Hodge-Dirac, Hodge-Laplacian and Hodge-Stokes operators in L^p spaces on Lipschitz domains

Abstract

This paper concerns Hodge-Dirac operators D = 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}), $\delta$ = d * and D is self-adjoint, thus having bounded resolvents (I + itD) --1 t$\in$R as well as a bounded functional calculus in L 2 ($\Omega$, {\lambda}). We investigate the range of values p H \textless{} p \textless{} 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 {\Delta} is the square of the Hodge-Dirac operator, i.e. --{\Delta} = D 2 , so it also has a bounded functional calculus in L p ($\Omega$, {\lambda}) when p H \textless{} p \textless{} p H. But the Stokes operator with Hodge boundary conditions, which is the restriction of --{\Delta} 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, p H S } \textless{} p \textless{} p H where p H S is the Sobolev exponent below p H , given by 1/p H S = 1/p H + 1/n, so that p H S \textless{} 2n/(n + 2). In 3 dimensions, p H S \textless{} 6/5. We show also that for bounded strongly Lipschitz domains $\Omega$, p H \textless{} 2n/(n + 1) \textless{} 2n/(n -- 1) \textless{} p H , in agreement with the known results that p H \textless{} 4/3 \textless{} 4 \textless{} p H in dimension 2, and p H \textless{} 3/2 \textless{} 3 \textless{} p H in dimension 3. In both dimensions 2 and 3, p H S \textless{} 1 , implying that the Stokes operator has a bounded functional calculus in L p ($\Omega$, {\lambda} 1) when $\Omega$ is strongly Lipschitz and 1 \textless{} p \textless{} p H .