Hodge-Dirac, Hodge-Laplacian and Hodge-Stokes operators in L^p spaces on Lipschitz domains
Résumé
This paper concerns Hodge-Dirac operators D = d + δ acting in L p (Ω, Λ) where Ω is a bounded open subset of R n satisfying some kind of Lipschitz condition, Λ is the exterior algebra of R n , d is the exterior derivative acting on the de Rham complex of differential forms on Ω, and δ is the interior derivative with tangential boundary conditions. In L 2 (Ω, Λ), δ = d * and D is self-adjoint, thus having bounded resolvents (I + itD) −1 t∈R as well as a bounded functional calculus in L 2 (Ω, Λ). We investigate the range of values p H < p < p H about p = 2 for which D has bounded resolvents and a bounded holomorphic functional calculus in L p (Ω, Λ). On domains which we call very weakly Lipschitz, we show that this is the same range of values as for which L p (Ω, Λ) has a Hodge (or Helmholz) decomposition, being an open interval that includes 2. The Hodge-Laplacian ∆ is the square of the Hodge-Dirac operator, i.e. −∆ = D 2 , so it also has a bounded functional calculus in L p (Ω, Λ) when p H < p < p H. But the Stokes operator with Hodge boundary conditions, which is the restriction of −∆ to the subspace of divergence free vector fields in L p (Ω, Λ 1) with tangential boundary conditions , has a bounded holomorphic functional calculus for further values of p, namely for max{1, p H S } < p < 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 < 2n/(n + 2). In 3 dimensions, p H S < 6/5. We show also that for bounded strongly Lipschitz domains Ω, p H < 2n/(n + 1) < 2n/(n − 1) < p H , in agreement with the known results that p H < 4/3 < 4 < p H in dimension 2, and p H < 3/2 < 3 < p H in dimension 3. In both dimensions 2 and 3, p H S < 1 , implying that the Stokes operator has a bounded functional calculus in L p (Ω, Λ 1) when Ω is strongly Lipschitz and 1 < p < p H .
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