While reading the book Regularity Theory of Elliptic PDE I’m confused with a theorem:
Thm. 2.30.
Let $\alpha \in (0,1)$ and $k \in N$ with $k \leq 2$, and let $\Omega$ be a bounded $C^{k, \alpha}$ domain of $R^n$. Let $u \in H^1(\Omega)$ be a weak solution to
$$ \Delta u = f \hspace{2pt} \in \hspace{2pt} \Omega $$ $$ u=g \hspace{2pt} \text{ on } \hspace{2pt} \partial \Omega$$
for some $f\in C^{k-2, \alpha}(\bar{\Omega}), g\in C^{k, \alpha}(\partial\Omega)$.
Then, $ u \in C^{k, \alpha}(\bar{\Omega})$ and
$$ \|u\|_{C^{k, \alpha}(\Omega)} \leq C(\|f\|_{C^{k-2, \alpha}(\Omega)}+\|g\|_{C^{k, \alpha}(\partial \Omega)})$$
Where C depending only on $\alpha, n, k \text{ and } \omega$.
The book didn’t give the proof but it says that it can be shown by techniques like:
after a blow-up, points near the boundary behave like in a local problem in the half-space (that is, the blow-up flattens ∂Ω), and we can reach a contradiction with Liouville’s theorem in the half-space.
But I fell into trouble at first because I don’t know how to do the blow-up to get the harmonic function in the half space. I tried to do it starting from that g is identically 0 and try to prove it for a $C^{k, \alpha}$ boundary, but if I set $u_r(x)=\frac{1}{r^2}u(rx)$ then I can’t even bound $D^k(u_r)$. What should I do to get the blow-up? Is there any reference I can send to?
