First Order Approach to $L^p$ Estimates for the Stokes Operator on Lipschitz Domains

Abstract

This paper concerns Hodge-Dirac operators $D_H = d + \delta$ 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 $\delta$ is the interior derivative with tangential boundary conditions. In $L^2(\Omega , \Lambda )$, $d' = \delta$ and $D_H$ is self-adjoint, thus having bounded resolvent ${\{(I + \mathrm{{it}}{D}_H)\}}_{\{t\in \mathbb {R}\}}$ 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 $D_H$ has bounded resolvents and a bounded holomorphic functional calculus in $L^p(\Omega , \Lambda )$.