# Question on optimal regularity for the elliptic Neumann problem

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?

• 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? – Michał Miśkiewicz Nov 13 '18 at 19:03
• @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 $u \in W^{1/2,2}$... – kaithkolesidou Nov 14 '18 at 8:44