 Solution

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