Let $\Omega \subset \mathbb{R}^3$ be a bounded region with the smooth boundary. Consider the Poisson equation with the Dirichlet boundary condition:
$$\begin{align*} - \nabla^2 \phi &= f \\ \phi \vert_{\partial \Omega} &= 0 \end{align*} $$
This has a unique weak solution in $H_{0}^{1}(\Omega)$.
Then $$\| \nabla \phi \|_{2} \le C \| f \|_{H^2}$$ for some constant $C$, independent of $\nabla \phi$ and $f$. (Here $\| \|_{2}$ denotes the $L^2$ norm and $\| f \|_{H^2} := \| f \|_{2} + \| \nabla f \|_{2} + \| \nabla^2 f \|_{2} $.)
I think
Estimation of gradients in Poisson's equation
is enough for explaining this.
However, I have no idea about analogous estimation for Neumann condition: $$\begin{align*} - \nabla^2 \phi &= f \quad \textrm{on} \ \Omega \\ \nabla \phi \cdot n &= 0 \quad \textrm{on} \ \partial \Omega \end{align*} $$ has a unique weak solution on $H^{1}(\Omega)$, provided that $\int_{\Omega} f =0$.
In this case, $\| \nabla \phi\|_{2} \le C \| f\|_{H^2}$.
The paper says that this is the 'trace estimate' but I have no idea about it... Any hint will be appreciated! Thank you.