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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.

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  • $\begingroup$ 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 $\endgroup$ – gerw Jan 14 at 11:01
  • $\begingroup$ I don't have access but I'll try to take a look at the book .Thanks $\endgroup$ – Tommaso Scognamiglio Jan 14 at 15:42

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