  We consider the generalized Stokes resolvent problem in an infinite layer
with Neumann boundary conditions. This problem arises from a free boundary
problem describing the motion of incompressible viscous one-phase fluid flow
without surface tension in an infinite layer bounded both from above and from
below by free surfaces. We derive a new exact solution formula to the
generalized Stokes resolvent problem and prove the $\mathscr{R}$-boundedness of
the solution operator families with resolvent parameter $\lambda$ varying in a
sector $\Sigma_{\varepsilon,\gamma_0}$ for any $\gamma_0>0$ and
$0<\varepsilon<\pi/2$, where $\Sigma_{\varepsilon,\gamma_0} =\{
\lambda\in\mathbb{C}\setminus\{0\} \mid |\arg\lambda|\leq\pi-\varepsilon, \
|\lambda|>\gamma_0 \}$. As applications, we obtain the maximal $L_p$-$L_q$
regularity for the nonstationary Stokes problem and then establish the
well-posedness locally in time of the nonlinear free boundary problem mentioned
above in $L_p$-$L_q$ setting. We make full use of the solution formula to take
$\gamma_0>0$ arbitrarily, while in general domains we only know the
$\mathscr{R}$-boundedness for $\gamma_0\gg1$ from the result by Shibata. As
compared with the case of Neumann-Dirichlet boundary condition studied by
Saito, analysis is even harder on account of higher singularity of the symbols
in the solution formula.
