In the problem, with $\Omega \subset \mathbb{R}^3$ a smooth bounded domain, and the given functions $h_1, h_2 \in C(\partial\Omega)$ \begin{equation}\nonumber%\label{eq:Pe}\tag{$P_{\varepsilon}$} \left\{ \begin{array}[c]{ll} -\Delta \chi + \Delta^2 \chi = 0, & \text{in } \Omega, \\ \chi = h_1, \; \Delta \chi = h_2, & \text{on } \partial\Omega\\ \end{array} \right. \end{equation} which admits a unique and regular solution $\chi \in H^2(\Omega)$, my question is, what can I say about $$\|\chi\|_\infty $$
and $$\|\Delta\chi\|_\infty$$
Exists some bound involving $h_1$ or $h_2$?
Thanks for any help!