On the Lp-Lq maximal regularity for Stokes equations with Robin boundary condition in a bounded domain

We obtain the Lp-Lq maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in Rn (n>2). The Robin condition consists of two conditions: v .u=0 and au+(beta)(T(u, p)v - (T(u, p)v, v)v)=h on the boundary of the domain with a, (beta)>0 and a+(beta)=1, where u and p denote a velocity vector and a pressure, T(u, p) the stress tensor for the Stokes flow and v the unit outer normal to the boundary of the domain. It presents the slip condition when (beta)=1 and non-slip one when a=1, respectively. The slip condition is appropriate for problems that involve free boundaries.