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

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