On the r-bounded solution operator and the maximal l_p-l_q 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^3−1/r_r domain of N-dimensional Euclidean space ℝ^N (N ⩾ 2, N < r < ∞). We prove the existence of R-bounded solution operator with spectral parameter λ varying in a sector Σ_ε,λ0 = {λ ∈ ℂ | | arg λ| ⩽ π − ε, |λ| ⩾ λ₀} (0 < ε < π/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 ∈ W¹_q (Ω) to the variational problem: (∇p,∇ϕ)_Ω = (f,∇ϕ)_Ω for any ϕ ∈ W¹_q′(Ω) with 1 < q < ∞ and q′ = q/(q − 1), where W¹_q (Ω) is a closed subspace of Ŵ¹_q,Γ (Ω) = {p ∈ L_q,loc(Ω) | ∇p ∈ L_q(Ω)^N, p|_Γ = 0} with respect to gradient norm ∥∇ · ∥_Lq(Ω) that contains a space W¹_q,Γ (Ω) = {p ∈ W¹_q (Ω) | p|_Γ = 0}, and Γ 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 (λ₀,∞), which was proved in Shibata [13]. Our assumption is satisfied for any q ∈ (1,∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer, straight cube, and exterior domains with W¹_q (Ω) = Ŵ¹_q,Γ (Ω).