On the maximal L _p-L _q regularity of the Stokes problem with first order boundary condition; model problems

In this paper, we proved the generalized resolvent estimate and the maximal L _p-L _q regularity of the Stokes equation with first order boundary condition in the half-space, which arises in the mathematical study of the motion of a viscous incompressible one phase fluid flow with free surface. The core of our approach is to prove the R boundedness of solution operators defined in a sector Σ _∈,γ0 = {λ ∈ C \ {0} | | arg λ| ≤ π - ∈, |λ| > γ _o} with 0 < ∈ < π/2 and γ _o ≥0. This R boundedness implies the resolvent estimate of the Stokes operator and the combination of this R boundedness with the operator valued Fourier multiplier theorem of L. Weis implies the maximal L _p-L _q regularity of the non-stationary Stokes. For a densely defined closed operator A, we know that what A has maximal L _p regularity implies that the resolvent estimate of A in λ ∈ Σ _∈,γ0, but the opposite direction is not true in general (cf. Kalton and Lancien[19]). However, in this paper using the R boundedness of the operator family in the sector Σ _{∈ λ0}, we derive a systematic way to prove the resolvent estimate and the maximal L _p regularity at the same time.