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




McIntosh, Alan
Monniaux, Sylvie

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Universidad Autonoma de Madrid


This paper concerns Hodge–Dirac operators D∥=d+δ– acting in Lp(Ω,Λ) where Ω is a bounded open subset of Rn satisfying some kind of Lipschitz condition, Λ is the exterior algebra of Rn, 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 L2(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 L2(Ω,Λ). We investigate the range of values pH<p<pH about p=2 for which D∥ has bounded resolvents and a bounded holomorphic functional calculus in Lp(Ω,Λ). On domains which we call very weakly Lipschitz, we show that this is the same range of values as for which Lp(Ω,Λ) has a Hodge (or Helmholz) decomposition, being an open interval that includes 2. The Hodge-Laplacian Δ∥ is the square of the Hodge–Dirac operator, i.e., −Δ∥=D∥2, so it also has a bounded functional calculus in Lp(Ω,Λ) when pH<p<pH. But the Stokes operator with Hodge boundary conditions, which is the restriction of −Δ∥ to the subspace of divergence free vector fields in Lp(Ω,Λ1) with tangential boundary conditions, has a bounded holomorphic functional calculus for further values of p, namely for max{1,pHS}<p<pH where pHS is the Sobolev exponent below pH, given by 1/pHS=1/pH+1/n, so that pHS<2n/(n+2). In 3 dimensions, pHS<6/5. We show also that for bounded strongly Lipschitz domains Ω, pH<2n/(n+1)<2n/(n−1)<pH, in agreement with the known results that pH<4/3<4<pH in dimension 2, and pH<3/2<3<pH in dimension 3. In both dimensions 2 and 3, pHS<1, implying that the Stokes operator has a bounded functional calculus in Lp(Ω,Λ1) when Ω is strongly Lipschitz and 1<p<pH.





Revista Matematica Iberoamericana


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