Generalized Stokes resolvent equations in an infinite layer with mixed boundary conditions

In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Omega(0) = Rn-1 x (-1, 1), n >= 2, in L-q-Sobolev spaces, 1 < q < infinity, with slip boundary condition of oil the "upper boundary" partial derivative Omega(+)(0) = Rn-1 x {1} and non-slip boundary condition on the "lower boundary" partial derivative Omega(-)(0) = Rn-1 x {-1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an L-q-theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L-2-spaces. (c) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.