Resolvent Estimates in (weighted) Lp spaces for the Stokes Operator in Lipschitz Domains

Abstract

Recently, by Z. Shen, resolvent estimates for the Stokes operator were established in L^p(Ω) when Ω is a Lipschitz domain in R^d, with d≥3 and |1/p-1/2|<1/(2d)+ε. This result implies that the Stokes operator generates a bounded analytic semigroup in L^p(Ω) in the case that Ω is a three-dimensional Lipschitz domain and 3/2-ε<p<3+ε. To fully understand the work of Z. Shen, a lot of background information is needed. In this thesis the resolvent estimates are studied in detail in the case d=3. In the end the results of Shen are extended to resolvent estimates in L^p(w,Ω), where Ω is a three-dimensional Lipschitz domain, |1/p-1/2|<1/6, and w∈A_2p/3∩RH_3/(3-p) is a weight function that belongs to an intersection of a Muckenhoupt weight class and satisfies a reverse Hölder inequality.