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We consider the generalized Stokes resolvent problem in an infinite layerwith Neumann boundary conditions. This problem arises from a free boundaryproblem describing the motion of incompressible viscous one-phase fluid flowwithout surface tension in an infinite layer bounded both from above and frombelow by free surfaces. We derive a new exact solution formula to thegeneralized Stokes resolvent problem and prove the $\mathscr{R}$-boundedness ofthe solution operator families with resolvent parameter $\lambda$ varying in asector $\Sigma_{\varepsilon,\gamma_0}$ 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 thewell-posedness locally in time of the nonlinear free boundary problem mentionedabove 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. Ascompared with the case of Neumann-Dirichlet boundary condition studied bySaito, analysis is even harder on account of higher singularity of the symbolsin the solution formula.

A solution formula and the R-boundedness for the generalized Stokes resolvent problem in an infinite layer with Neumann boundary condition