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I asked a similar question before but didn't get any responses. So I will attempt again (the prior question was regarding Holder continuity).

Let $ \Omega$ denote a cube in $ R^n$ and consider solving $$- \Delta u(x) + u(x) = f(x) $$ in $ \Omega$ with $ \partial_\nu u=0$ on $ \partial \Omega$.

Suppose $ f \in L^p(\Omega)$ where $ 1 <p< \infty$. Then does one have that $ u \in W^{2,p}(\Omega)$ along with the desired estimate?

I am aware of the book by Grisvard "Elliptic_Problems_in_Nonsmooth_Domains".
Since I am on a cube it seems one can just reflect and prove the above result ??

Any comments on whether one even thinks this result is true would be greatly appreciated. Also what about if $ f $ is Holder continuous of order $ 0 < \alpha<1$. Can we get the usual $ C^{2,\alpha}$ regularity? (again i managed to "prove" this to myself but i not convinced).

thanks

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  • $\begingroup$ Can you please explain why the results by Grisvard are not sufficient? He also considers spaces of Hölder continuous functions btw. $\endgroup$
    – Delio Mugnolo
    Nov 25, 2015 at 8:00
  • $\begingroup$ They seem to cover very general cases and since I have the very special case I thought someone probably just knows the answer off the top of their head. Honestly I attempted to decipher the results but gave up before I could come to a conclusion... $\endgroup$
    – Math604
    Nov 25, 2015 at 16:20
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    $\begingroup$ The philosophy in that book is that if anything can go wrong in a domain with a nonsmooth domain, it is because of sharp edges/corners, which means that for convex domains the situation will generally be benign. That said, the result you are looking for is Thm. 3.2.1.3 in the case $p=2$, but I had no time to look for the general case. $\endgroup$
    – Delio Mugnolo
    Nov 25, 2015 at 18:25
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    $\begingroup$ Thanks for the comment. I saw some papers online and it appears the value of $p$ actually plays a role. They have results that say provided $p$ is less than some geometrical quantity then one has the desired $W^{2,p}$ estimate (this is for both the Dirichlet and Neumann problem). In the case of the Neumann problem it appears that one is not really able to compute this geometrical quantity. Putting this aside it seems that since I am on a cube one can do even reflections to obtain the desired regularity from interior estimates (but maybe i am completely missing something). $\endgroup$
    – Math604
    Nov 25, 2015 at 18:49
  • $\begingroup$ In the case of domains smooth enough I have never seen a dependence on $p$ - even the spectrum and the eigenfunctions of all these operators is identical! But I do know instances of $p$-dependence in the case of certain nonlinear operators, like the $p$-Laplacian (but here $p$ has nothing to do with "your $p$"). So I would be interested in the kind of results you are mentioning. $\endgroup$
    – Delio Mugnolo
    Nov 26, 2015 at 9:23

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