First Order Approach to Lp Estimates for the Stokes Operator on Lipschitz Domains

This paper concerns Hodge-Dirac operators D-H = d + delta acting in L-p(Omega, Lambda) where Omega is a bounded open subset of R-n satisfying some kind of Lipschitz condition, Lambda is the exterior algebra of R-n, d is the exterior derivative acting on the de Rham complex of differential forms on Omega, and delta is the interior derivative with tangential boundary conditions. In L-2(Omega, Lambda), d(') = delta and D-H is self-adjoint, thus having bounded resolvent {(I + itD(H))}({t is an element of R}) as well as a bounded functional calculus in L-2(Omega, Lambda). We investigate the range of values p(H) < p < p(H) about p = 2 for which DH has bounded resolvents and a bounded holomorphic functional calculus in L-p(Omega, Lambda).