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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.

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  • $\begingroup$ I think you need to specify some boundary conditions in order to get a global estimate. $\endgroup$
    – Jose27
    Dec 14 '16 at 20:45
  • $\begingroup$ @Jose27 Is a Lipschitz boundary sufficient? $\endgroup$
    – 0xbadf00d
    Dec 14 '16 at 22:04
  • $\begingroup$ I meant boundary conditions on $p $, not restrictions on the domain ( although these play a role too). $\endgroup$
    – Jose27
    Dec 14 '16 at 22:13
  • $\begingroup$ @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? $\endgroup$
    – 0xbadf00d
    Dec 14 '16 at 22:41
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$H^2$-regularity of $p$ is valid if $\Lambda$ is a bounded, polyhedral set, see chapter 4 of Grisvard's book "Elliptic problems in nonsmooth domains".

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  • $\begingroup$ 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 $\mathbb R^3$)? $\endgroup$
    – 0xbadf00d
    Feb 2 '17 at 10:14
  • $\begingroup$ 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." $\endgroup$
    – gerw
    Feb 2 '17 at 11:45
  • $\begingroup$ It's rather vague, isn't it? $\endgroup$
    – 0xbadf00d
    Feb 2 '17 at 13:46
  • $\begingroup$ Maybe. I havn't read the preceding theorems. $\endgroup$
    – gerw
    Feb 2 '17 at 17:56
  • $\begingroup$ Sorry for replying to this old post, but can we show that if $f\in H^k(\Lambda)$, then for $p\in H_0^1(\Lambda)$ satisfying $(2)$ it holds $p\in H^{k+1}(\Lambda)$? $\endgroup$
    – 0xbadf00d
    Feb 8 at 8:04

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