On the r-bounded solution operator and the maximal lp-lq regularity of the stokes equations with free boundary condition

Abstract

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 W3−1/rr domain of N-dimensional Euclidean space ℝN (N ⩾ 2, N < r < ∞). We prove the existence of R-bounded solution operator with spectral parameter λ varying in a sector Σε,λ0 = {λ ∈ ℂ | | arg λ| ⩽ π − ε, |λ| ⩾ λ0} (0 < ε < q