The folowing theorem
Let $u$ be a solution to $ \left\{ \begin{array}{ccccc} \Delta u &=& f \chi_{\{u\neq 0\}} &\mbox{in}& B_1,\\ u &=& g &\mbox{on}& \partial B_1, \end{array} \right. $ in a suitable weak sense and assume furthermore that $f=\Delta v$ where $v\in C^{1,1}(B_1)$ and that $g \in C( \partial B_1)$. Then $u \in C^{1,1}(B_{1/2})$ and $\|D^{2}u\|_{L^{\infty}(B_{1/2})} \le C( \|u\|_{L^{1}(B_1)} + \|D^{2}v\|_{L^{\infty}(B_1)}),$ where $C$ depends on the dimension.
This theorem suggests that there is a relation like $\|u\|_{C^{1,1}(B_{1/2})} \le C_n\|D^{2}u\|_{L^{\infty}(B_{1/2})}$. Is this what happens? You can find the details in reference 1 and I found a more simple case in the proof of theorem 1.1 on the page 11 in reference 2. Thank you.