TY - JOUR

T1 - Hodge-Dirac, Hodge-Laplacian and Hodge-Stokes operators in Lp spaces on Lipschitz domains

AU - McIntosh, Alan

AU - Monniaux, Sylvie

N1 - Publisher Copyright:
© European Mathematical Society.

PY - 2018

Y1 - 2018

N2 - This paper concerns Hodge-Dirac operators D = d + δ acting in Lp(Ω, A) where Ω is a bounded open subset of Rn satisfying some kind of Lipschitz condition, A 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, A), δ = d and D is self-adjoint, thus having bounded resolvents {(I + itD )-1}tER as well as a bounded functional calculus in L2(Ω, A). 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,A). On domains which we call very weakly Lipschitz, we show that this is the same range of values as for which Lp(Ω, A) 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 = D2, so it also has a bounded functional calculus in Lp(Ω, A) 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,A1) 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, A1) when Ω is strongly Lipschitz and 1 < p < pH.

AB - This paper concerns Hodge-Dirac operators D = d + δ acting in Lp(Ω, A) where Ω is a bounded open subset of Rn satisfying some kind of Lipschitz condition, A 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, A), δ = d and D is self-adjoint, thus having bounded resolvents {(I + itD )-1}tER as well as a bounded functional calculus in L2(Ω, A). 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,A). On domains which we call very weakly Lipschitz, we show that this is the same range of values as for which Lp(Ω, A) 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 = D2, so it also has a bounded functional calculus in Lp(Ω, A) 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,A1) 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, A1) when Ω is strongly Lipschitz and 1 < p < pH.

KW - Hodge-Dirac operator

KW - Hodge-Laplacian

KW - Hodge-Stokes operator

KW - Lipschitz domains

KW - Potential operators

UR - http://www.scopus.com/inward/record.url?scp=85060882656&partnerID=8YFLogxK

U2 - 10.4171/rmi/1041

DO - 10.4171/rmi/1041

M3 - Article

SN - 0386-6157

VL - 55

SP - 1711

EP - 1753

JO - Journal of the Society of Powder Technology, Japan

JF - Journal of the Society of Powder Technology, Japan

IS - 12

ER -