Resolvent Estimates in L p for the Stokes Operator in Lipschitz Domains

Estimates for the Stokes Operator in Lipschitz Domains

We study the Stokes operator A in a threedimensional Lipschitz domain Ω. Our main result asserts that the domain of A is contained in W 1,p 0 (Ω)∩W (Ω) for some p > 3. Certain L∞-estimates are also established. Our results may be used to improve the regularity of strong solutions of Navier-Stokes equations in nonsmooth domains. In the appendix we provide a simple proof of area integral estimate...

Weighted Estimates in L for Laplace’s Equation on Lipschitz Domains

Let Ω ⊂ Rd, d ≥ 3, be a bounded Lipschitz domain. For Laplace’s equation ∆u = 0 in Ω, we study the Dirichlet and Neumann problems with boundary data in the weighted space L2(∂Ω, ωαdσ), where ωα(Q) = |Q−Q0|α, Q0 is a fixed point on ∂Ω, and dσ denotes the surface measure on ∂Ω. We prove that there exists ε = ε(Ω) ∈ (0, 2] such that the Dirichlet problem is uniquely solvable if 1 − d < α < d − 3 +...

Resolvent Estimates of the Dirac Operator

The L Dirichlet Problem for the Stokes System on Lipschitz Domains

We study the Lp Dirichlet problem for the Stokes system on Lipschitz domains. For any fixed p > 2, we show that a reverse Hölder condition with exponent p is sufficient for the solvability of the Dirichlet problem with boundary data in LpN (∂Ω, R d). Then we obtain a much simpler condition which implies the reverse Hölder condition. Finally, we establish the solvability of the Lp Dirichlet prob...

The L∞-Stokes semigroup in exterior domains

The Stokes semigroup on a bounded domain is an analytic semigroup on spaces of bounded functions as was recently shown by the authors based on an a priori L∞-estimate for solutions to the linear Stokes equations. In this paper, we extend our approach to exterior domains and prove that the Stokes semigroup is uniquely extendable to an analytic semigroup on spaces of bounded functions.