Consider $B_2^+$ the half ball in $R^N$ and consider $ (-\Delta)^s u = f(x) $ in $B_2^+$ with $ u=0$ outside. Is there any references where someone tries to use an odd extension of $u$ across $ x_N=0$ to obtain boundary regularity? Looking at the optimal boundary regularity it appears that this trick will fail; I am curious how it fails.
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$\begingroup$ Well, it will fail because the resulting function doesn't solve the PDE on the original domain, since you've changed it far away and the equation is nonlocal. I would recommend arxiv.org/abs/1207.5985, which uses some basic barrier arguments to get optimal regularity and discusses the solution on a half-space along the way. $\endgroup$– user378654Commented Jul 28, 2023 at 22:22
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$\begingroup$ Thanks for the comment and of course I aware of those papers but I have never looked at details in them (I have read the results). But i will look at the half space stuff since it will give me some much needed insight. The reason i asking is about trying an odd reflection with a fractional term and a $ \Delta$; but i guess there is no hope of that giving much. thanks $\endgroup$– Math604Commented Jul 29, 2023 at 2:07
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