On sectoriality of degenerate elliptic operators

Let $c_{kl}\in W^{1,\mathrm{\infty }}(\mathrm{\Omega },\mathbb{C})$ for all $k,l \in {1, \ldots, d}$and$\Omega \subset $\mathbb{R}$^d$ be open with Lipschitz boundary. We consider the divergence form operator $ A_p = - \sum_{k,l=1}^d \partial_l (c_{kl} , \partial_k) $in$L_p(\Omega)$whenthecoefficientmatrixsatisfies$(C(x) , \xi, \xi) \in \Sigma_\theta$forall$x \in \Omega$and$\xi \in $\mathbb{C}$^d$,where$\Sigma_\theta$bethesectorwithvertex0andsemi-angle$\theta$inthecomplexplane\mathrm{.}Weshowthatasectorialestimateholdfor$A_p$forall$p$ in a suitable range. We then apply these estimates to prove that the closure of $-A_p$ generates a holomorphic semigroup under further assumptions on the coefficients. The contractivity and consistency properties of these holomorphic semigroups are also considered.