$H^2$-regularity of a solution to a Poisson equation

Question

Let

• $d\in\mathbb N$
• $\Lambda\subseteq\mathbb R^d$ be bounded and open
• $f\in L^2(\Lambda)$
• $p\in L_{\text{loc}}^2(\Lambda)$ admit a weak gradient $\nabla p\in L^2(\Lambda,\mathbb R^d)$

Assuming $$\Delta p=f\;,\tag 1$$ i.e. $$\langle\nabla\phi,\nabla p\rangle_{L^2(\Lambda,\:\mathbb R^d)}=-\langle\phi,f\rangle_{L^2(\Lambda)}\;\;\;\text{for all }\phi\in C_c^\infty(\Lambda)\;,\tag 2$$ are we able to conclude $p\in H^2(\Lambda)$?

I know that this true under some regularity assumptions on $\partial\Lambda$. Since I'm particularly interested in a cube, the only regularity of $\partial\Lambda$ I'm willing to assume is being Lipschitz.

Answer 1

$H^2$-regularity of $p$ is valid if $\Lambda$ is a bounded, polyhedral set, see chapter 4 of Grisvard's book "Elliptic problems in nonsmooth domains".