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