2016/08/05 by Mcintosh, Alan, Monniaux, Sylvie · 1 citation
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1608.01797
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 \textless p \textless 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 Δ is the square of the Hodge-Dirac operator, i.e. --Δ = D 2 , so it also has a bounded functional calculus in L p (Ω, λ) when p H \textless p \textless p H. But the Stokes operator with Hodge boundary conditions, which is the restriction of --Δ 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 max1, 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 Ω, 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 (Ω, λ 1) when Ω is strongly Lipschitz and 1 \textless p \textless p H .