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

Abstract

This paper concerns Hodge–Dirac operators $D\_{{}^\Vert}=d+\underline{\delta}$ acting in $L^p(\Omega, \Lambda)$ where $\Omega$ is a bounded open subset of ${\mathbb{R}}^n$ satisfying some kind of Lipschitz condition, $\Lambda$ is the exterior algebra of ${\mathbb{R}}^n$, $d$ is the exterior derivative acting on the de Rham complex of differential forms on $\Omega$, and $\underline{\delta}$ is the interior derivative with tangential boundary conditions. In $L^2(\Omega,\Lambda)$, $\underline{\delta} = {d}^\*$ and $D\_{{}^\Vert}$ is self-adjoint, thus having bounded resolvents ${({\rm I}+itD\_{{}^\Vert})^{-1}}{t\in{\mathbb{R}}}$ as well as a bounded functional calculus in $L^2(\Omega,\Lambda)$. We investigate the range of values $p\_H < p < p^H$ about $p=2$ for which $D{{}^\Vert}$ 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 $\Delta\_{{{}^\Vert}}$ is the square of the Hodge–Dirac operator, i.e., $-\Delta\_{{}^\Vert}={D\_{{}^\Vert}}^2$, so it also has a bounded functional calculus in $L^p(\Omega,\Lambda)$ when $p\_H < p < p^H$. But the Stokes operator with Hodge boundary conditions, which is the restriction of $-\Delta\_{{}^\Vert}$ 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${1,{p\_H}\_S} < p < p^H$ 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 $\Omega$, $p\_H < 2n/(n+1) < 2n/(n-1) < p^H$, 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 $L^p(\Omega,\Lambda^1)$ when $\Omega$ is strongly Lipschitz and $1 < p < p^H$.