Finiteness of non-symmetric and symmetric cohomologies associated with Jackson integrals of type $BC_n$ is studied. Explicit bases of the cohomologies are also stated. It is shown that the integrals using these bases satisfy holonomic systems of linear $q$-difference equations with respect to the parameters.

We prove the $L_p$-$L_q$ maximal regularity of solutions to the Neumann problem for the Stokes equations with non-homogeneous boundary condition and divergence condition in a bounded domain. And as an application, we consider a free boundary problem for the Navier-Stokes equation. We prove a locally in time unique existence of solutions to this problem for any initial data and a globally in time unique existence of solutions to this problem for some small initial data.