I am doing exercise 10 of Colding-Minicozzi's minimal surfaces book which goes as
Find a sequence of functions $u_j:D_1\to\mathbb{R}$ on the unit disk $D_1$ satisfying the minimal surface equation with $|Du_j|\le 1$ on $D_1$ but $\sup_{D_1} |D^2u_j|\to\infty$.
This problem is compared with exercise 9 which goes as:
$u_j:D_1\to\mathbb R$ satisfies the minimal surface equation with $|Du_j|\le 1$. Use standard elliptic theory to prove a uniform bound $|D^2u(0)|\le C$.
I am ok with Exercise 9 which follows from De Giorgi-Nash theorem and interior Schauder estimate.
For exercise 10, I was thinking of constructing some $\varphi_j$ with $|\varphi_j|_{1;D_1}$ bounded and $|D^2\varphi_j|_{D_1}\to \infty$ and solve the equation $Qu_j=0$ in $D_1$, $u_j=\varphi_j$ on $\partial D_1$ where $Qu=0$ is the minimal surface equation.
What I have to do is to get a bound on $|Du_j|$. However, boundary gradient estimate of $u_j$ depends on $|D^2\varphi_j|$ cf. Theorem 14.2 or Corollary 14.3 of Gilbarg-Trudinger so my approach does not quite work. Is there any way around this?