$H^{1/2}$ regularity for the pressure in stationary Stokes problem in bounded domains I am wondering for my research if, given the classical Stokes problem in a bounded domain $\Omega\subset \mathbb{R}^n$, $n=2,3$,
\begin{align}
- \Delta v + \nabla p &= f \text{ in }\Omega\\
\nabla \cdot v &= 0 \text{ in } \Omega\\
v &= 0 \text{ on } \partial\Omega
\end{align}
the following result holds true:
$$
\Vert p\Vert_{H^{1/2}(\Omega)}\leq C\Vert f\Vert_{H^{-1/2}(\Omega)},
$$
for some positive $C>0$.
I think so by interpolation between the well known results in $H^1$ and $L^2$ (see, e.g., Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems):
$$
\Vert p\Vert_{H^{1}(\Omega)}\leq C\Vert f\Vert,
$$
$$
\Vert p\Vert_{L^2(\Omega)}\leq C\Vert f\Vert_{H^{-1}(\Omega)},
$$
but I could not find any reference about it...could someone help me?
Thanks!
 A: There is likely a more elementary way to do this, but one way is to use complex interpolation. This is covered for instance in the text:
Taylor, Michael E., Partial differential equations. I: Basic theory, Applied Mathematical Sciences 115. New York, NY: Springer (ISBN 978-1-4419-7054-1/hbk; 978-1-4419-7055-8/ebook). xxii, 654 p. (2011). ZBL1206.35002.
In particular in Chapter 4, Section 2, it is shown in (2.18) that we have
$$ [H^{s}(\Bbb R^n),H^{t}(\Bbb R^n)]_{\theta} = H^{\theta s + (1-\theta)t}(\Bbb R^n), $$
where $[\cdot,\cdot]_{\theta}$ is the space obtained via complex interpolation. For general $\Omega$ this depends on your definition of the fractional spaces, but see Proposition 3.1 and (4.13) in the later sections.
Now assuming we have a unique existence theory in the scales you mentioned, we can consider the operator $f \mapsto p$ which bounded as a map $L^2(\Omega) \to H^1(\Omega)$ and as a map $H^{-1}(\Omega) \to L^2(\Omega).$ Then by complex interpolation (see Proposition 2.1) it is bounded as a map $H^{s-1}(\Omega) \to H^{s}(\Omega)$ for all $s \in [0,1].$
