Square roots of elliptic second order divergence operators on strongly Lipschitz domains: $L^p$ theory

Abstract.

We study $L^p$ estimates for square roots of second order elliptic non necessarily selfadjoint operators in divergence form $L=- { {\mathrm div} (A \nabla)}$ on Lipschitz domains subject to Dirichlet or to Neumann boundary conditions, pursuing our work [4] where we considered operators on ${\mathbb R}^n$. We obtain among other things $\|L^{1/2}f\|_p \le c\|\nabla f\|_p$ for all $1< p < \infty$ if L is real symmetric and the domain bounded, which is new for $1< p < 2$. We also obtain similar results for perturbations of constant coefficients operators. Our methods rely on a singular integral representation, Calderón-Zygmund theory and quadratic estimates. A feature of this study is the use of a commutator between the resolvent of the Laplacian (Dirichlet and Neumann) and partial derivatives which carries the geometry of the boundary.