# Regularity for weak solution of Poisson problem in a rectangle

Let $$\Omega=(0,1)^2$$. Let $$u$$ be a weak solution of $$\Delta u=f$$ con $$f \in L^2(\Omega)$$ e $$u \in H^1_0(\Omega)$$. I would like to prove that $$u \in H^2(\Omega)$$.

I know that $$u \in H^2_{loc}(\Omega)$$ because of elliptic interior regularity. I tried to adapt the demonstration of that fact and to find compact subsets $$K_n$$ with $$\|u\|_{H^2(K_n)}$$ uniformly bounded, but I was not able to go any further.

As a reference to understand what I know about the subject(which is very few) I've studied the chapter about that of Evans book and of Brezis book.

• Concerning this subject, I would recommend Grisvard's "Elliptic Problems in Nonsmooth Domains". If you have access, you can download it at doi.org/10.1137/1.9781611972030 – gerw Jan 14 at 11:01
• I don't have access but I'll try to take a look at the book .Thanks – Tommaso Scognamiglio Jan 14 at 15:42