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

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.

• I think you need to specify some boundary conditions in order to get a global estimate. – Jose27 Dec 14 '16 at 20:45
• @Jose27 Is a Lipschitz boundary sufficient? – 0xbadf00d Dec 14 '16 at 22:04
• I meant boundary conditions on $p$, not restrictions on the domain ( although these play a role too). – Jose27 Dec 14 '16 at 22:13
• @Jose27 It would be fine for me, if we assume Dirichlet boundary conditions, $p\in H_0^1(\Lambda)$. Can we conclude $p\in H^2(\Lambda)$ with that assumption, for the general $\Lambda$ of the question? – 0xbadf00d Dec 14 '16 at 22:41

$H^2$-regularity of $p$ is valid if $\Lambda$ is a bounded, polyhedral set, see chapter 4 of Grisvard's book "Elliptic problems in nonsmooth domains".
• I guess you've got Theorem 4.3.1.4 in mind. However, it's only stated for a polygon in $\mathbb R^2$. Is there a similar result for polyhedra in higher dimensions (in particular, in $\mathbb R^3$)? – 0xbadf00d Feb 2 '17 at 10:14
• What about section 8.1? On p. 371, it is stated "This shows again the regularity of $u$ in $H^2(\Omega)$ when $\Omega$ is a convex polyhedron." – gerw Feb 2 '17 at 11:45