In this paper, we consider the boundary value problem of Stokes operator arising in the study of free boundary problem for the Navier-Stokes equations with surface tension in a uniform W-r(3-1/r) domain of N-dimensional Euclidean space R-N (N >= 2, N < r < infinity). We prove the existence of R-bounded solution operator with spectral parameter. varying in a sector Sigma(epsilon,lambda 0) = {lambda is an element of C vertical bar vertical bar arg lambda| <= pi - epsilon, vertical bar lambda vertical bar >= lambda(0)} (0 < epsilon < pi/2), and the maximal L-p-L-q regularity with the help of the R-bounded solution operator and the Weis operator valued Fourier multiplier theorem. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely it is assumed the unique existence of solution p is an element of W-q(1) (Omega) to the variational problem: (del p, del phi)(Omega) = (f, del phi)(Omega) for any phi is an element of W-q'(1) (Omega) with 1 < q < infinity and q' = q/(q - 1), where W-q(1) (O) is a closed sub-space of (W) over cap (q),(1)(Gamma) (Omega) = {p is an element of L-q,L-loc(Omega)vertical bar del p is an element of L-q(Omega)(N), p vertical bar Gamma = 0} with respect to gradient norm parallel to del.parallel to L-q (Omega) that contains a space W-q,Gamma(1) (Omega) = {p is an element of W-q(1) (Omega) vertical bar p vertical bar Gamma = 0}, and Gamma is one part of boundary on which free boundary condition is imposed. The unique solvability of such weak Dirichlet-Neumann problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to spectral parameter varying in (lambda(0),infinity), which was proved in Shibata [13]. Our assumption is satisfied for any q is an element of(1,infinity) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer, straight cube, and exterior domains with W-q(1)(Omega) = (W) over cap (q),(1)(Gamma) (Omega).
