Considering the problem \begin{equation} \left\{ \begin{array}[c]{11} \Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\ \Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\ \end{array} \right. \end{equation}
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$ and $h_1$ and $h_2$ are given functions in $\partial\Omega$.
By Riesz theorem the system admits a unique and regular solution $\chi \in H^2(\Omega)$.
I want to find some estimative for the $\|\Delta \chi\|_{\infty}$ in terms of $\|\chi\|_{\infty}$.
By the Maximum Principle we have $$\|\Delta\chi - \chi \|_{\infty} = \|h_2 - h_2\|_{\infty}$$ and by the other hand $$\|\Delta\chi - \chi \|_{\infty} \leq \|\Delta \chi\|_{\infty} + \|\chi\|_{\infty}.$$
I was tryin to argue something like "Then exists some $c\in \mathbb{R}^+$ such that" $$c\|\chi\|_{\infty} \geq \|h_2 - h_2\|_{\infty}$$ but I'm stuck on the passage of proving the existence of this constant $c$.(I don't know if this is thruth!!!!)
Another way of thinking is to search an argument to prove that I can estimate $$\|\Delta \chi\|_{\infty} \leq A\|\chi\|_{\infty} $$ for some $A$ depending of some parameters.
The main goal here is to find an estimate of $\|\chi\|_{\infty}$ in terms of $h_1$ and $h_2$.
If anyone have ideas, I'll be grateful!