I recently studied the Schauder estimates with the boundary and checked the wiki page. The following is the link (the boundary estimate is on the bottom part):
https://en.wikipedia.org/wiki/Schauder_estimates
I don't see whether the domain $\Omega$ needs to be bounded. Thus, I am curious whether this boundary estimate would work in the following unbounded domain case:
Suppose $\Delta u = f$ in $\Omega$, and $u = 0$ on $\partial \Omega$, where $\Omega = \mathbb{R}^3_+$ (the upper-half plane). Then
$|u|_{2, \alpha; \Omega} \leq C (|u|_{0, \Omega} + |f|_{0, \alpha; \Omega})$,
where the constant $C$ depends on dimension, $\alpha$, coeffients and $\Omega$.
I would appreciate any answers and comments, Thank you.