Generalized resolvent estimates of the stokes equations with first order boundary condition in a general domain

In this paper, we prove unique existence of solutions to the generalized resolvent problem of the Stokes operator with first order boundary condition in a general domain Ω of the N-dimensional Eulidean space ℝ^N, N ≥ 2. This type of problem arises in the mathematical study of the flow of a viscous incompressible one-phase fluid with free surface. Moreover, we prove uniform estimates of solutions with respect to resolvent parameter Λ varying in a sector ∑_σ, _λ0 = {λ ∈ ℂ |arg λ| < π-σ, |λ| ≥ λ₀}, where 0 < σ < π/2 and λ₀ ≥ 1. The essential assumption of this paper is the existence of a unique solution to a suitable weak Dirichlet problem, namely it is assumed the unique existence of solution p ∈ Ŵ¹_q,Γ, (Ω) to the variational problem: (∇_p, ∇ _φ) = (f, ∇_φ) for any φ ∈ Ŵ¹_q',Γ(Ω). Here, 1 < q < ∞, q' = q/(q-1), Ŵ¹_q,Γ(Ω) is the closure of Ŵ¹_q,Γ(Ω) = {p ∈ Ŵ¹_q(Ω) |p|Γ = 0} by the semi-norm ||∇ ̇ ||L_q(Ω), and Γ is the boundary of Ω. In fact, we show that the unique solvability of such a Dirichlet problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to resolvent parameter varying in (λ₀, ∞). Our assumption is satisfied for any q ∈ (1, ∞) by the following domains: whole space, half space, layer, bounded domains, exterior domains, perturbed half space, perturbed layer, but for a general domain, we do not know any result about the unique existence of solutions to the weak Dirichlet problem except for q = 2.