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 $L^2(\Omega ,\Lambda )$, $\underline{\delta }=d^{\ast }$ and $D_{{}^{\Vert }}$ is self-adjoint, thus having bounded resolvents $\{(\mathrm{I}+it{D}_{{}^{\Vert }}{)}^{-1}{\}}_{t\in \mathbb{R}}$ 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 $1<p<p^H$.