Let $u \in W^{1,2} (\Omega)$ be the weak solution to the following Poisson's equation with Robin Boundary condition.
\begin{equation} -\Delta u= f \ \ \ \ \text{ in $\Omega$},\\ \frac{\partial u }{\partial \nu} + \tau u = \xi \ \ \text{ on $ \partial \Omega$}. \end{equation}
Here $\Omega$ is a smooth domain, $ f \in L^p (\Omega)$, for $ p \in [2, \infty)$. $\tau$ and $\xi$ are smooth functions on $\partial \Omega$ and $ \tau (x) \geq 0$ on $\partial \Omega$, but $\tau$ is not a zero function. Then can we say $u \in W^{2,p}(\Omega)$?
The book ''Elliptic Problems in Nonsmooth Domains'' by Grisvard says, $u \in W^{2,p}(\Omega)$, provided $\tau (x) > 0$ on $\partial \Omega$. But in the case of Grisvard, the domain is non-smooth. But I am considering a smooth domain. So I hope the condition on $\tau $ can be generalized. Any reference would be appreciated.