# Regularity up to the boundary for the Poisson problem

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 \left\{ \begin{aligned} -\Delta u =f \text{ in } \Omega\\ u=0 \text{ on } \partial\Omega \end{aligned} \right. 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?

• I am not sure if you would find it there, but Adams: Sobolev Spaces is a good reference for Sobolev spaces' results.
– Alan
Oct 18, 2015 at 18:24
• Nope, there is nothing in Adams about regularity theory... Oct 18, 2015 at 18:27
• you can llok to Gilbard Trudinger or the courant lecture notes by Han Lin
– Paul
Oct 18, 2015 at 19:15
• Gilbarg and Trudinger provide the result as a product of the Schauder theory which assumes Hölder regularity of the boundary. It seems that this is also the case with Han and Lin Oct 18, 2015 at 20:02

P. Grisvard, Elliptic Problems in Nonsmooth Domains, 1985: Thm. 2.5.1.1 (you even need to impose less regularity on $\partial\Omega$).
• Are you missing the explanation of why the solution is $W^{2,p}$ in the very first place? Well, it also follows directly from Thm. 2.4.2.6 from the same book. Oct 19, 2015 at 8:57