It seems that the following assertion is widely accepted:


For $k\in\mathbb N$, $p\geq 2$, $\Omega \subset \mathbb R^n$ bounded with $\partial\Omega\in C^{k+2}$ and $f\in W^{k,p}(\Omega)$, the weak solution $u\in H^1\_0(\Omega)$ of the problem
$$
\begin{equation}
\left\{
\begin{aligned}
-\Delta u =f \text{ in } \Omega\\
u=0 \text{ on } \partial\Omega
\end{aligned}
\right.
\end{equation}
$$
satisfies $u\in W^{k+2,p}(\Omega)$ and $\|u\|\_{W^{k+2,p}}\leq C\_{\Omega,k,p}\|f\|\_{W^{k,p}}$ for some $C\_{\Omega,k,p}>0$.


The above is proved in Evans using difference quotients for $p=2$. For $k=0$ it appears to be true due to an interpolation argument (Theorem 7.1 of Giaquinta's and Martinazzi's book on regularity theory). For Hölder continuous domains one can use the classical Schauder theory. But is there a reference for the complete result?



Regularity up to the boundary for the Poisson problem
P. Grisvard, Elliptic Problems in Nonsmooth Domains, 1985:
Thm. 2.5.1.1 (you even need to impose less regularity on $\partial\Omega$).


