Hodge-Dirac, Hodge-Laplacian and Hodge-Stokes operators in Lp spaces on Lipschitz domains

This paper concerns Hodge-Dirac operators D-parallel to = d + (delta) under bar 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) under bar is the interior derivative with tangential boundary conditions. In L-2(Omega, Lambda), (delta) under bar = d* and D-parallel to is self-adjoint, thus having bounded resolvents {(I + itD(parallel to))(-1)}(t is an element of R) as well as a bounded functional calculus in L-2(Omega, Lambda). We investigate the range of values pH < p < p(H) about p = 2 for which D has bounded resolvents and a bounded holomorphic functional calculus in L-p (Omega, Lambda). On domains which we call very weakly Lipschitz, we show that this is the same range of values as for which L-p (Omega, Lambda) has a Hodge (or Helmholz) decomposition, being an open interval that includes 2. The Hodge-Laplacian a dagger(parallel to) is the square of the Hodge-Dirac operator, i.e. -a dagger(parallel to) = D-parallel to(2) , so it also has a bounded functional calculus in L-p(Omega, Lambda) when pH < p < p(H). But the Stokes operator with Hodge boundary conditions, which is the restriction of -Delta(parallel to) to the subspace of divergence free vector fields in L-p(Omega, Lambda(1)) with tangential boundary conditions, has a bounded holomorphic functional calculus for further values of p, namely for max{1, pH(S)} < p < p(H) where pH(S) is the Sobolev exponent below pH, given by 1/pH(S) = 1/pH + 1/n, so that pH(S) < 2n/(n + 2). In 3 dimensions, pH(S) < 6/5. We show also that for bounded strongly Lipschitz domains Omega, pH < 2n/(n + 1) < 2n/(n - 1) < p(H) , in agreement with the known results that pH < 4/3 < 4 < p(H) in dimension 2, and pH < 3/2 < 3 < p(H) in dimension 3. In both dimensions 2 and 3, pH(S) < 1, implying that the Stokes operator has a bounded functional calculus in L-p (Omega, Lambda(1)) when Omega is strongly Lipschitz and 1 < p < p(H).