The Stokes problem with Navier slip boundary condition: Minimal fractional Sobolev regularity of the domain

We prove well-posedness in reflexive Sobolev spaces of weak solutions to the stationary Stokes problem with Navier slip boundary condition over bounded domains $\mathrm{\Omega }$ of ${\mathbb{R}}^n$ of class $W_s^{2-1\mathrm{/}s}$, $s>n$. Since such domains are of class $C^{1,1-n\mathrm{/}s}$, our result improves upon the recent one by Amrouche-Seloula, who assume $\mathrm{\Omega }$ to be of class $C^{1,1}$. We deal with the slip boundary condition directly via a new localization technique, which features domain, space and operator decompositions. To flatten the boundary of $\mathrm{\Omega }$ locally, we construct a novel $W_s^2$ diffeomorphism for $\mathrm{\Omega }$ of class $W_s^{2-1\mathrm{/}s}$. The fractional regularity gain, from $2-1\mathrm{/}s$ to $2$, guarantees that the Piola transform is of class $W_s^1$. This allows us to transform $W_r^1$ vector fields without changing their regularity, provided $r\le s$, and preserve the unit normal which is H"older. It is in this sense that the boundary regularity $W_s^{2-1\mathrm{/}s}$ seems to be minimal.