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I 'm reading a paper at the moment and I have a really hard time understanding the following:

Let $U \subset \mathbb R^3$ be open, bounded and connected with a $C^2-$ regular boundary $\partial U$. Consider the following problem:

  • $\Delta u=0$ in $U$
  • $\frac{\partial u}{\partial \nu}=u-f$ on $\partial U$

where $\frac{\partial u}{\partial \nu}$ denotes the directional derivative and $f$ is a function such that $f\in L^2(\partial U)$.

By optimal regularity theory it follows $u \in W^{1-1/2,2}(U)$.

To begin with I never heard before the term "optimal regularity" and after some research I didn't find something clear and useful. I can't understand why the sentence in bold holds and I would really appreciate if somebody could enlighten me.

Moreover, what book do you suggest for studying this type of regularity?

Thanks in advance

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  • $\begingroup$ This is not some magical notion. I guess "optimal regularity" means here that (a) it can be proved in general that $u \in W^{1/2,2}(U)$, (b) for each $\varepsilon > 0$, $u \in W^{\varepsilon+1/2,2}(U)$ doesn't hold in general. With which part do you have problems? $\endgroup$ – Michał Miśkiewicz Nov 13 '18 at 19:03
  • $\begingroup$ @MichałMiśkiewicz Excuse me but I don't understand your comment. I suppose too, that it's not a magical notion, however I don't see how one can deduce the estimate $u \in W^{1/2,2}$... $\endgroup$ – kaithkolesidou Nov 14 '18 at 8:44

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