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Assume

$$ Lu=f\quad \text{in } [0,1]^d\\ u=0 \quad\text{ on } \partial[0,1]^d $$ for some strongly-elliptic operator $L$, and $f\in H^k$$, k\geq -1$. Do we have $u\in H^{k+2}$? I can only find the result for smooth domains.

I am pretty sure the answer is yes, for the Laplacian I can derive it myself. However, all proofs for general $L$ rely on "flattening the boundary", i.e. a $C^{k+2}$ boundary.

Bananach
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  • This reference seems to include a discussion on elliptic operators on torus https://www.springer.com/cda/content/document/cda_downloaddocument/9783319146478-c2.pdf?SGWID=0-0-45-1506284-p177198215 – Yuxin Wang Dec 07 '18 at 18:45
  • @Yuxin, if I understand Sections 11 and 16 of that reference correctly, regularity is shown for $-1\leq k\leq -1+\epsilon$ for general Lipschitz domains. Do you know if more is possible for the hypercube, i.e. can $k$ be chosen arbitrarily large? – Bananach Dec 08 '18 at 06:30
  • @Bananach Did you end up finding an answer to this question? – atzol Nov 18 '21 at 21:28
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    @atzol I didn't – Bananach Nov 18 '21 at 23:01

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