On the R-Bounded Solution Operator and the Maximal Lp-Lq Regularity of the Stokes Equations With Free Boundary Condition

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).